COMMENTS ON MY PAPERS. (A reference like [99] means paper nr.99 in my list of publications.)

8. (with J.Milnor and F.P.Peterson) SEMICHARACTERISTICS AND COBORDISM, 1969.

I did the work on this paper during a two months stay in Oxford (fall of 1968). During my first meeting with Atiyah, he and Singer explained to me the following question on the (Kervaire) semicharacteristic. A compact smooth oriented manifold M of dimension 4n+1 has a semicharacteristic c(M,p)=\sum_{i\in[0,2n]}\dim H^i(M,k)\mod 2 with respect to a field k of characteristic p\ge0. At the time it was known that the obstruction to the existence of two independent vector fields on M is equal to c(M,2) if M is spin [E.Thomas, Bull.Amer.Math.Soc.1969] and to c(M,0), without assumption (Atiyah-Singer). The question was whether there is a relation between c(M,0) and c(M,2) which would make clear that the Atiyah-Singer result implies the Thomas result. (A few months earlier they asked F.Peterson at MIT the same question.) Then I and (independently) F.Peterson and J.Milnor found (different) formulas for c(M,0)-c(M,2); one of those formulas expressed c(M,0)-c(M,2) as the Stiefel-Whitney number w_2w_{4n-1} which clearly vanishes for spin manifolds. The initial proofs of both formulas used Wall's results on the structure of the oriented cobordism group (that is the formulas were checked on the generators of that group) but in the final version Wall's results are not used. The result of this paper is used in [Atiyah,Singer, Ann.Math. 1971] and is generalized in [10]. (3/24/2011)

9. REMARKS ON THE HOLOMORPHIC LEFSCHETZ FORMULA, 1969.

I did the work on this paper during a two months stay in Oxford (fall of 1968). The main result of the paper is the following rigidity result: the holomorphic Lefschetz number of a circle action on a compact complex manifold is constant. (This is of interest only in the case where the manifold is non-Kaehler, when the induced action on the Dolbeault cohomology may be nonconstant.) The argument that I used in this paper has played a role in the proof of the vanishing of the \hat A-genus of a 4k-dimensional spin manifold with nontrivial circle action given in [Atiyah,Hirzebruch, in Essays on topology and related topics, 1970], where there is a reference to my result (but not to the paper itself). There is a similar reference in: [Bott,Taubes, Jour.Amer.Math.Soc. 2(1989)]. (7/30/2010)

10. (with J.Dupont) ON MANIFOLDS SATISFYING w_1^2=0, 1971.

This paper was written in 1970 during my stay (1969-71) at IAS. It contains a generalization of the result on semicharacteristic in [8] where the orientability assumption w_1=0 is weakened to w_1^2=0 (w_1 is the first Stiefel-Whitney class). This was used in the paper [Davis and Milgram, Trans.AMS, 1989].
The appendix of this paper is a study of the symmetric power SP^nX where X is a compact unorientable smooth 2-manifold whose first rational Betti number is g. We show that SP^nX is a (2n-g)-dimensional bundle over a g-dimensional real torus with fibre RP^{2n-g}, the real projective space of dimension 2n-g. In particular if X is a projective plane (g=0) then SP^nX=RP^{2n}; if X is a Klein bottle (g=1) then SP^nX is a RP^{2n-1}-bundle over the circle. (SP^nY for Y a compact Riemann surface was studied earlier in [Macdonald, Topology 1962].) We also show that S^{\infinity}X is a product of a g-dimensional torus with RP^{\infinity}. Our result SP^nX=RP^{2n} is reinterpreted in [Arnold, Topological content of the Maxwell theorem on multiple representations of spherical functions, Topological methods in nonlinear analysis 7(1996)].
This paper contains also a study of a cobordism ring G_* based on closed manifolds together with an element \Gamma in H^1(M,Z/4Z) which reduces to the first Stiefel Whitney class by reduction mod2. Our explicit determination of this ring relies on earlier work of [C.T.C.Wall, Ann.Math.1960]; a special role in our description is played by the manifolds SP^nX where X is a Klein bottle and n is a power of 2. (There are two natural choices for \Gamma but they represent the same element in G_*.) (7/14/2011)

11. NOVIKOV'S HIGHER SIGNATURE AND FAMILIES OF ELLIPTIC OPERATORS, 1972.

This paper was written in 1970 during my stay (1969-71) at IAS. I used it as my Ph.D. thesis at Princeton University (may 1971). The main contribution of this paper is to introduce the analytic approach (based on the index theorem) to attack the Novikov's conjecture on higher signature. That conjecture states that, if one multiplies the Hirzebruch L-class of a compact oriented manifold M with a cohomology class which comes from the cohomology of the classifying space of the fundamental group of M and then one integrates the result over M, one obtains a homotopy invariant of M. In this paper I introduce, for M of even dimension with fundamental group Z^n, a family of elliptic operators on M. These operators are obtained by twisting the Atiyah-Singer signature operator by a variable flat vector bundle on M coming from a unitary one dimensional representation of the fundamental group. While the index of each of these operators is the same as that of the untwisted operator it turns out the family of operators has an interesting index in the K-theory of the parameter space of the space of flat bundles considered and I showed that this index is on the one hand a homotopy invariant and on the other hand from it one can recover the whole Novikov higher signature thus proving Novikov's conjecture in this case.
Another contribution of this paper is to formulate a version of the Hirzebruch signature theorem in which cohomology is taken with coefficients in a flat vector bundle with a flat hermitian form which is not necessarily positive definite. In this case Hirzebruch's original proof (with constant coefficients) does not work but the Atiyah-Singer theorem can be used instead. In the paper I show that from this "twisted" signature theorem one can derive various examples where Novikov's conjecture holds for certain nonabelian fundamental groups.
The analytic approach of this paper has been extended by Mischenko and Kasparov to the case where the fundamental group is a discrete subgroup of Lie groups and then by Connes, Moscovici,Gromov, Higson and others to even more general fundamental groups. See [Ferry,Ranicki,Rosenberg: Novikov signatures, index theory and rigidity, London Math.Soc.Lect.Notes, 1995] for a review of these developments.
The twisted signature theorem of this paper is used in: [Gromov and Lawson, Ann.of Math. 1980], [Atiyah, Math.Annalen 1987], [Gromov, in "Functional analysis on the eve of the 21st century, II", Progress in Math. 132,Birkhauser 1996]. (7/11/2011)

14. INTRODUCTION TO ELLIPTIC OPERATORS, 1974.

This (mainly expository) paper is based on a lecture that I gave at a Trieste summer school in 1972. It contains the definition of elliptic operators and their index. The only part which is perhaps non-standard is the definition of analytical index as a homomorphism ind:K(BT^*M,ST*M)@>>>Z where T^*M is the cotangent bundle of a compact manifold M, BT^*M is its unit disk bundle, ST^*M is its unit sphere bundle and K() is K-theory. The usual (Atiyah-Singer) definition of ind is via the theory of pseudo-differential operators. But in this paper I show that if we are willing to increase Z to Z[1/2], one can define ind:K(BT^*M,ST*M)@>>>Z[1/2] in a more elementary way, using only differential operators.(7/15/2011)

17. ON THE DISCRETE SERIES REPRESENTATIONS OF THE CLASSICAL GROUPS OVER A FINITE FIELD, 1974.

This paper represents my talk at the ICM in August 1974. I was originally invited to give a talk in the Algebraic Topology section but I requested to change to the Algebraic Groups section. In 1973 the only cases where cuspidal representations of a reductive group over a finite field were constructed were GL_n(F_q) (by J.A.Green), Sp_4(F_q) (by B.Srinivasan) and G_2(F_q) (by B.Chang,R.Ree). In 1973, after my study [13] of the "Brauer lifting" of the standard n-dimensional representation of GL_n(F_q), I tried to find the constituents of the Brauer lifting X of the standard representation (of dimension N) of a symplectic orthogonal or unitary group G(F_q). The result that I found is that X=X_1+X_2+...+X_N where X_i is plus or minus an irreducible representation, X_1 is up to sign a cuspidal representation (new at the time) of dimension |G(F_q)| divided by the order of a "Coxeter torus" and by the order of a maximal unipotent subgroup; moreover X_i for i>1 were noncuspidal and could be explicitly described as components of certain induced representations from analogous cuspidal representations of Coxeter type of smaller classical groups or GL_n by determining explicitly the relevant Hecke algebras. Thus this method gives a way to approach at least the "Coxeter series" of cuspidal representations of a classical group. This is what is explained in the first part of this paper. (The proofs of the results in the first part were never published since they were superseded by later developments.)
In the second part of the paper I described my joint work with Deligne (done during the spring 1974 at IHES) in which l-adic cohomology is used to construct representations of G(F_q) where G is a connected reductive group over F_q. This method was first used by Tate and Thompson [Tate, Algebraic cycles and poles...,1965] who observed that the obvious action of the unitary group U_3(F_{q^2}) on the projective curve x^{q+1}+y^{q+1}+z^{q+1}=0 over \bar F_q induces an action on H^1 which is the (interesting) irreducible representation of degree q^2-q of U_3(F_{q^2}). Around 1973 Drinfeld observed that the cuspidal representations of SL_2(F_q) can be realized in the cohomology with compact support of the curve xy^q-x^qy=1 over \bar F_q by taking eigenspaces of the obvious action of T={t\in\bar F_q^*;t^{q+1}=1} which acts on the curve by homothety. (I learned about this fact from T.A.Springer in 1973.) (Note that Drinfeld's curve can be viewed as the (open) part of the Tate-Thompson curve where z is nonzero. This open part is stable under SL_2(F_q) times T viewed as a subgroup of U_3(F_{q^2}).) The main result of this section was the introduction for any element w in the Weyl group of G, of two new algebraic varieties: the variety X_w of Borel subgroups B of G such that B and its image under the Frobenius map F are in relative position w and the finite principal covering \tilde X_w of X_w whose group is the group T_w of rational points of an F-stable torus of type w of G. (Note that the Drinfeld curve is a special case of \tilde X_w in the case G=SL_2 and the Tate-Thompson curve is the compactification of an X_w in the case G=GL_3 with a nonsplit F_q-structure.) These varieties admit natural action of G(F_q) and the principal covering above gives rise to G(F_q)-equivariant local systems on X_w one for each character \theta of T_w. By passing to cohomology with compact support with coefficients in such a local system one obtains representations of G(F_q). By taking alternating sums one obtains certain virtual representations R(w,\theta) of G(F_q) indexed by the various \theta. At the time when this paper was written (summer 1974) we conjectured that R(w,\theta) for \theta generic are up to sign irreducible characters which provide a solution of the Macdonald conjecture. But we could only prove that their character at regular semisimple elements is as predicted by Macdonald's conjecture. (10/05/2010)

18. SUR LA CONJECTURE DE MACDONALD, 1975.

By my joint work with Deligne (spring 1974) described in [17], a conjectural solution to the Macdonald conjecture for the irreducible representations of G(F_q) was known in terms of the the virtual representations R(w,\theta) defined by the subvarieties X_w, \tilde X_w of the flag manifold and their cohomology with compact support. But it was not clear how to prove the irreducibility of R(w,\theta) for \theta generic or how to compute the degree of R(w,\theta). In the fall 1974 (at Warwick) I completed the proof of the fact that the virtual representations R(w,\theta) defined in [17] are indeed a solution to the Macdonald conjecture. In this paper (written in late 1974) I sketched this proof; it is based on the following principle:
(*) Assume that H is a finite group acting on an algebraic variety Y in such a way that the space of orbits Y/H is again an algebraic variety Y'; assume further that there is a partition of Y into finitely many locally closed H-stable pieces Y_i and on each Y_i the action of H extends to an action of a connected algebraic group H_i. Then Y',Y have the same Euler characteristic.
The main observation of this paper is that (*) is applicable in the following two cases:
(A) Y=\tilde X_w/P(F_q),Y'= X_w/P(F_q) where P is a parabolic defined over F_q;
(B) Y=(\tilde X_w\times X_{w'})/G(F_q),Y'=(X_w\times X_{w'))/G(F_q) where w,w' are Weyl group elements.
Now (*) in case (A) implies easily the expected formula for the degree of R(w,\theta) and (*) in case (B) implies easily an explicit formula for the inner product of an R(w,\theta) with an R(w',\theta') from which the desired irreducibility result follows. The proof of the degree formula and that of the inner product formula given later in [22] are quite different: they use a disjointness theorem.(10/03/2010)

19. DIVISIBILITY OF PROJECTIVE MODULES OF FINITE CHEVALLEY GROUPS BY THE STEINBERG MODULE, 1976.

This paper was written during my stay at IHES in the spring of 1974. The motivation for this paper was to find evidence for the Macdonald conjecture for a connected reductive group G defined over F_q. (However, by the time this paper appeared, Macdonald's conjecture was proved.) If T is a maximal torus of G defined over F_q and \theta is a character of T(F_q) then the induced representation R=Ind_{T(F_q)}^{G(F_q)}(\theta) is defined. If \theta is in general position and if R(T,\theta) is the irreducible representation of G(F_q) provided by Macdonald's conjecture (assumed to hold) then R is isomorphic to R(T,\theta)\otimes S where S is the Steinberg representation. Therefore if we can prove apriori that R is isomorphic to S tensor some virtual representation then this would be evidence for the Macdonald conjecture. Now note that R can be viewed as a representation coming from a projective G(F_q)-module over the ring of integers in a suitable p-adic field. Hence it would be enough to show that any such projective G(F_q)-module is "divisible" by S. This is what is shown in this paper. (10/01/2010)

20. A NOTE ON COUNTING NILPOTENT MATRICES OF FIXED RANK, 1976.

This paper was written in early fall 1974. At that time the series of representations of a reductive group attached to a maximal torus over F_q were already constructed (in the joint work with Deligne, see [17]) but their irreducibility was not yet proved (it was proved shortly afterwards in [18]). But in the case of the even nonsplit orthogonal group over F_q and the series corresponding to the Coxeter torus the character was explicitly computable (in this case I could compute the Green functions, by some computations which later became part of [23]) hence in this case irreducibility could be proved directly by using orthogonality of the explicit Green functions; to do this it was necessary to know the number of unipotent elements u such that u-1 has fixed rank. (It turned out out that the Green function on a unipotent element u depended only on the rank of u-1.) This number is computed in this paper. The result of this paper gives a new proof (for classical groups over F_q) for Steinberg's theorem on the total number of unipotent elements. (For GL_n that theorem is due to Fine and Herstein and independently to Ph.Hall.) (6/29/2011)

22. (with P.Deligne) REPRESENTATIONS OF REDUCTIVE GROUPS OVER FINITE FIELDS, 1976.

This paper (written during the first half of 1975) contains a detailed study of the varieties X_w,\tilde X_w associated in [17] (by me and Deligne) to an element w in the Weyl group of a connected reductive group G defined over a finite field F_q and of the associated virtual representations R(w,\theta) of G(F_q). Here \theta is a character of the finite "torus" T_w of type w. In Section 3 a Lefschetz type fixed point formula for a transformation of finite order of an algebraic variety is given. (This formula was already used implicitly in [17].) This is used in Section 4 to prove a formula for the character of R(w,\theta) assuming that the Green functions of G and smaller groups are known. In Section 6 the disjointness theorem is proved: the virtual representations R(w,\theta), R(w',\theta') are disjoint unless \theta,\theta' are conjugate after extension of the ground field (and composition with the trace); it is also shown that the equivalence classes of various \theta as above can be viewed as semisimple conjugacy classes defined over F_q in the "dual" group G^* (at least if G has connected centre). The proof of the disjointness uses the possibility of extending the action of a finite group on (\tilde X_w\times\tilde X_{w'})/G(F_q) to the action of a higher dimensional group, not on the whole variety, but separately on each piece of a partition of the variety in pieces stable under the finite group (compare with (*) in the comments to [18]). The disjointness theorem has several applications (which were proved in a different way in [18]): the degree formula for R(w,\theta); the inner product formula for R(w,\theta),R(w',\theta'); the orthogonality formula for Green functions; the character formula on semisimple elements. In Corollary 7.7 it is shown that any irreducible representation of G(F_q) appears in some R(w,\theta); a completely different proof of this result based on the theory of perverse sheaves was later given in [178]. Combining this result with the disjointness theorem one obtains a canonical map from the set of irreducible representations of G(F_q) (up to isomorphism) to the set of semisimple conjugacy classes in G^*(F_q) (at least if G has connected centre); this is an initial but crucial step in the classification of irreducible representations of G(F_q). In Section 9 it is shown that X_w is affine assuming that q is greater than the Coxeter number. Recently it has been shown that X_w is affine without restriction on q assuming that w has minimal length in its twisted conjugacy class [Orlik and Rapoport, J.Algebra, 2008] and [He, J.Algebra, 2008], see also [Bonnaf\'e and Rouquier, J.Algebra, 2008]. In 9.16 it is shown that any Green function evaluated at a regular unipotent element is equal to 1. In Section 10 (assuming that the centre of G is connected) it is shown how to parametrize explicitly the irreducible components of the Gelfand Graev representations and that these components are explicit linear combinations of R(w,\theta). In Section 11 the results of the paper are extended to Ree and Suzuki groups. In this case the inner product formula cannot be handled by the methods of this paper in the case q=\sqrt{2) or q=\sqrt{3), but it can be handled by the proof of the inner product formula given later in [30]. (10/04/2010)

23. ON THE GREEN POLYNOMIALS OF CLASSICAL GROUPS, 1976.

This paper was written in the summer of 1975, after the completion of [22]. In [22] a general study of the variety X_w (all Borel subgroups in relative position w with their transform under Frobenius in a reductive group G over F_q) was made. In the present paper I tried to study in detail the first nontrivial class of examples of the variety X_w, namely the case where G is a classical group and w is a Coxeter element of minimal length. In this case I obtained explicit formulas for (a) the number of rational points of X_w over any extension of the ground field) and for (b) the Green functions (alternating sums of traces of a unipotent element of G(F_q) on the cohomology with compact support of X_w). In the case of symplectic groups, (b) solved a conjecture of B.Srinivasan. For other groups or other w an explicit closed formula for Green functions is still not available.
This paper was a preparation for my next project [25] in which I studied X_w for w a (twisted) Coxeter element of minimal length for a general G. (10/25/2010)

24. ON THE FINITENESS OF THE NUMBER OF UNIPOTENT CLASSES, 1976.

The work on this paper was started during a visit to IHES in December 1974 and was completed during another visit to IHES in December 1975. Let G be a connected reductive group defined over F_q, let P be a parabolic subgroup of G (not necesarily defined over F_q) and let L be a Levi subgroup of P defined over F_q. In this paper I define for any virtual representation \r of L(F_q) a virtual representation R(L,P,\r) of G(F_q). In the case where P is defined over F_q this is the usual induced representation from P(F_q) to G(F_q) where \r is viewed as a virtual representation of P(F_q). In the case where L is a maximal torus defined over F_q and \r is a character of T(F_q), this reduces to the construction in [22].
One of the main results of this paper is an inner product formula for two virtual representations R(L,P,\r),R(L',P',\r') under a genericity assumption. One consequence of this is that, under a genericity assumption, R(L,P,\r) is irreducible (up to sign) for \r irreducible. This irreducibilty result plays a key role in my later papers [29],[57]; it allows one to construct new irreducible representations starting with known irreducible representations of L(F_q). Another consequence is that the number of unipotent conjugacy classes in G is finite, answering a conjecture of Steinberg at the ICM in 1966. Previously this finiteness result was known only for classical groups or for exceptional groups in good characteristic. A few years later (1980) Mizuno gave another proof of the finiteness result for exceptional groups in bad characteristic based on extensive computations. I believe that the proof given in this paper is still the only proof of finiteness which does not use classification.
After this paper was written, Deligne stated a refinement of the inner product formula for R(L,P,\r),R(L',P',\r') (without genericity assumptions) as an analogue of Mackey's theorem (generalizing the case where L,L' are maximal tori, known from [22]) and proved it assuming that q is large. Deligne's proof remains unpublished. Around 1985 I proved a character sheaf version of this formula (see [65]); in view of the results of [89] this implies the refined inner product formula for representations (again for large q). A version of Deligne's proof appeared in [Bonnaf\'e, J.Alg. 1998]. Recently [Bonnaf\'e and Michel, J.Alg.2011] gave a proof of this formula with an extremely mild assumption on q, using computer calculation. The general case is still not proved. The (refined) inner product formula would imply that R(L,P,\r) is independent of P. (5/27/2011)

25. COXETER ORBITS AND EIGENSPACES OF FROBENIUS, 1976.

The work on this paper was done in late 1975. This paper continues the project started in [23] to study in detail the variety X_w of [22] in the case where w is a (twisted) Coxeter of minimal length in the Weyl group of a connected reductive group G defined over F_q. (The case of Suzuki and Ree groups is also treated in the paper.) One of the main results of this paper is a construction of several new unipotent cuspidal representations of G(F_q) in the case where G is exceptional. Let d be the smallest integer \ge1 such that X_w is stable under F^d, the d-th power of the Frobenius map. In this paper I give
(a) an explicit computation of the eigenvalues of F^d on H^*_c(X_w) and an explicit formula for the dimensions of its eigenspaces;
(b) a proof that F^d acts semisimply on H^*_c(X_w) and its eigenspaces are irreducible, mutually nonisomorphic G(F_q)-modules.
In the case where G is a Suzuki or Ree group of type B_2 or G_2 with q=\sqrt{Q} where Q is an odd power of m (m=2 or 3), the variety X_w (w a simple reflection) is an affine curve and its compactification \bar X_w is a smooth projective curve defined over F_Q such that \bar X_w-X_w consists of Q^m+1 points. On the other hand by theorem 3.3(i) of this paper, X_w has no F_Q-rational points. It follows that the number of F_Q-rational points of \bar X_w ie equal to Q^m+1 points. Note also that the genus of \bar X_w is determined explicitly from (a) or [22]. In a letter to me dated May 11, 1983, J.-P.Serre made the following remarks.
(1) \bar X_w has the maximum number of F_q-rational points compatible with its genus.
(2) If a smooth curve over F_Q has Q^m+1 rational points and has the same genus as \bar X_w then it has the same zeta function as \bar X_w (which is determined from (a)).
Due to property (1), these curves have been used to produce Goppa (error correcting) codes. See [N.Hurt: Many rational points; coding theory and algebraic geometry, Kluwer, 2003]. (11/09/2010)

29. IRREDUCIBLE REPRESENTATIONS OF FINITE CLASSICAL GROUPS, 1977.

The work on this paper was started at Warwick during the summer of 1976 and completed during my visit to MIT in the fall 1976. This paper contains the classification and degrees of the irreducible complex representations of classical groups (with connected centre) other than GL_n, over a finite field. It relies on: the use of the "cohomological induction" [22],[23]; the use of the dimension formulas for the irreducible representations of Hecke algebras of type B with two parameters [Hoefsmit, UBC Ph.D.Thesis 1974] (of which I learned from B.Chang during my visit to Vancouver at ICM-1974). This paper establishes what was later called "the Jordan decomposition" for the representations of classical groups (with connected centre). It also establishes the parametrization of unipotent representation for these groups in terms of some new combinatorial objects, the "symbols" and the classification of unipotent cuspidal representations of classical groups (for example Sp_{2n}(F_q) has such a representation if and only if n=k^2+k which is then unique). Also it is shown that the endomorphism algebra of the representation induced from a unipotent cuspidal (or more generally isolated cuspidal) representation to a larger classical group is an Iwahori-Hecke algebra (anticipating a later result of [Howlett-Lehrer, Inv.Math.1980]) and giving also precise information on the values of the parameters of that Iwahori-Hecke algebra. For this we need to count in terms of generating functions the number of conjugacy classes in a classical group with connected centre. This together with an inductive hypothesis and the methods outlined above give a way to predict the number of isolated cuspidal representations. The degrees of these isolated cuspidal representations can be guessed using the technique of symbols by "interpolation" from the degrees of noncuspidal representations. To prove that these guessed are correct we need to calculate the sum of squares of the (guessed) degrees of unipotent representations which is perhaps the most interesting part of this paper. To do this I find explicit formulas (for each irreducible representation E of the Weyl group) of the polynomial d_E(q) whose coefficients record the multiplicities of E in the various cohomology spaces of the flag manifold. (I do this first for GL_n and then reduce the case of classical groups to that of GL_n.) Then I show that the (guessed) degree polynomials can be expressed as linear combinations of the d_E(q) with constant coefficients of the form plus or minus 1/2^s. This anticipates the notion of family of representations of the Weyl group and the role of the nonabelian Fourier transform [34] (which in this case happens to be abelian.) Here the use of the technique of symbols (introduced in this paper) is crucial. It is remarkable that suitable variations of the notion of symbol (used here in connection with unipotent representations) were later shown to be exactly what one needs to describe explicitly the Springer correspondence (including the generalized one) for classical groups [59],[61] and for classical Lie algebras in characteristic 2 [Xue, 2009]. (9/05/2010)

30. REPRESENTATIONS OF FINITE CHEVALLEY GROUPS, 1978.

This paper represents lectures that I gave in August 1977 at Madison,Wisconsin. Let G be a connected reductive group defined over F_q. Among other things, in this paper I give two refinements (see (a),(b) below) of the inner product formula for the virtual representations R(w,\theta) (see [22]) of G(F_q):
(a) a proof (in 2.3) which applies equally well to the Ree and Suzuki groups with q=\sqrt{2) or q=\sqrt{3) which were not covered by earlier proofs in [18],[22];
(b) a proof (see 3.8) that, if w,w' are in the Weyl group W and X_w,X_{w'} are the varieties of [22] then |((X_w times X_{w'})/G(F_q))(F_{q^s})| is equal to the trace of a linear transformation h\to T_whT_{w'^{-1}} of the Hecke algebra with parameter q^s. (Assume that G is split over F_q.)
Note that (b) is a refinement of the inner product formula for R(w,\theta) (in the case where \theta=1) since for that formula one needs the Euler characteristic of (X_w times X_{w'})/G(F_q) which is the limit of |((X_w times X_{w'})/G(F_q))(F_{q^s})| as s goes to 0.
Also I show that to any unipotent representation of G(F_q) one can attach an eigenvalue of Frobenius well defined up to an integer power of q. This result was later used in [Digne and Michel, C.R.Acad.Sci.Paris,1980] and by [Asai, Osaka J.Math.,1983]. In 3.34 these eigenvalue of Frobenius are described in all cases arising in type \ne E_8.
On page 26 (see (d)) it is shown that X_w is irreducible if and only if for any simple reflection s, some element in the Frobenius orbit of s appears in a reduced expression of w; this result has been rediscovered in [Bonnaf\'e and Rouquier, C.R.Acad.Sci.Paris,2006] (another proof is given in [Goertz, Repres.Theory, 2009]).
In 3.13 and 3.16 an explicit formula for the sum of squares of unipotent representations of G(F_q) is given. This is used in 3.24 to classify the unipotent representations in the case where G is split E_6 or E_7 (it turns out that all cuspidal unipotent representations arise from the analysis in [25]). In the case where G is nonsplit E_6, triality D_4 or F_4, a classification of unipotent representations is again given assuming that q is large; in these cases there are cuspidal unipotent representations which do not arise from the analysis in [25].
On page 24 (see (b)) (assuming that Frobenius acts on W as conjugation by the longest element w_0) I define for each w\in W a bijective morphism t_w:X_{w_0}@>>>X_{w_0} as follows: t_w(B)=B' where B\in X_{w_0} and B' is defined by pos(B,B')=w,pos(B',Frob(B))=w^{-1}w_0 (so that B'\in X_{w_0}); I show that the t_w define a homomorphism of the braid group in the group of permutations of X_{w_0}. Moreover I show that after passage to cohomology one obtains a representation of the Hecke algebra of W with parameter -q on H^*_c(X_{w_0}). Several years later (around 1982) I used a similar idea in the case where G is of type D_4 so that W has simple reflections s_0,s_1,s_2,s_3 with s_1,s_2,s_3 commuting and w=s_1s_0s_2s_0s_3s_0\in W. (This is unpublished but there is a reference to it in [Brou\'e,Malle, Ast\'erisque 1993, 5A] and [Brou\'e,Michel,Progr.in Math.141,1997, page 114].) Let Z(w) be the centralizer of w in W. I defined three permutations A,B,C of X_w into itself (similar to t_w above) such that A,B,C commute with Frobenius and ABC=BCA=CAB=Frobenius. The maps A,B,C are associated to three generators a,b,c of Z(w) which satisfy abc=bca=cab=w. This suggests that the "braid group" corresponding to Z(w) (a complex reflection group) should have the relation ABC=BCA=CAB. Indeed, this later appeared as a special case of the relations of such "braid groups" given in [Brou\'e,Malle, Ast\'erisque 1993]. The idea in this example was further pursued in [Digne,Michel, Nagoya Math.J.,2006] and in [203].(5/26/2011)

31. (with W.M.Beynon) SOME NUMERICAL RESULTS ON THE CHARACTERS OF EXCEPTIONAL WEYL GROUPS, 1978.

For any irreducible representation E of a Weyl group the fake degree d_E(q) of E is defined in [30] as the polynomial in q which records the multiplicities of E in the various cohomology spaces of the flag manifold. After the polynomials d_E(q) were explicitly computed in [29] in the case of classical groups, it was natural to try to compute them for simple exceptional groups. This is what is done in the present paper, using a computer and the known character tables of W (but we found and corrected some errors in the character table for type E_8). These computations were later used in [34]. One observation of this paper is that the polynomials d_E(q) are palindromic apart from a small number of exceptions in type E_7 (for E of degree 512) and E_8 (for E of degree 4096). (10/01/2010)

33. ON THE REFLECTION REPRESENTATION OF A FINITE CHEVALLEY GROUP, 1979.

The work on this paper was done in the spring of 1977; the results were presented at an LMS Symposium on Representations of Lie Groups in Oxford (july 1977). I will explain the main result of this paper using concepts which were developed several years after the paper was written (theory of character sheaves). Let G be a connected reductive group over an algebraic closure of a finite field F_q with a fixed F_q-split rational structure and Frobenius map F:G \to G. For each w in the Weyl group one can consider (following [22]) the variety X_w of Borel subgroups B of G such that B,FB are in position w. Then G(F_q) acts natural on the l-adic cohomology H^i_c(X_w). We can also consider (as in the theory of character sheaves) for each g in G the variety Y_{w,g} of Borel subgroups B of G such that B,gBg\i are in position w. The union over g in G of these varieties maps naturaly to G and we can take the direct image K_w with compact support of the sheaf \bar Q_l under this map. Then {}^pH^iK_w are perverse sheaves on G. Now for any irreducible representation E of the Weyl group we denote by E_q the corresponding irreducible representation of G(F_q) which appears in H^0_c(X_1) (functions on flag manifold of G(F_q)) and we denote by E_1 the simple perverse sheaf on G corresponding to E which appears in K_1 (a perverse sheaf on G with W-action). The main result of this paper is that for any w we have \sum_i(-1)^i(E_q:H^i_c(X_w)=(-1)^{\dim G}\sum_i(-1)^i(E_1:{}^pH^iK_w) where (:) denotes multiplicity. Here the left hand side can be interpreted as the value of the character of E_q on a regular semisimple element in a maximal torus of type w. This result does not compute these character values but it shows that these values are universal invariants which make sense also over complex numbers. Now the multiplicity (E_1:{}^pH^iK_w) does not change when E_1,{}^pH^iK_w are restricted to the variety of regular semisimple elements of G. But after this restriction E_1,{}^pH^iK_w become local systems and (E_1:{}^pH^iK_w) is equal to the corresponding multiplicity of local systems which can be considered independently of the theory of perverse sheaves. It is in this form that the result above appears in the present paper where the varieties Y_{w,g} are introduced only for g regular semisimple (in which case they are shown to be smooth of dimension equal to the length of w). Thus this paper can be viewed as a precursor of the theory of character sheaves which was developped in [63-65,68,69]. As an application I determine explicitly the value of the character of the "reflection representation" of G_(F_q) (constructed earlier by Kilmoyer) on a regular semisimple elements of type w assuming that G is of type A,D, or E. Namely it is shown that this value is equal to the trace of w on the reflection representation of W. At the time when the paper was written this result was new for types D,E. (04/11/2011)

34. UNIPOTENT REPRESENTATIONS OF A FINITE CHEVALLEY GROUP OF TYPE E_8, 1979.

This paper was written in the spring of 1978, soon after my arrival to MIT (january 1978). This paper introduces a new type of Fourier transform (the "non-abelian Fourier transform"). It is a unitary involution of the vector space of functions on a set M(G) associated to a finite group G; here M(G) is the set of all pairs (x,r) where x is an element of G (up to conjugacy) and r is an irreducible representation of the centralizer of x (up to isomorphism). About ten years later:
-I found [77] an interpretation of the "non-abelian Fourier transform" as the "character table" of the equivariant complexified K-theory convolution algebra K_G(G) (where G acts on itself by conjugation): this (commutative) algebra has a natural basis indexed by M(G) and its (one dimensional) representations are also indexed by M(G) hence its character table is defined;
-physicists [Dijkgraaf, Vafa, E.Verlinde, H.Verlinde, Comm.Math.Phys.1989], [Dijkgraaf, Pasquier, Roche, Nuclear Phys.1990] rediscovered this Fourier transform (possibly with a twist by a 3-cocycle);
-Drinfeld explained it in terms of his "double" of the group algebra of G.
In this paper the "non-abelian Fourier transform" is used to complete the classification and computation of degrees of the unipotent representations of finite Chevalley groups (started in [29],[30]). Note that for the analogous problem for classical groups, the standard (abelian) Fourier transform is sufficient.
It is remarkable that the "non-abelian Fourier transform" enters in an essential way in subsequent works in representation theory: the multiplicity formulas in the virtual representations R(T,\theta) of [22], see [57]; the analogous multiplicity formulas for character sheaves [63-65,68,69]; the relation of character sheaves to irreducible characters, see [71,102] and [Shoji, Adv.Math.1995].
This paper (see Section 8) also introduces the concept of "special representation" of a Weyl group (which was further developed in [36]) and that of "family" of unipotent representations (which contains as a particular case the notion of family of irreducible representations of a Weyl group). The concept of special representation of a Weyl group was suggested by the calculations in [29]. It is nowadays used extensively in the representation theory of reductive groups over real or complex numbers. The partition of the set of irreducible representations of a Weyl group into families turns out to be the same as the partition defined later in terms of the two-sided cells [37] of the Weyl group.(4/05/2011)

35. (with N.Spaltenstein) INDUCED UNIPOTENT CLASSES, 1979.

Let G be a connected reductive group over an algebraically closed field, let L be a Levi subgroup of a parabolic subgroup P of G and let C be a unipotent class of G. In this paper we associate to L,C a unipotent class C' of G (said to be induced by C) it is the unique unipotent class of G whose intersection with CU_P is dense in CU_P (here U_P is the unipotent radical of P). In this paper we show that C' does not depend on the choice of P and that C'\cap CU_P is a single P-conjugacy class. When C={1} then C' reduces to the Richardson class defined by L. We give two proofs for the independence on P; one of these depends on some results on representations of a reductive group over a finite field and on the Lang-Weil estimates; the other is more elementary but uses some case by case arguments.
In this paper we also introduce the idea of truncated induction for representations of Weyl groups generalizing a construction of Macdonald. We show that the Springer representation attached to an induced unipotent class is obtained from the Springer representation of the original unipotent class by truncated induction. This has been used in subsequent works (such as [36], [48]) to compute the Springer correspondence in certain cases arising from exceptional groups.(11/02/2010)

36. A CLASS OF IRREDUCIBLE REPRESENTATIONS OF A WEYL GROUP, 1979.

In this paper (written in the summer of 1978) I give an alternative definition of the class S_W of irreducible representations of a Weyl group W of a complex adjoint group G (introduced in [34] and later called "special representations").
We have two commuting involutions A,B of Irr(W): A is tensoring by the sign representations and B is the q=1 specialization of an involution of the set of irreducible representations of the Hecke algebra given by the action of the Galois group which takes \sqrt{q} to -\sqrt{q}. (Note that B is the identity for W of classical type and it is almost the identity in general.) Let T=AB=BA. In this paper it is shown that
(i) S_W is preserved by the truncated induction [35] from a parabolic subgroup, and
(ii) S_W is preserved by T;
moreover, S_W is characterized by (i)-(ii) and the fact that it contains the unit representation. This result has the following consequence for the two-sided cells (introduced later in [37]) of W. Let w_0 be the longest element of W and let c be a two-sided cell of W (we assume W\ne{1}). Then c':=cw_0=w_0c is again a two sided cell and either c or c' meets a proper parabolic subgroup of W. (This kind of result allowed me (in the later work [57]) to analyse unipotent representations inductively by using "truncated induction" from a proper parabolic subgroup and "duality" (which interchanges c,c').)
The class S_W is explicitly computed in each case (using the formalism of symbols [29] for classical types), the results of [31] on "fake degrees" and the results of [35] on Springer representations.
In this paper (Sec.9) I formulate the idea of "special unipotent class" of G (although I did not use the word "special"): these are unipotent classes in 1-1 correspondence with the special representation of W (under the Springer correspondence). Since the set of special representations of W admits a natural involution (given by T above) it follows that the set of special unipotent classes admits a natural involution. (For example the class {1} is interchanged with the regular unipotent class.) Later, [Spaltenstein, LNM 946, III, Springer Verlag 1982], motivated by this paper (as mentioned in [loc.cit., p.210]) gave a definition of a subset of the set of unipotent classes of G and an order reversing involution of this subset; this definition is based on properties of the partial order of the set of unipotent classes and is somewhat unsatisfactory [loc.cit., p.210] for exceptional types. One can show that the subset defined in [loc.cit.] is the same as the set of special unipotent classes but this was stated in [loc.cit.] not as a fact but as an analogy.
In this paper I also define a class \bar S_W of irreducible representations of W (which contains S_W, but unlike S_W, depends on the underlying root system). The representations in \bar S_W are obtained by truncated induction [35] from special representations of subgroups of W which are Weyl groups of Borel-de Siebenthal subgroups (=centralizers of semisimple elements) of the dual group G^* of G.
I believe that the most interesting and unexpected contribution of this paper is the statement that \bar S_W is in bijection with the set of unipotent classes in G via Springer's correspondence when G is of classical type and conjecturally in general; for exceptional groups this was verified in [48], see also 13.3 of [57]. (The details of the proof for classical groups appeared only 25 years later in [188].) This has the following consequence: there is a natural map from the set of special unipotent classes of a Borel-de Siebenthal subgroup of G^* (or its dual) to the set of unipotent classes in G; moreover, all unipotent classes in G appear in this way. This map has been later interpreted in terms of representation theory in [Barbasch-Vogan, Primitive ideals and orbital integrals..., Math.Ann.1982] (for complex groups) and in [100] (for groups over F_q and character sheaves). (2/20/2012)

37. (with D.Kazhdan) REPRESENTATIONS OF COXETER GROUPS AND HECKE ALGEBRAS, 1979.

The work on this paper was done in late 1978 and early 1979. It was an outgrowth of [38] (which was written after this paper but was conceived before it). My motivation for this work came from the desire to construct explicitly representations of the Hecke algebra H with parameter q attached to a Weyl group W. A basis B of a vector space with W-action is said to be "good" if for any simple reflection s of W and any b\in B we have either sb=-b or sb=b+\sum_{b'\in B;b'\ne b,sb'=-b'}a_{b,b',s}b' where a_{b,b',s} are integers. Similarly, a basis B of a vector space with H-action is said to be "good" if for any simple reflection s of W and any b\in B we have either T_sb=-b or T_sb=qb+\sqrt{q}sum_{b'\in B;b'\ne b,T_sb'=-b'}a_{b,b',s}b' where a_{b,b',s} are scalars. In late 1977 I showed that if u is a unipotent element in a semisimple group G over C and \BB_u is the variety of Borel subgroups containing u, then in the Springer representation of W on H_{top}(\BB_u) the basis given by the irreducible components of \BB_u is good with a_{b,b',s}\ge0. This appeared in a letter I sent to Springer (March 1978). The same idea appeared in [Hotta, J.Fac.Sci.Univ.Tokyo, 1982] where the letter above is cited. In the case where u is subregular (with G of type ADE), H_{top}(\BB_u) could be identified with the reflection representation of W with the basis formed by simple roots. In Kilmoyer's MIT thesis (which became a part of [Curtis,Iwahori,Kilmoyer,Publ.Math.IHES, 1971]) an explicit q-deformation of the reflection representation of W to a representation of H with a good basis is found. This suggested that the bases of H_{top}(\BB_u) (as above) may admit a q-analog which are good bases for an H-action. This is so for for G=SL_4,SL_5. Also, in [38] a good basis (b_w) for the regular representation of W was constructed in terms of irreducible components of the Steinberg variety Z of triples of G and because of the analogy between Z and \BB_u it was natural to ask whether this W-action could be deformed in a simple way to an H-action with a good basis. This project has failed but it still produced a good basis (c_w) of H (not a deformation of (b_w) in general).
I will try to explain the definition of the elements c_w in a way somewhat different from the paper. Let (T_w) be the standard basis of H and let w_0 be the longest element of W. Let b_w^* be the image of b_w by the involution w\to\sgn(w)w and let c_w^* be the image of c_w (to be defined) by the involution T_s\to -qT_s^{-1} of H. Since b_{w_0}=\sum_y\sgn(y)y satisfies sb_{w_0}=-b_{w_0} for all s, we expect that T_sc_{w_0}=-c_{w_0} for all s, so that c_{w_0}=\sum_y\sgn(y)T_y and c_{w_0}^*=\sum_yT_y. Similarly we expect c_s^*=T_s+1 if s is a simple reflection. Note that T_sT_{w_0}=qT_{sw_0}+(q-1)T_{w_0}. Here the right hand side has some coefficient involving -1 but if you replace T_s by T_s+1 we obtain (T_s+1)T_{w_0}=qT_{sw_0}+qT_{w_0} and now the right hand side has only coefficients q. Similarly (\sum_yT_y)T_{w_0}=\sum_yq^\nu T_y, \nu=l(w_0), l=length, and the right hand side involves only coefficients q^\nu. This suggests a way to characterize c_{w_0}^*. Assume that W is of type A_3 with generators 1,2,3. We have
T_{2132}T_{w_0}=(q^4-3q^3+4q^2-3q+1)T_{312312}
+(q^4-3q^3+3q^2-q)T_{13213}+(q^4-3q^3+3q^2-q)T_{32312}+(q^4-3q^3+3q^2-q)T_{12312}
+(q^4-2q^3+q^2)T_{3212}+(q^4-2q^3+q^2)T_{1232}+(q^4-2q^3+q^2)T_{2312}+(q^4-2q^3+q^2)T_{3213}
+(q^4-2q^3+q^2)T_{1213}+(q^4-q^3)T_{213}+(q^4-q^3)T_{123}+(q^4-q^3)T_{321}+(q^4-q^3)T_{312}+q^4T_{13}
and again the right hand has several coefficients involving a small power of q. As in the case of T_s we can hope that by adding to T_{2132} a linear combination of the T_y (y strictly less than 2132) with coefficients sums of powers of q very close to 1, the resulting sum times T_{w_0} is a linear combination of T_{y'} with coefficients sums of powers of q very close to q^4. There is a unique way to that:
(T_{2312}+T_{231}+T_{232}+T_{312}+T_{212}+T_{23}+T_{32}+T_{12}+T_{21}+
T_{13}+T_1+T_3+(q+1)T_2+(q+1))T_{w_0}=
(q^4+q^3)T_{312312}+(q^4+q^3)T_{13213}+q^4T_{32312}+q^4T_{12312}
+q^4T_{3212}+q^4T_{1232}+q^4T_{2312}+q^4T_{3213}+q^4T_{1213}+q^4T_{213}+
q^4T_{123}+q^4T_{321}+q^4T_{312}+q^4T_{13}.
We then take
c_{2312}^*=T_{2312}+T_{231}+T_{232}+T_{312}+T_{212}+T_{23}+T_{32}+T_{12}+
T_{21}+T_{13}+T_1+T_3+(q+1)T_2+(q+1).
This procedure works in general and leads to a basis (c_w^*) of H hence to a basis (c_w). (More explicitly, c^*_w=\sum_{y\le w}P_{y,w}(q)T_y is characterized by T_{w_0}c^*_w=\sum_{y\le w}q^{l(w)}P'_{y,w}(q^{-1})T_{w_0y} where P_{y,w},P'_{y,w} are polynomials in q of degree at most (l(w)-l(y)-1)/2 if y\ne w and P_{w,w}=P'_{w,w}=1.) In this paper it is shown that the basis (c_w) is good for both the left and right H-module structure (and in fact the scalar a_{b,b',s} is independent of s whenever it is nonzero. However the basis (b_w) is not obtained for q=1 from (c_w), even in type A, as shown by [Kashiwara,Y.Saito, Duke Math.J. 1997]. Note also that the definition of c_w^* is applicable to any Coxeter group by replacing the operation of multiplication by T_{w_0} by the operation \bar which replaces T_x by T_{x^{-1}}^{-1} and q by q^{-1} (which is what appears in the paper). In terms of \bar, c^*_w=\sum_{y\le w}P_{y,w}T_y is characterized by \bar c^*_w=q^{-l(w)}\sum_{y\le w}P'_{y,w}T_y where P_{y,w},P'_{y,w} are polynomials in q of degree at most (l(w)-l(y)-1)/2 if y\ne w and P_{w,w}=P'_{w,w}=1. (In the paper it is required that P_{y,w}=P'_{y,w} but this is actually automatically true.) Also in the paper left cells, right cells and two sided cells are introduced for any Coxeter group and the left cells in type A are determined explicitly. The coefficient of T_y in c_w^* is a polynomial P_{y,w} in q with constant term 1. The inversion formula 3.1 shows that the inverse of the triangular matrix (P_{y,w}) is the triangular matrix which has again the entries P_{y,w} in another indexing and with some sign changes. This inversion formula was later generalized in a beautiful way in [Vogan, Duke Math.J. 1982] to the case of symmetric spaces, in which case a passage to the Langlands dual of G is necessary. In this paper it is observed that the nontriviality of P_{y,w} is very closely related to the failure of local Poincar\'e duality on a Schubert variety.
The fact that the equivalence relation on the set of irreducible representations of W given by the two-sided cells (of this paper) seemed to coincide with the equivalence relation defined by the families (introduced earlier in [34] in conection with the representation theory of finite reductive groups), suggested that the P_{y,w} may have a representation theory significance. In this paper, a conjecture is stated to the effect that P_{y,w}(1) should be equal to the multiplicity [L_y:M_w] of a simple module L_y in a Verma module M_w over the Lie algebra of G. Some evidence for the conjecture came from the fact that the matrix [L_y,M_w] was known in the literature for rank \le 3 (Jantzen) and the P_{y,w} could be explicitly computed in rank \le 3 and they matched the [L_y,M_w]. Another evidence came from [Joseph, W-module structure on the primitive spectrum...,1979] which showed among other things that the basis of the regular representation of W given by \sum_y\sgn(y)\sgn(w)[L_y,M_w]y is good.
A bridge between the two sides of the conjecture was established in [39] via local intersection cohomology. As a consequence the conjecture became the equality between [L_y,M_w] and the Euler characteristic of a certain local intersection cohomology space. In this form the conjecture was established in [Beilinson,Bernstein, C.R.Acad.Sci.Paris, 1981] and [Brylinski,Kashiwara, Invent.Math. 1981].(4/04/2011)

38. (with D.Kazhdan) A TOPOLOGICAL APPROACH TO SPRINGER'S REPRESENTATIONS, 1980.

This paper was written in 1979 but the work on it was done in late summer of 1978 (except Sec.7). In 1976, Springer defined an action of the Weyl group W on the cohomology H^*(\BB_u) of the variety \BB_u of Borel subgroups containing a unipotent element u of a reductive algebraic group over C, using methods in characteristic p>0. Moreover in a letter to me (1977) Springer defined an action of W times W on the cohomology H^*_c(Z) of the Steinberg variety Z of triples (u,B,B') where u is a variable unipotent element and B,B' are Borel subgroups containing u; in the same letter he conjectured that the representation in H_c^{top}(Z) is the regular representation. In this paper an elementary construction of the Springer representation of W on H^{top}(\BB_u) and of the Springer representation of W\times W on H_c^{top}(Z) is given and the conjecture of Springer mentioned above is proved. The construction in this paper is based on an explicit homotopy equivalence s_i from \BB_u to \BB_u for any simple reflection in W. We were expecting (but unable to prove) that the maps s_i give a representation of W in the group of homotopy equivalences modulo homotopy of \BB_u; we could only prove this after passage to cohomology and only in top degree. The stronger statement has been established later in [Rossmann, J.Funct.Analysis 1991]. This paper's use of the Steinberg variety Z of triples reappeared in [72] in connection with the study of representations of an affine Hecke algebra. (4/04/2011)

39. (with D.Kazhdan) SCHUBERT VARIETIES AND POINCAR\'E DUALITY, 1980.

The work on this paper was done in early 1979. The appendix to [37] showed that the nontriviality of the polynomials P_{y,w} of [37] (for ordinary Weyl groups) was very closely related to the failure of local Poincare duality on a Schubert variety. It looked like the computations made in that appendix were actually computations of local intersection cohomology in case of an isolated singularity or in the case where one meets the singular locus for the first time. (R.Bott has suggested to Kazhdan that the results in that appendix could be related to intersection cohomology. On the other hand I have attended a lecture of MacPherson on intersection cohomology at Warwick in 1977 which dealt with the failure of Poincar\e duality, as did the appendix to [37], and I was wondering about the connection between the two.) However, a preprint of Goresky, MacPherson gave a different value for the local intersection cohomology than what we found in our case. Therefore Kazhdan and I arranged to meet MacPherson (at Brown U.) to clarify this point. It turned out that the Goresky-MacPherson preprint had a misprint and in fact it should have matched our computation. After this Kazhdan and I tried to identify all of P_{y,w} with the local intersection cohomology of a Schubert variety and we succeeded in doing so (using results of Deligne). This is what is done in this paper. This can be viewed as a step in the proof of the conjecture on multiplicities in Verma modules in [37]. It also gives a proof of the positivity of coefficients of P_{y,w} (for Weyl groups and affine Weyl groups); this is still the only known proof of this positivity property in these cases. The idea to consider the affine Schubert variety (attached to an element in the affine Weyl group) as an algebraic variety also appears (perhaps for the first time) in this paper.(10/02/2010)

40. SOME PROBLEMS IN THE REPRESENTATION THEORY OF FINITE CHEVALLEY GROUPS, 1980.

This paper is based on a talk given in july 1979 at the Santa Cruz Conference on Finite Groups. It states several problems. Problem I states as a conjecture the multiplicity formula for unipotent representations in the virtual representations R(T,1) of [22]. This was solved in [42],[45],[46],[57], (the last three make use of the results in [39]). Problem II is about assigning a unipotent support to an irreducible representation. This was solved in large characteristic in [100] and later in general in [Geck,Malle, Trans.Amer.Math.Soc.,2000]. Problem III relates the families [34] of irreducible representations of the Weyl group with the two-sided cells [37]; it has been solved in [Barbasch,Vogan, Math.Ann.1982 and J.Alg. 1983]. Problem IV is a conjecture on the characters of irreducible modular representations of a semisimple group in characteristic p>0 in terms of the polynomials [37] attached to the affine Weyl group of the Langlands dual group. Problem V states that the unipotent classes of a semisimple group in large characteristic are in bijection with the two-sided cells (see [37]) of the affine Weyl group of the Langlands dual group. This was solved in [86]. (9/17/2010)

41. HECKE ALGEBRAS AND JANTZEN'S GENERIC DECOMPOSITION PATTERNS, 1980.

In this paper I introduce and study a certain module over an affine Hecke algebra, which I now call the periodic module. For simplicity I define it here in type A_1. Let E be an affine euclidean space of dimension 1 with a given set P of affine hyperplanes (points) which is a single orbit of some nontrivial translation of E. Then the group \Omega of affine transformations of E generated by the reflections with respect to the various H in P is an infinite dihedral group. The connected components of E-P are called alcoves; they form a set X on which \Omega acts simply transitively. Let S be the set of orbits of \Omega on P. It consists of two elements. If s\in S then s defines an involution A\to sA of X where sA is the alcove \ne A such that A and sA contain in their closure a point in the orbit s. The maps A\to sA generate a group of permutations of X which is a Coxeter group (W,S) (an affine Weyl group of type A_1 acting on the left on X). We assume that for any two alcoves A,A' whose closures contain exactly one common point (in P) we have a rule which says which of the two alcove is to left (or to the right) of the other in a manner consistent with translations. Let v be an indeterminate. Let \HH be the Hecke algebra attached to W,S and let M be the free Z[v,v^{-1}]-module with basis X. There is a unique \HH-module structure on M such that for s\in S,A\in X we have T_sA=sA if sA is to the right of A and T_sA=v^2sA+(v^2-1)A if sA is to the left of A. For each H\in P let e_H\in M be the sum of the two alcoves in X whose closures contain H. Let M^0 be the \HH-submodule of M generated by the elements e_H. Now in the paper the higher dimensional analogue of the situation above is studied. The analogue of X and the \HH-modules M,M^0 are introduced. A bar involution of M^0 is introduced; it is semilinear with respect to the bar involution [37] on \HH. A canonical basis of the Z[v,v^{-1}]-module M^0 is constructed using the bar operator on M^0 by a method similar to that of [37] (but the construction is more intricate). This canonical basis is indexed by the alcoves in X. The polynomials which give the coefficient of an alcove B in the basis element corresponding to an alcove A are periodic with respect to a simultaneous translation of A and B. They can be related to the polynomials attached in [37] to W; this relation proves a periodicity property for these last polynomials the proof of which was the main motivation for this paper. (An analogous periodicity property for the multiplicities in the Weyl modules of a simple algebraic group in characteristic p was first pointed out by [Jantzen, J.Algebra, 1977] and the periodicity result of this paper provided support for the conjecture in [40] on these multiplicities). Shortly after writing this paper I found the folowing geometric interpretation of the results of this paper. Let G be a simply connected almost simple group over C. Let K=C[[t]]. Let U be the unipotent radical of a Borel subgroup of G and let I be an Iwahori subgroup of G(K). Then the set of double cosets U(K)\G(K)/I is (noncanonically) the affine Weyl group and (canonically) the set X of alcoves as above. (A closely related statement is contained in [Bruhat,Tits, Groupes r\'eductifs sur un corps local, Publ.IHES,1972, Prop.(4.4.3)(1).) This led me to the statement that the periodic polynomials of this paper can be interpreted as local intersection cohomologies of the (semiinfinite) U(K)-orbits on G(K)/I. This statement appears without proof in [59]; a proof appears in [Finkelberg,Mirkovic, Semiinfinite flags I; Feigin,Finkelberg,Kuznetsov,Mirkovic, Semiinfinite flags II, Transl.of A.M.S,1999]. (2/19/2012)

42. ON THE UNIPOTENT CHARACTERS OF THE EXCEPTIONAL GROUPS OVER FINITE FIELDS,1980.

In this paper I determine the multiplicities of the unipotent representations of an exceptional group over a finite field F_q in the virtual representations R(w,1) of [22], assuming that q is large. These multiplicity formulas were conjectured in [40]. The proof given in the paper uses the formulas (known at the time) for the dimensions of unipotent representations and unlike the later proof [54] (where the restriction on q was removed) it does not use intersection cohomology methods. The method of this paper does not seem to be strong enough in the case of classical groups which was treated later in [45],[46] using intersection cohomology methods. (6/25/2011)

43. ON A THEOREM OF BENSON AND CURTIS, 1981.

Let H be the Hecke algebra over Q associated to the Weyl group W and to the parameter q, a power of a prime. In 1964 Iwahori conjectured that H is isomorphic to the group algebra Q[W]. Tits showed (in an exercise in Bourbaki) that this conjecture holds if Q is replaced by its algebraic closure. In 1972 Benson and Curtis showed that Iwahori's conjecture was true as originally stated but Springer found a gap in the proof (for type E_7). (The Benson-Curtis proof was correct for types other than E_7,E_8.) Springer in fact showed that a 512 dimensional irreducible representation of H (of type E_7) is not defined over Q: its character definitely involves a square root of q.
In this paper (written in 1980) I construct an isomorphism of H with Q[W] when the scalars are extended to Q[\sqrt q]. The key new observation is as follows. Consider the vector space apanned by the elements of a fixed two-sided cell of W. There is a left action on this space for the Hecke algebra H with parameter q in which the basis elements are identified with the elements of the new basis [37] of H; there is also a right action on this space for the Hecke algebra H' with another parameter q' in which the basis elements are identified with the elements of the new basis [37] of H'. The two actions obviously commute with each other if q=q' but surprisingly they also commute with each other when q,q' are independent. The proof is based on some properties of primitive ideals in an enveloping algebra. The isomorphism I construct is explicit unlike those in earlier approaches. The use of the theory of primitive ideals can nowadays be eliminated and replaced by the use of the "a-function" introduced in [60]. This paper also gives W-graphs (in the sense of [37]) for the left cell representations of H in the noncrystallographic case H_3 and an example analogous to the 512 dimensional representation (for E_7) is pointed out in type H_3.(9/24/2010)

44. GREEN POLYNOMIALS AND SINGULARITIES OF UNIPOTENT CLASSES, 1981.

In this paper I find a relation between:
(1) the local intersection homology groups of the closure of a unipotent class in GL_n;
(2) the local intersection homology of an affine Schubert variety in an affine grassmannian of type A;
(3) the character value at a unipotent element of an irreducible unipotent representation of GL_n(F_q).
The connection (1)-(3) is a precursor of the theory of character sheaves which was developped in [63-65,68,69]. The connection (2)-(3) implies that the groups in (2) can be described in terms of multiplicities of weights for the finite dimensional representations of GL_n(C) which was an inspiration for the paper [53] (a generalization from GL_n to general reductive G). This paper also formulated the idea (used in many subsequent papers) that the Springer resolution is a small map and uses this idea to give a new definition of the Springer representations of a Weyl group in terms of intersection cohomology (unlike previous definitions this was valid in arbitrary characteristic). This shows in particular that the direct image of the constant sheaf under the Springer resolution is a perverse sheaf up to shift. Conjecture 2 of this paper was subsequently proved by [Borho,MacPherson, C.R.Acad.Sci.Paris 1981].
The method introduced in this paper to construct Springer's representations has been used in later papers:
(a) to construct the "generalized Springer correspondence" [59];
(b) to construct analogues of the Springer representation over parameter spaces which yield representations of graded affine Hecke algebras [81].
(c) to construct a version of Springer representations for affine Weyl groups [125];
(d) to construct a Weyl group action on the cohomology of certain quiver varieties [149].
I will now comment on Remark (a) on p.177 of this paper. Let X be the variety of unipotent elements in a connected reductive group. In Remark (a) I state that (in sufficiently large characteristic:)
(*) "X is rationally smooth";
moreover I state that this is an unpublished result of Deligne. This was based on (an apparently faulty recollection of) a discussion I had with Deligne in 1974 in which he explained to me how to get Steinberg's theorem that
(&) "the number of rational points of X over F_q is a power of q" in a geometric way using monodromies. (Note that (&) follows easily from (*).) However, later Deligne denied ever having proved (*).
In the spring of 1981 I proved (*) or rather the following statement which at the time was known to be equivalent to (*):
(**) The space of Weyl group invariants on the i-th cohomology of the Springer fibre at u\in X has dimension 1 if i=0 and 0 if i>0.
My proof of (**) used the results of Springer and Kazhdan on Green functions. (This proof is not published but its existence is mentioned in [Borho,MacPherson, Ast\'erisque 101-102, 1983].) Note that (*) is not known in small characteristic. Also the analogue of (*) for nilpotent elements in the Lie algebra (or its dual) in small characteristic is not known. (7/26/2011)

45,46. UNIPOTENT CHARACTERS OF THE SYMPLECTIC AND ODD ORTHOGONAL GROUPS OVER A FINITE FIELD, 1981; UNIPOTENT CHARACTERS OF THE EVEN ORTHOGONAL GROUPS OVER A FINITE FIELD, 1982.

The first of these paper was conceived during a visit to the Australian National University, Canberra (Jan.1981); the second one was written later in 1981. Let G(F_q) be a group as in the title. In [30, Conj.4.3] I conjectured the precise pattern which gives the multiplicities of the various unipotent representations in the virtual representations R_w of [22] or equivalently in the linear combinations R_E of the R_w with coefficients given by an irreducible character E of the Weyl group; namely the pattern should be the same as the pattern [29] describing the dimensions of unipotent representations as linear combinations of fake degrees. This conjecture is established in this paper. The main new technique in the proof is the use of the local intersection cohomology of the closures of the varieties X_w of [22] which I show that is the same as the the local intersection cohomology of a Schubert variety and hence [29] is computable in terms of Hecke algebras. Another new technique used in the paper is the systematic use of the leading coefficients of character values of the Hecke algebra. These techniques were later generalized to any reductive group (see [57]).(5/29/2011)

47. (with P.Deligne) DUALITY FOR REPRESENTATIONS OF A REDUCTIVE GROUP OVER A FINITE FIELD, 1982.

In 1977 I found a definition of an operation D in the complex representation group of a reductive group over F_q which to any representation E associates \sum_P(sgn_P)ind_P aa^*res_P(E) where P runs over the parabolic subgroups over F_q containing a fixed Borel subgroup over F_q, ind_P is induction from P(F_q) to G(F_q), a is lifting from P(F_q)/U_P(F_q) to P(F_q) (L=Levi of P over F_q), a^* is the adjoint of a and res_P is restriction to P(F_q); sgn_P is a sign. It was known at the time (Curtis) that D takes the unit representation to the Steinberg representation. If E is cuspidal then a^*res_P(E) is zero if P\ne G hence DE=\pm E. But my main motivating example was one which I encountered in [13] where G=GL_n(F_q), the complex numbers are replaced by F_q and E is the natural representation of G on F_q^n. In that case DE can be defined as above and can be viewed as a reduction mod p of a cuspidal complex representation of G of dimension (q-1)(q^2-1)...(q^{n-1}-1); this was the main observation on which the work [13] was based. I conjectured that over complex numbers D takes any irreducible E to an irreducible representation (up to sign) and that D^2=1. In 1977 I communicated this conjecture to D.Alvis and C.W.Curtis (at the Corvallis Conference) and (separately) to N.Kawanaka. My conjecture was proved (at the level of characters) by [Alvis, Bull.AMS,1979], [Curtis,J.Algebra,1980] and independently by [Kawanaka,Invent.Math.,1982]. In the present paper a version of D at the level of representations (rather than characters) is given. As an application another proof of the conjecture is given. The operation D played a key role in my later work [57] where it was used to analyse unipotent representations inductively (in conjunction with "truncated induction" from a proper parabolic subgroup). An analogous operation plays a key role in the classification of character sheaves. In [A.M.Aubert, Trans.AMS,1995] a study of a p-adic analogue of the operation D defined in this paper is given. (6/26/2011)

48. (with D.Alvis) ON SPRINGER'S REPRESENTATIONS FOR SIMPLE GROUPS OF TYPE E_n (n=6,7,8), 1982.

Let G be as in the title (over C). In this paper we compute the Springer representation of the Weyl group W of G corresponding to any unipotent class and the local system C on it. There are three tools that are used in the proof: (a) the compatibility of truncated induction with the Springer correspondence [35]; (b) the conjecture (2) in [44] which was just proved by Borho-MacPherson; (c) an induction formula for the total Springer representation for a unipotent element contained in a proper Levi subgroup. Moreover, using the induce/restrict tables of Alvis we showed that the class of irreducible representations of W thus obtained coincides with the class \bar S_W introduced in [36], thereby completing the proof of the conjecture at the end of [36] (which at the time of [36] was already known for classical types and G_2). In the appendix (by Spaltenstein) the rest of the Springer correspondence (involving irreducible local systems \ne C) is determined. (2/19/2012)

50. A CLASS OF IRREDUCIBLE REPRESENTATIONS OF A WEYL GROUP II, 1982.

This paper was written in early 1981. Let W be a Weyl group and let Irr W be the set of irreducible representations of W (up to isomorphism). In [34] a partition of Irr W into subsets called families was described. The definition was such that the degrees of unipotent representation of a finite Chevalley groups were linear combinations of fake degrees of objects of Irr W in a fixed family. In the present paper an elementary definition of families is given. More precisely a collection of possibly reducible representations (called cells and in later papers, constructible representations) is defined by induction. Namely it is required that by applying a certain kind of truncated induction to a cell of a proper parabolic subgroup one obtains a cell of W; moreover by tensoring a cell by the sign representation of W one obtains again a cell. The cells are obtain by applying a succession of such operation starting with the trivial one dimensional representation of W. In this paper the cells of any W are explicitly determined. It is shown that any E in Irr W appears in some cell; every cell contains a unique special representation (in the sense of [36]) which in fact has multiplicity one; and two cells have a common irreducible component if and only if they contain the same special representation. Therefore we can define an equivalence relation on Irr W as follows: E,E' in Irr W are equivalent if there exist cells c,c' such that E appears in c, E' appears in c' and c,c' have the same special component. The equivalence classes are called families. In the paper it is conjectured that the cells of W are exactly the representations of W that are carried by the left cells of W (in the sense of [37]). This conjecture was proved in [70]. Using the results of this paper one can give a new definition of the involution of the set of special representations of W (see the comments to [36]) which bypasses the consideration of a Galois group action: namely the involution maps a special representation E in a family f to the unique special representation in the family (f tensored by sign).(2/24/2012)

51. (with D.Vogan) SINGULARITIES OF CLOSURES OF K-ORBITS ON A FLAG MANIFOLD, 1983. The work on this paper was done in late 1980. Its main object of study was the local intersection cohomology (l.i.c.) of the closure of a K-orbit where K is the identity component of an involution of a complex reductive group G. At the time it was known from the work of Beilinson and Bernstein that this l.i.c. is closely related to the computation of multiplicities in standard module of the various irreducible representations of a real reductive group attached to the involution in the same way as the l.i.c. of Schubert varieties was known to be closely related to multiplicities in Verma modules. The problem of determining the l.i.c. in the present case was a generalization of the problem of determining the l.i.c. of Schubert varieties solved in [39]. But the method of [39] did not work in the present case, partly due to the presence of non-trivial equivariant local systems (of order two) on the K-orbits. Unlike in [39] in this paper the connection with the representation theory of real groups is used in the computation; also the purity theorem of Gabber (which was not available at the time of [39]) plays a key role in the proof. The main result of this paper is that the l.c.i. are described in terms of some new polynomials P_{\gamma,\delta}, where each of \gamma and \delta is a K-orbit together with a K-equivariant irreducible local system on it, which are explicitly computable and which generalize the polynomials P_{y,w} of [37]. (Further work by Fokko Du Cloux has made possible the computation of P_{\gamma,\delta} on a computer.) This paper also contains an interpretation of the product in the Hecke algebra and in certain modules over it in terms of convolution in derived categories (involving operations of inverse image, direct image and tensor product in derived categories). This interpretation which has become part of the folklore has been also found around the same time by MacPherson, see [Springer, Sem.Bourbaki 589, 1982]. A proof of the results of this paper which is purely geometric (that is it does not rely on representation theory of real groups) has been later found by [Mars and Springer, Represent. Th., 1998].(10/03/2010)

52. (with P.Deligne) DUALITY FOR REPRESENTATIONS OF A REDUCTIVE GROUP OVER A FINITE FIELD, II, 1983.

Let G be a connected reductive group over F_q. In this paper it is shown that the "duality operator" $D$ of [47] applied to the virtual representation R(T,\theta) in [22] is equal (up to sign) to R(T,\theta). The proof is based on an inner product formula between R(T,\theta) and a R(L,r) (as in [24]) where L is a Levi subgroup over F_q of a parabolic (not necessarily over F_q) and r is a representation of L(F_q). The proof of this orthogonality formula given in the paper contains a (not very serious) error. The corrected proof (which I supplied to Digne and Michel at their request) appears in the book [Digne,Michel, Representations of finite groups of Lie type, 1991, 11.13]. (7/10/2011)

53. SINGULARITIES, CHARACTER FORMULAS AND A q-ANALOG OF WEIGHT MULTIPLICITIES, 1983.

This paper was written in 1981 and presented at the Luminy Conference on Analysis and Topology on Singular Spaces (July 1981). In this paper I find a very close connection between
-the category A of finite dimensional representations of a complex simply connected group G and
-the category A' of G^*[[\e]])-equivariant perverse sheaves on the affine Grassmannian associated to the Langlands dual G^* of G.
In more detail, let \L^+ be the set of dominant weights of G. For x\in\L^+, let V_x be the finite dimensional representation of G corresponding to x and let m_y(V_x) be the multiplicity of y\in\L^+ in V_x. Let M_x be the element of the (extended) affine Weyl group W of G^* which has maximal length in the double coset of x with respect to the usual Weyl group W_f. Let H be the affine Hecke algebra of W and let (C_w)_{w\in W} be the basis [37] of H. For x\in\L^+ we set \gamma_{M_x}=\pi^{-1}C_{M_x} where \pi=q^{-\nu/2}\sum_{w\in W_f}q^{l(w)}; \nu is the number of positive roots. For x\in\L^+ let \bar O_x be the closure of the G^*[[\e]]-orbit coresponding to x in the affine Grassmannian and let \Pi_x be the corresponding simple object of A'. For w,w' in W let P_{w',w} be the polynomial defined in [37].
Here are the main results of this paper.
(I) For x,y in \L^+, we have m_y(V_x)=P_{M_y,M_x}(1). (Thus the weight multiplicities m_y(V_x) are related to the dimension of stalks of \Pi_x.)
(II) For x,y in \L^+, we have \gamma_x\gamma_y=\sum_{z\in\L^+}c_{x,y,z}\gamma_z where c_{x,y,z} are natural numbers (apriori they are only polynomials in q). An equivalent statement is that the convolution \Pi_x*\Pi_y is a direct sum of objects \Pi_z (z\in\L^+) without shifts; or that the map which defines this convolution is semismall.
(III) For x,y,z in \L^+, the number c_{x,y,z} in (II) is equal to the multiplicity of V_z in the tensor product V_x\otimes V_y.
(IV) For x in \L^+, the vector space V_x is isomorphic to the total intersection cohomology of \bar O_x.
(Statement (IV) appears in the last line of this paper; note that the odd intersection cohomology of \bar O_x is zero.)
Statement (II) is called the "miraculous theorem" in [Beilinson-Drinfeld, Quantization of Hitchin integrable system... (1991), 5.3.6]. It is equivalent to the fact that A' is a monoidal category under convolution. Statement (III) suggests that this monoidal category is equivalent to A with its obvious monoidal structure and statement (IV) suggests the definition of a fibre functor for A' which would enter in the construction of such an equivalence. The tensor equivalence of A and A' was established in [Gi]=[Ginzburg, arxiv:alg.geom./9511007] based on the results of this paper (using (II) and the fibre functor above), except that the commutativity isomorphism for A' given in [Gi] was incorrect and was later provided by Drinfeld (whose construction is sketched in [MV]=[Mirkovic,Vilonen, Math.Res.Lett.2000]). Thus the equivalence of A,A' as monoidal categories (now known as the "geometric Satake equivalence") has been established by combining the ideas of this paper with those of Ginzburg and Drinfeld. A version of the the geometric Satake equivalence in positive characteristic is established in [MV].
My motivation for writing this paper was as follows. First I noticed that my conjecture [40] relating the characters of modular representations of a semisimple group in characteristic p with the polynomials P_{w',w} as above implies (I). Hence I tried to prove (I) to provide evidence for the conjecture in [40]. (At that time I already knew that (I) is true in type A, as a consequence of [44].)
Now by (I) each weight multiplicity appears by setting q=1 in a polynomial in q with positive coefficients; hence that polynomial can be viewed as a "q-analog of weight multiplicities", hence the title of the paper. Subsequently, a (partly conjectural) interpretation of these q-analogs was given purely in terms of representations of G in [R.K.Gupta (later Brylinsky), JAMS 1989]; this was later confirmed in [Joseph,Letzter,Zelikson, JAMS 2000]. In this paper I also introduce a q-analogue of the Kostant partition function and prove that it is equal to the q-analogue of weight multiplicities in the stable range.
The proof of (II) given in this paper relies on some dimension estimates in [41] involving semiinfinite geometry in disguise.
Another result of this paper is a description of the affine Grassmannian as an ind-variety (as a subset of the set of selfdual orders in a simple Lie algebra over C((\e)) given by explicit equations). (11/19/2012)

54. SOME EXAMPLES OF SQUARE INTEGRABLE REPRESENTATIONS OF SEMISIMPLE P-ADIC GROUPS, 1983.

Let G be the group of rational points of a simple split adjoint algebraic group over a nonarchimedean local field whose residue field has q elements. This paper introduces the notion of unipotent representation of G; these are the irreducible admissible representations of G whose restriction to some parahoric subgroup of G contains a unipotent cuspidal representation of the "reductive" quotient of G. Let U be the subset of the set U' of unipotent representations of G formed by the Iwahori-spherical representations of G. This paper formulates a refinement for the Deligne-Langlands conjecture according to which U is in finite to one correspondence with the set of pairs (s,u) where s,u are a semisimple element and a unipotent element (up to conjugacy) in the complex "dual" group such that su=u^qs.
My refinement of the conjecture was to add a third parameter: an irreducible representation \rho of the group A(s,u) of connected components of the simultaneous centralizer of s,u on which the centre of the dual group acts trivially. In this paper I state the conjecture that the triples (s,u,\rho) as above are in canonical bijection with U' and that U is in bijection with the set of triples (s,u,\rho) such that \rho appears in the cohomology of the variety X of Borel subgroups containing s and u.
A good thing about the refined conjecture is that it indicates that the representations of the affine Hecke algebra may be constructed geometrically in terms of a space like X. The connection with geometry became even stronger after the equivariant K-theoretic approach of [66] was found and led to the solution of the (refined) conjecture in [67],[72]. The idea of the refined conjecture came from the experiments performed in this paper: I construct explicitly (using W-graphs) the reflection representation and some closely related representations of the affine Hecke (and I show that they are often square integrable by some very complicated computation); these representations correspond conjecturally to the subregular unipotent element and this provides evidence for the refined conjecture. In these examples I also found that the weight structure of the representations I construct can be interpreted in terms of the geometry of the varieties X above, further reinforcing the idea that the geometry of X should play a role in the proof of the conjecture.
In this paper I introduce a description of the affine Weyl group of type A as a group of periodic permutations of the integers. This point of view was later used extensively in [Shi, The Kahdan-Lusztig cells in certain affine..., Springer LNM 1179, 1986]. I also give a conjecture giving the number of left cells in each two sided cell of an affine Weyl group of type A, which was later proved by [Shi, loc.cit.] and a conjecture describing explicitly the two sided cells of an affine Weyl group of type A, which I later proved in [62].
Another result of this paper is the construction of an imbedding of a Coxeter group of type H_4 (resp. H_3) into the Weyl group E_8 (resp. D_6) which has the property of sending any simple reflection to the product of two commuting simple reflections and any element of length n to an element of length 2n (this is part of a general result about imbedding of Coxeter groups, see 3.3.) This imbedding has been rediscovered ten years later in [Moody & Patera, J.of Physics,A, 1993].
The argument in 2.8 has been used in the later papers [67,72] to prove square integrability of certain geometrically defined representations of an affine Hecke algebra. The last sentence in 2.11 was later proved in [78]. (4/16/2013)

56. OPEN PROBLEMS IN ALGEBRAIC GROUPS, 1983.

In the summer of 1983 I participated to a Taniguchi conference in Katata, Japan. The participants were asked to write up a list of open problems. Here are some of the problems on my list.
(1) Let W be an affine Weyl group. Then the number of left cells contained in the two-sided cell corresponding under the bijection in [86] to the conjugacy class of a unipotent element u in a reductive group over C of dual type to that of W is equal to the dimension of the part ot the cohomology of the Springer fibre at u invariant under the action of the centralizer of u.
(2) Let W be as in (1). We identify W with the set of (closed) alcoves in an euclidean space in the standard way. Let A,B be two alcoves in the same two sided cell. Show that A,B are in the same left cell if and only if there exists a sequence of alcoves A=A_0,A_1,...,A_n=B (all in the same two sided cell) such that A_i,A_{i+1} share a codimension 1 face for i=0,1,...,n-1. Show that the union of alcoves in a left cell is a contractible polyhedron. Show that similar results hold for a finite Weyl group by replacing the euclidean space with the corresponding triangulated sphere. (6/06/2011)

57. CHARACTERS OF REDUCTIVE GROUPS OVER A FINITE FIELD, 1984.

Let G be a connected reductive group with connected centre defined over F_q. The main contribution of this book (written in 1982) is the classification of the irreducible representations of G(F_q) and the computation of their multiplicities in the virtual representations R(w,\theta) of [22]. (Earlier, this kind of results were known for unipotent representations with q large, see [42],[45],[46]; the classification (but not the multiplicities ) for classical groups with any q was also known [29]).
Let L be a G-equivariant line bundle over the flag manifold of G and let L-0 be the complement of the zero section of L. In Ch.1 contains the computation of the local intersection cohomology of L-0 with coefficients in certain "monodromic" local systems on some smooth subvarieties of L-0, in terms of the polynomials [37] for the Weyl group of the centralizer of a semisimple element s in the dual group. This computation generalizes results of [39] which correspond to the case s=1. Since these local intersection cohomology groups were known (by Beilinson and Bernstein) to compute multiplicities in Verma modules with regular rational highest weight, the computation in Ch.1 was a new instance of a connection between representations of a group and geometry of the dual group. This computation is used in [Beilinson-Bernstein, A proof of Jantzen's conjecture, Adv.Sov.Math.1993]. An affine generalization of this computation is given in [117] where it is used as one of the steps in the proof of the character formula for quantum groups of nonsimplylaced type at a root of 1. Finally this computation is used in Ch.2 of this book to determine the local intersection cohomology of the closures of the varieties X_w of [22] with coefficients in local systems associated with the covering \tilde X_w of X_w described in [22]. Again the result is expressed in terms of the polynomials of [37] for the Weyl group of the centralizer of a semisimple element in the dual group. (8/01/2011)

58. CHARACTERS OF REDUCTIVE GROUPS OVER FINITE FIELDS, 1984.

This is based on my talk at the ICM-1982 held in Warsaw in 1983 (the 1982 event was postponed due to the martial law). This paper is an exposition of the main results of [57] (written in 1982) which were under the assumption of connected centre. But in the present paper that assumption was removed. In order to remove two words: "connected centre" from my paper I had to do two months of intensive work (june/july 1983) mainly with the case of Spin_{4n}. These computations with spin groups (not included in the paper where no proofs were given) have been published 25 years later in [180] with some earlier hints given in [83].(9/05/2010)

59. INTERSECTION COHOMOLOGY COMPLEXES ON A REDUCTIVE GROUP, 1984.

This paper was written in late 1982 and early 1983. Let G be a connected reductive group over an algebraically closed field of characteristic p. Let X be the (finite) set of all pairs (C,E) where C is a unipotent class in G and E is G-equivariant irreducible local system on C (up to isomorphism). In the late 1970's Springer showed that (for large p) there is a natural bijection between a certain subset X_0 of X and the set Irr(W) of irreducible representations of the Weyl group W of G. In [44] I gave a new definition of the Springer representations of W using intersection cohomology methods which is valid without restriction on p, but the proof that it induces a bijection between X_0 and Irr(W) was first given for arbitrary p in this paper, using a study of sheaves on the variety of semisimple classes. In this paper I show (extending the method of [44]) that a suitable enlargement of Irr(W) is in canonical bijection ("generalized Springer correspondence") with X itself. The enlargement is a disjoint union of sets of the form Irr(W_i) where W_i is a collection of Weyl groups (one of which is W). Of particular interest are the objects of X for which the corresponding W_i is {1}. These are the "cuspidal local systems" (c.l.s.) which are introduced, studied and classified in this paper. A G-equivariant local system E on a unipotent class C of G is a c.l.s. if for any proper parabolic P of G with unipotent radical U_P and any unipotent g in P, the d-th cohomology with compact support of C\cap gU_P with coefficients in E is zero (where d is dim(C) minus the dimension of the conjugacy class of g in P/U_P); note that if d is replaced by d'>d then the corresponding vanishing property holds for any E. A new feature of this paper is the explicit combinatorial description of the generalized Springer correspondence in terms of some objects closely related to the "symbols" in [29]. This was new even for the ordinary Springer correspondence which was previously known only in the form of an algorithm (Shoji), rather than by a closed formula. In the case where G is a spin group with p odd, the generalized Springer correspondence gives a combinatorial interpretation of the Jacobi triple product formula (see Section 14).
Another new result of this paper was a definition of "admissible complexes" on G, a class of perverse sheaves on G whose existence was conjectured in [57, 13.7,13.8] where the required class of perverse sheaves was defined for G=GL_n. One of the main ingredients in the definition of admissible complexes is the notion of c.l.s. (see above) extended from unipotent classes to "isolated classes". The admissible complexes on G reemerged in another incarnation (as "character sheaves") in the series [63-65,68,69]. (5/26/2011)

60. CELLS IN AFFINE WEYL GROUPS, 1985.

Let W be a Weyl group or an affine Weyl group. This paper develops some techniques for computing the left/two-sided cells [37] of W. The main contribution of this paper is the definition of the function a:W\to(natural numbers). For w\in W, I define a(w) essentially as the order of the worst pole of the coefficient of C_w (the Hecke algebra element of [37]) in a product T_xT_y of two (variable) standard basis elements of the Hecke algebra. I show that a is constant on the two-sided cells of W. When W is of affine type the fact that a(w) is well defined needs a proof (given in the paper); in fact I show that a(w) is at most the number N of positive roots. Therefore the set W_*={w\in W;a(w)=N} is of particular significance. Let W_! be the set of all products abc where a,b,c\in W, the length of abc is the sum of the lengths of a,b,c and b has length N and is contained in a finite parabolic subgroup of W. In the paper it is shown that W_! is contained in W_*; in particular, W_* contains "almost all" elements of W. In this paper, using the function a, I describe explicitly the decomposition of the affine Weyl group W of type A_2,B_2,G_2 into left/two-sided cells in terms of a picture in which W is viewed as the set of alcoves in a decomposition of an euclidean plane and each alcove is colored according to the two-sided cell to which it belongs. (The picture that I found for B_2, which reminds one of the Union flag (resp. for G_2) has been adopted as the logo of the Conference on Representation Theory in Tokyo 2001 (resp. in Nagoya 2006 and Nagoya 2010); the picture for G_2 also appears on some t-shirts made by T.Tanisaki.) It turns out that, for affine A_2,B_2,G_2 the number of two-sided cells is 3,4,5; this was one of the pieces of evidence which led to my conjecture [40] (restated in this paper) on the relation between two-sided cells and unipotent classes. From the results of this paper one can see that in rank 2 one has W_*=W_! and W_* is a single two-sided cell. This was later extended to arbitrary rank in [Shi, J.Lond.Math.Soc. 1987] and [B\'edard, Commun.in Alg. 1988].(8/05/2011)

62. THE TWO-SIDED CELLS OF THE AFFINE WEYL GROUP OF TYPE A, 1985.

The results of this paper were presented at a conference at MSRI in May 1984. In early 1983 I have learned from R.Carter about the remarkable work of his Ph.D. student J.Y.Shi (at Warwick) in which Shi determined explicitly the left cells of the affine Weyl group W of type A_n; it turned out that Shi's methods were not sufficient to determine the two-sided cells of W (for which I formulated a conjecture in [54]). After I introduced the function a on W in [60], I realized that the results of Shi together with the use the function a and its properties are sufficient to determine the two-sided cells of W. This is what is done in this paper; see also [Shi, The Kazhdan-Lusztig cells in certain affine..., Springer LNM 1179, 1986]. (8/04/2011)

66. EQUIVARIANT K-THEORY AND REPRESENTATIONS OF HECKE ALGEBRAS, 1985.

The work on this paper was done at the Tata Institute, Bombay, in Dec. 1983. At the time when this paper was written, the parameter q of a Hecke algebra was viewed as a number, an indeterminate, a Tate twist or a shift in a derived category. One of the main contributions of this paper is to formulate the idea (new at the time) to view q as the generator of the equivariant K-theory of a point with respect to the circle group and that various modules of the affine Hecke algebra H can be realized in terms of equivariant K-theory with respect to a group containing the circle group as a factor. More specifically in this paper I show that the principal series representations of H admits a description in terms of equivariant K-theory as above and conjectures are formulated for a description in the same spirit of other H-modules attached to nilpotent elements.
The idea to use equivariant K-theory to study affine Hecke algebras was subsequently developed in the papers [67], [72] (with Kazhdan) and in [Chriss,Ginzburg, Representation Th.and complex geometry, 1997]. The same idea was later used
-by Garland-Grojnowski (and later by Varagnolo-Vasserot) to realize the Cherednik (double affine Hecke) algebra;
-by Nakajima to realize geometrically an affine quantum group.(7/31/2010)

73. CELLS IN AFFINE WEYL GROUPS,II, 1987.

This paper is a continuation of [60]. Let W be a Weyl group or an affine Weyl group. One of the main contributions of this paper is a definition (in terms of the function a of [60]) of a set D of involutions of W (which I call distinguished involutions). The definition was inspired in part by a conjecture of [A.Joseph, J.Algebra 1981] for finite W, which in fact follows from the results of this paper. For finite W, one can identify D with the set of Duflo involutions defined in the theory of primitive ideals; but I don't know a similar identification for affine W. In this paper I show that each left cell contains exactly one element of D and that the set of left cells in W is finite (hence D is also finite). Note that the set of left cells in a more general Coxeter group can be infinite, see [R.B\'edard, Communications in Alg. 1986 and 1989]. The second main contribution of this paper is the definition of the asymptotic Hecke ring J of W. This is a Z-module with basis t_w indexed by w\in W in which the multiplication constants are obtained from those for the new basis [37] of the Hecke algebra by some strange way to make q tend to 0 (involving the a-function of [60]). It is not immediately clear that J is associative (it is so, due to [43]); this ring has a rather non-obvious unit element namely the sum of all t_d where d runs through D. (Here the finiteness of D is used). It is also shown that the Hecke algebra admits a natural algebra homomorphism \phi into the algebra J with scalars suitably extended (this is again based on [43]). (8/07/2011)

78. (with C.DeConcini and C.Procesi) HOMOLOGY OF THE ZERO SET OF A NILPOTENT VECTOR FIELD ON A FLAG MANIFOLD, 1988.

The work on this paper was done during my sabbatical leave in Rome (1985/86). Let g be the Lie algebra of a connected reductive group over C, let N be a nilpotent element of g and let B_N be the variety of Borel subalgebras of g that contain N. At the time this paper was written it was known that the rational homology of B_N is zero in odd degrees. (The most difficult case, that of type E_8, was done by [Beynon,Spaltenstein, J.Algebra 1984] based on computer calculation and then in my paper [69] without computer calculation.) In this paper we prove a stronger result namely that the integral homology of B_N is zero in odd degrees and has no torsion in even degrees. The key case is that where N is distinguished. There are separate proofs for the case of classical groups (where we show the existence of a cell decomposition) and in the exceptional case (where we are unable to prove the existence of a cell decomposition but instead we give an alternative argument based on blow ups and downs which in a sense gives a more precise result than for the classical groups).
It would be interesting to complete the results of this paper by 1) extending the method used for exceptional groups (connectedness of a certain graph) to classical groups and 2) showing that the cell decomposition also exists for exceptional groups.
In this paper we also show that the Chow group of B_N is the same as the integral homology. This has the consequence that the K-theory of coherent sheaves on B_N is computable, which is a necessary ingredient of [140,143] and also of [Bezrukavnikov,Mirkovic, arxiv:1001.2562].(7/31/2010)

79. QUANTUM DEFORMATIONS OF CERTAIN SIMPLE MODULES OVER ENVELOPING ALGEBRAS, 1988.

In 1986, A. Borel wrote to me a letter pointing out the interesting new work of Jimbo in which quantized enveloping algebras (q.e.a) were introduced. As a result of this letter I gave a course (1986/87) at MIT on q.e.a. and this paper came out of it. In this paper the divided powers E_i^{(n)}, F_i^{(n)} are introduced for the first time by replacing the denominator n! of the classical divided powers by a q-analogue of n! (depending on i). The choice of denominator was such that the formulas for the action of E_i^{(n)}, F_i^{(n)} on the standard simple modules of quantum sl_2 were as simple as possible and also the quantum Serre relations can be written in a form which is as simple as possible. Using these divided powers, in this paper I define a Q[q,q^{-1}]-form of the q.e.a. (In later papers [90], [91] this is refined to a Z[q,q^{-1}]-form which has become one of the ingredients in the definition of the canonical basis [92].) Using this I show that a simple integrable module of a Kac-Moody Lie algebra can be deformed to a module over the corresponding q.e.a. This paper also contains the first appearance of the braid group action on a q.e.a. at least in the simply laced case (but the proofs appeared only in [107]).(9/01/2010)

80. (with D.Kazhdan) FIXED POINT VARIETIES ON AFFINE FLAG MANIFOLDS. 1988

My motivation for this paper was as follows. Let G be a semisimple adjoint group over C with Lie algebra \fg. Since [40] I knew that (conjecturally) the nilpotent classes of \fg are in bijection with the two-sided cells of the affine Weyl group W_{af} of G^* (Langlands dual); moreover experiments showed that dim H^*(B_x)^{A(x)} (B_x="Springer fibre" at a nilpotent x, A(x)=group of components of the centralizer of x in G) is equal to the number of left cells in the corresponding two-sided cell. For example if G is of type E_8 and x is the sum of nonzero root vectors in \fg corresponding to all simple roots (a subregular element), then B_x has 8 irreducible components (all lines), H^*(B_x)^{A(x)}=H^*(B_x) is 9 dimensional and there are 9 left cells in the corresponding two-sided cell c. If we now take the affine analogue x' of a subregular nilpotent in \fg (the sum of nonzero root vectors in \fg((t)) corresponding to all affine simple roots) and if we replace B_x by the set B'_{x'} of Iwahori subalgebras of \fg((t)) that contain x' we see that B'_{x'} has exactly 9 irreducible components (all lines). So the number of left cells in c can now be interpreted not as a dimension of a vector space but as a number of elements in a set attached to x' (the set of irreducible components of B'_{x'}). This gave me some hope of finding an analogous relation in more general cases. Although this hope remained unfulfilled it motivated my interest in investigating sets of the form B'_{x'}.
In this paper it is shown that if N\in\fg((t)) is, like x' above, regular semisimple and topologically nilpotent (that is lim N^k=0 as k\to\infinity) then B'_N (defined as for x') is a finite or countable (but locally finite) union of projective algebraic varieties all of the same dimension; moreover if N is in addition elliptic then B'_N is itself an algebraic variety. In the paper a conjectural formula for \dim(B'_N) is given and it is shown how to reduce the proof of this formula to the case where N is elliptic. (The case where N is elliptic was settled in [R.Bezrukavnikov, Math.Res.Lett. 1996].) Also, it is shown that if x\in\fg is nilpotent then for an "open dense" subset S(x) of x+t\fg[[t]], all elements N\in S(x) are regular semisimple (and of course topologically nilpotent), \dim B'_N=\dim B_x and the conjugacy class in the Weyl group which parametrizes the Cartan subalgebra of \fg((t)) containing N depends only on x. (For example if x\in\fg is subregular nilpotent then x' as above belongs to S(x); note that x\to N is an affine analogue of the process of induction [35]. ) This gives a map \Psi from nilpotent orbits of \fg to the set of conjugacy classes in the Weyl group. In the paper this map is described explicitly in type A (where it is a bijection) and in the cases arising from a nilpotent element of \fg whose centralizer in G is connected, unipotent. For example if G has type E_8 and x is regular/subregular/subsubregular then \Psi(x) contains an element of order 30/24/20. The map \Psi was later computed for G of type B,C,D in [N.Spaltenstein, Ast\'erisque 168(1988)], [N.Spaltenstein, Arch.Math.(Basel) 1990], and in many cases in exceptional types in [N.Spaltenstein, Adv.Math. 1990]. But there are several cases in exceptional groups where \Psi(x) remains uncomputed. In [207] another map between the same two sets is defined using completely different considerations (based on [199] where a map in the opposite direction is defined using properties of Bruhat decomposition). The map in [207] is computable in all cases and I expect it to be the same as \Psi.
The varieties B'_N introduced in this paper play a key role in the work of Ngo B.C. on the fundamental lemma.
I would like to state the following problem. Let x\in\fg be a distinguished nilpotent element and let N\in S(x) (so that B'_N is a well defined algebraic variety containing B_x, see Cor.2 in Sec.3 and 9.2). Let X_N be the set of irreducible components of B'_N. Show that A(x) acts naturally on X_N, that \dim H^*(B_x)^{A(x)}=\card(X_N/A(x)) and that the set of left cells in the two-sided cell of W_{af} corresponding to x is in natural bijection with X_N/A(x).
For example if G is of type G_2 (resp. F_4) and x is subregular then A(x)=S_3 (resp. S_2) and \dim H^*(B_x)^{A(x)}=3 (resp. 5); on the other hand B'_N is a Dynkin curve of type affine E_6 (resp. affine E_7) which has a natural S_3 (resp. S_2) action whose fixed point set on the set of irreducible components has cardinal 3 (resp. 5). (10/23/2010)

81. CUSPIDAL LOCAL SYSTEMS AND GRADED HECKE ALGEBRAS, I, 1988.

The work on this paper was done in late 1987. In this paper I introduce a graded analogue of affine Hecke algebras (with possibly unequal parameters) associated to any root system. (After this paper was written I learned of the paper [Drinfeld, Funkt.Anal.Appl.1986] where a similar algebra was introduced for a root system of type A and with the grading being disregarded.)
Another new idea of this paper is to define equivariant homology. While in Borel's definition of equivariant cohomology with respect to an action of an algebraic group G, any classifying space of G can be used, the definition that I give for equivariant homology is more subtle: it exploits the fact that the classifying space of G can be approximated by smooth varieties. (The same idea appeared independently in the definition of equivariant derived category given in [Bernstein and Lunts, LNM, Springer Verlag 1994].)
This paper contains also a new application of the theory of character sheaves. Originally this theory was supposed to provide a machine to compute the values of the irreducible characters of a reductive group over a finite field. But in this paper character sheaves (cuspidal with unipotent support) are used to construct geometrically representations of a graded Hecke algebra (which ultimately leads to representations of a p-adic group [123],[155]). In fact in this paper I give a geometric realization of certain graded Hecke algebras in terms of equivariant homology of a space with group action and with a local system associated to a cuspidal local system with unipotent support.(10/03/2010)

84. MODULAR REPRESENTATIONS AND QUANTUM GROUPS, 1989.

The work on this paper was done in the spring of 1987 and the results were presented at a US-China conference at Tsinghua University, Beijing in the summer of 1987. This paper introduces a new concept: that of the quantum group U_\zeta (\zeta is a primitive m-th root of 1 in the complex numbers) obtained from the Q[q,q^{-1]}]-form of the quantum group introduced in [79] (which involves q-analogues of divided powers) by specializing q=\zeta. This paper also formulates the idea (new at the time) that, in the case where m is a prime number p, the representation theory of U_\zeta is governed by laws similar to those of the rational representation theory of a semisimple algebraic group G over a field of characteristic p. Most of the paper is concerned with providing evidence for this idea. For example I prove an analogue for U_\zeta of the Steinberg tensor product theorem [Steinberg, Nagoya J.Math. 1963]. Thus I show that a "Weyl module" of U_\zeta with highest weight \lambda=\lambda_0+p\lambda_1 (where \lambda_0 has coordinates strictly less than p) is the tensor product of the Weyl module of U_\zeta with heighest weight \lambda_0 with a U_\zeta-module which may be viewed as the simple U_1-module with highest weight \lambda_1. (Implicit in this statement is the existence of a "quantum Frobenius homomorphism" from U_\zeta to the classical enveloping algebra U_1 which is also one of the main new observations of this paper.) The key to this tensor product theorem is the following property of the Gaussian binomial coefficients specialized at \zeta: if N,R are integers and N=N_0+pN_1,R=R_0+pR_1 where N_i,R_i are integers, 0\le N_0\le p-1,0\le R_0\le p-1 then [N,R]=[N_0,R_0](N_1,R_1) for q=\zeta. Here [N,R],[N_0,R_0] are Gaussian binomial coefficients and (N_1,R_1) is an ordinary binomial coefficient. Similarly, the key to the classical Steinberg theorem is the following congruence (which I learned in my student days from Steenrod's book "Cohomology operations"): if N,R are integers and N=N_0+pN_1+p^2N_2+...,R=R_0+pR_1+p^2R_2+... where N_i,R_i are integers, 0\le N_i\le p-1,0\le R_i\le p-1 then (N,R)=(N_0,R_0)(N_1,R_1)(N_2,R_2)... mod p. In this paper I also formulate a conjecture describing the character of an irreducible finite dimensional U_\zeta-module in terms of the polynomials [37] attached to the affine Weyl group of the Langlands dual group, similar to the conjecture that I stated in [40], Problem IV. This conjecture (which is now known to hold) is one of the steps in the solution of Problem IV in [40].
After this paper appeared G.D.James formulated [G.D.James,1990] the idea (similar to that of this paper) that the representation theory of the Hecke algebra of type A_{n-1} at a p-th root of 1 is very closely related to the representation theory of the symmetric group S_n over a field of characteristic p. (9/26/2010)

86. CELLS IN AFFINE WEYL GROUPS IV, 1989.

The conjectural bijection in 10.8 is proved in [Bezrukavnikov, Represent.Th. 2003] with an important contribution by [Ostrik, Represent.Th. 2000]. A weaker form of Conjecture 10.5 is proved in [Bezrukavnikov, Ostrik, in Adv.Studies Pure Math.40 Mat.Soc.Japan 2004]. For type A, Conjecture 10.5 is proved in [Xi, Mem.Amer.Math.Soc. 157(2002)].

[88]. ON QUANTUM GROUPS, 1990.

This paper (written in early 1989) consists of two parts. In the first part it is shown that from a quantum group associated to a positive definite symmetric Cartan matrix one can recover in a natural way the Hecke algebra attached to the same Cartan matrix. Namely, an explicit construction of the q-analog of the adjoint representation is given (together with an explicit basis which can now be interpreted as the canonical basis [92] of that representation) and it is shown that the braid group acts naturally on this representation so that the induced action on the 0-weight space satisfies the relations of the Hecke algebra. In the second part two conjectures are formulated. Conjecture 2.3 predicts an equivalence of categories between a certain category C of representations of a quantum group at a root of 1 and a certain category C' of representations of an affine Lie algebra at a negative central charge related to the order of the root of 1. (This conjecture was later proved in [108,109,115,116].) Conjecture 2.5(b) (resp.2.5(c)) predicts a character formula for the simple objects in C' (resp. C) in terms of the polynomials [37] for an affine Weyl group analogous to a conjecture I made in [40] for modular representations of a semisimple group in characteristic p. Conjecture 2.5(b) has been already stated in [84] but the present paper suggests that one could prove it if one could prove Conjectures 2.3 and 2.5(c). Eventually that was indeed the way that Conjecture 2.5(b) was proved. (Conjecture 2.5(c) was proved by [Kashiwara and Tanisaki, Duke Math.J. 1995]).(2/26/2012)

[90]. FINITE DIMENSIONAL HOPF ALGEBRAS ARISING FROM QUANTIZED UNIVERSAL ENVELOPING ALGEBRAS, 1990.

This paper was written in the spring of 1989. Let A=Z[v,v^{-1}]. This paper introduces a new object: the A-form {}_AU (and {}_AU^+) of a quantized enveloping algebra U of simplylaced type (and its plus part U^+). While the definition does not need new ideas (compared to the definition of the Q[v,v^{-1}]-form of U or U^+, already introduced in [79]) the problem that arises is to show that one gets a well behaved object, for example that {}_AU^+ is a "lattice" in U^+. This property is established in the present paper by constructing an A-basis for {}_AU^+ which is also a basis of U^+. This basis is defined in terms of the braid group action introduced in [79]. The proof relies on explicit calculations involving (in particular) the roots of E_8. In this paper I also introduce for an integer N>0 a new Hopf algebra of finite dimension N^{number of roots} which can be viewed as the Hopf algebra kernel of the quantum Frobenius map of [84]. This Hopf algebra is sometimes referred to as the "small quantum group".
This paper is a step toward the construction of the canonical basis of U^+ which was achieved in [92]. Indeed the lattice {}_AU^+ is one of the key ingredients in the definition of the canonical basis of U^+ given in [92].
(10/24/2010)

89. GREEN FUNCTIONS AND CHARACTER SHEAVES, 1990

I got the main idea for this paper during a visit at the College de France (May 1988) where I gave a series of lectures on character sheaves. The paper was completed in the fall of 1988 when I was visiting IAS, Princeton. This paper is a step in the program (initiated in [64, p.226]) of relating (for a connected reductive group G defined over F_q of characteristic p), the characters of representations of G(F_q) and the characteristic functions of character sheaves on G which are "defined" over F_q. A part of this program would be to show that the Green functions of G(F_q) (defined in [22]) can be expressed in terms of character sheaves. In this paper I show that this is indeed so assuming that q is large (no restriction on p). The corresponding result for large p was known at the time (it could be deduced from the work of Springer and Kazhdan). The assumption that q is large enough was later removed by [Shoji, Adv.in Math.1995]. Moreover in this paper it is shown that the "generalized Green functions" associated to the "induction" functor R_{L,P}^G of [24] can be expressed in terms of character sheaves assuming that q is large enough and p is good. This was new even for large p. In fact the assumption that p is good can now be removed in view of the cleanness property [204]. The methods and results in this paper were used in [Shoji, Adv.in Math.1995 and 1996] to study my conjecture [64, p.226] on the relation of irreducible characters of G(F_q) and character sheaves. (7/12/2011)

[91]. QUANTUM GROUPS AT ROOTS OF 1, 1990.

In this paper the definition of the braid group action in [79], the results of [90] about the Z[v,v^{-1}]-form of U^+ ("lattice property") and the definition of the small quantum group in [90] are extended to the nonsimplylaced case. The case of G_2 was particularly complicated since (unlike the other rank two cases) there are no simple explicit formulas for the commutation of two divided powers of "root vectors" and for this reason the argument becomes involved. Also the quantum Frobenius homomorphism (which is almost explicit in [84]) is made explicit. In the Appendix (joint work with M.Dyer) the "Poincar\'e-Birkhoff-Witt basis" of U^+ corresponding to any reduced expression of the longest Weyl group element is introduced, using the braid group action and the computations in rank 2 from the main body of the paper. Note that the basis introduced in [89] is a special case of this PBW basis; the appendix allows one to simplify some arguments in [89]. Later, these PBW bases turned out to be another of the key ingredients in the definition of the canonical basis of U^+ (in the simplylaced case) given in [92].(9/25/2010)

[92]. CANONICAL BASES ARISING FROM QUANTIZED ENVELOPING ALGEBRAS, 1990.

The results of this paper were obtained while I was giving a course (MIT, fall 1989) on quantum groups and in particular on Ringel's work [Ringel, Hall algebras and quantum groups, Inv.Math.1990] and were presented in that course. This paper introduces a rather miraculous object: the canonical basis for U^+, the plus part of a quantized enveloping algebra U of type A,D,E. This is done by two methods (which lead to the same basis):
(1) an algebraic one based on the following three ingredients:
(i) an integer form of U^+ which I introduced earlier [79],[90],
(ii) a bar involution of U^+ and
(iii) a basis at infinity of U^+ coming from any PBW basis, see [91] (remarkably, the basis at infinity defined by a PBW basis is independent of the PBW basis);
(2) a topological method based on the local intersection cohomology of the orbit closures in the moduli space of representations of a quiver. Now even in the approach (1), there is a (minimal) use of the elementary representation theory of quivers (not intersection cohomology and not in the statements but in the proofs). Note that (1) (resp. (2)) bear some superficial similarity with things which appeared in the study of Hecke algebras [37] (resp. [39]); in that study the role of PBW bases is played by the (single) standard basis of the Hecke algebra. One of the remarkable properties of the canonical basis is that it induces a basis in each finite dimensional irreducible module of U. This paper introduces also a natural piecewise linear structure for the canonical basis that is, a finite collections of bijections of the canonical basis with N^n (N=natural numbers, n=number of positive roots) so that any two of these bijections differ by composition with a bijection of N^n with itself given by a composition of operations which involve only the sum or difference of two numbers or the minimum of two numbers. Later, I found that exactly the same pattern appears in a rather different context: the parametrization of the totally positive semigroup attached to a group of type A,D,E, see [119]. A similar pattern exists in the nonsimplylaced case, see [193].
Theorem 8.13 gives what is I believe the first purely combinatorial formula for the dimension of a finite dimensional irreducible representation (and its weight spaces) of a simple Lie algebra of type A,D,E (it expresses the dimension as the result of counting the number of elements of an explicit set defined using the piecewise linear structure above); the previously known dimension formulas gave the dimension as a ratio of two integers which is not obviously an integer (Weyl) or as a difference of two integers which is not obviously positive (Kostant). Subsequently another purely combinatorial formula was found by Littelman using his paths. The remarks on Fourier transform in Sec.13 are a precursor of [97].
Another proof of the existence of the canonical basis (valid also for Kac-Moody Lie algebras) was later given in [Kashiwara, Duke Math.J. 1991] by a purely algebraic method which again uses (i),(ii) above; and in [97] which generalizes the intersection cohomology approach (2).
The following question remains open: consider the entries of the transition matrix from the canonical basis of U^+ to the PBW basis of U^+ attached to a reduced expression of the longest Weyl group element. Do these entries have a geometric interpretation (in particular are they positive?). This paper answers this question only for certain special reduced expressions (adapted to an orientation of the Coxeter graph) when these entries are interpreted as local intersection cohomology of orbit closures.
For further history of this paper, see Nagoya Math.J.2006. (04/16/2013)

95. CANONICAL BASES ARISING FROM QUANTIZED ENVELOPING ALGEBRAS,II, 1990.

In May 1990 I attended a conference in Kyoto about common trends in mathematics and physics. At this conference Kashiwara announced his new proof of the existence of the canonical basis which I introduced in [92]. This paper was written after this conference. One of the main results is the introduction of a new family of "quiver varieties" attached to a graph. It is shown that these varieties are equidimensional. (Each one is in fact a Lagrangian variety in a symplectic vector space.) Moreover the union Z of the sets of irreducible components of these varieties is endowed with certain geometrically defined maps E_k:Z\to Z (see 8.8); k is a vertex of the graph. In this paper it is conjectured (10.2) that the crystal graph of the plus part of the quantized enveloping algebra corresponding to the graph can be geometrically realized as the set Z together with our maps E_k:Z\to Z (there are also maps F_k:Z\to Z but they are essentially inverse to E_k hence they need not be separately constructed). This conjecture was proved by [Kashiwara and Y.Saito, Duke Math.J. 1997]. (5/04/2011)

97. QUIVERS,PERVERSE SHEAVES AND QUANTIZED ENVELOPING ALGEBRAS, 1991,

Let U^+ be the plus part of the quantized enveloping algebra corresponding to a symmetric Cartan matrix C. After writing the paper [92] on the canonical basis of U^+ in the case where C is positive definite, I tried to consider the similar problem for a general C. The main problem was to find an appropriate definition for the class X of irreducible perverse sheaves on the space of representations of fixed dimension D of a quiver attached to C which should constitute the canonical basis. If C is positive definite X consists of all G-equivariant simple perverse sheaves (G=product of GL_n's); but in the indefinite case there are infinitely many G-equivariant simple perverse sheaves which is not what X should be. I first tried [95] to define X by imposing in addition to G-equivariance a condition on the characteristic variety namely that it should be contained in the explicit Lagrangian variety \Lambda defined in [95]. (This last condition is automatic in the positive definite case.) But I was not able to develop the theory from this definition. Instead I adopted a definition from the theory of character sheaves, namely X is defined as the collection of simple perverse sheaves which appear (up to shift) as direct summands of the direct image of the constant sheaf under the projection maps from certain spaces which consists of a representation of dimension D of the quiver and a "flag" of a fixed type compatible with the representation. This makes X finite for any prescribed D. With this definition the collection of the various X when D varies can be viewed as a basis of an algebra over Z[q,q^{-1}] in which multiplication is an analogue of induction of character sheaves (q appears as the shift). In this paper I prove that the resulting algebra is a Z[q,q^{-1}]-form of U^+ and that the basis of U^+ provided by the perverse sheaves does not depend on the orientation of the quiver; hence it is a canonical basis of U^+. I also show that this algebra has something close to a comultiplication (it is defined as an analogue of restriction of character sheaves). The structure constants of both the multiplication and "comultiplication" are in N[q,q^{-1}]. Another result of this paper is a new realization of the algebra U^+ (for v=1) in terms of convolution of certain constructible functions on the Lagrangian variety \Lambda (as above). This realization actually plays a role in the proofs in this paper. (7/11/2011)

98. (with J.M.Smelt) FIXED POINT VARIETIES IN THE SPACE OF LATTICES, 1991

Let V be a vector space of dimension n over C[[\epsilon]] with a basis e_1,...,e_n. Let I be the space of Iwahori subalgebras of SL(V) (an affine flag manifold). Let N be the linear map from V to V which sends e_i to e_{i+1} for i=1,...,n-1 and sends e_n to \epsilon e_1. Let t be a positive integer relatively prime to n. In this paper we study the space X_t={B\in I;N^t\in B} (by [80], X_t is a projective algebraic variety over C). It is shown that the Euler characteristic of X_t is \chi(X_t)=t^{n-1} and that X_t can be paved with affine spaces. After this paper appeared, I defined a generalization of N^t for any simple Lie algebra g over C; namely for an integer t\ge1 prime to the Coxeter number h we write t=ah+b,1\le b\le h-1, and let N_t=\epsilon^a\sum_{\alpha:root}c_\alpha e_\alpha where e_\alpha are the root vectors and c_\alpha=1 if the height of \alpha is b, c_\alpha=\epsilon if the height of \alpha is h-b, c_\alpha=0 if the height of \alpha is not b or h-b; then N_t is a topologically nilpotent regular semisimple elliptic element of Coxeter type. Let X_t be the variety of Iwahori subalgebras of g[[\epsilon]] that contain N_t (a projective variety); I conjectured that the Euler characteristic of X_t is \chi(X_t)=t^{rank(g)}, which in type A reduces to the formula in this paper. This conjecture was proved in [Fan, Transformation Groups, 1996]. The result on paving was generalized in [Goresky,Kottwitz,MacPherson, Represent.Th.,2006]. In this paper there is also an explicit formula for the Euler characteristic in the case where the space of Iwahori subalgebras is replaced by that of maximal parahoric algebras (type A); this was generalized to arbitrary g in [Sommers, Nilpotent orbits and ...(Ph.D.Thesis at MIT), 1997]. The formula for \chi(X_t) in this paper plays a role in [Berest,Etingof,Ginzburg, IMRN, 2003]. The variety X_t (type A) and its paving in this paper also plays a role in [Laumon, Fibres de Springer et jacobiennes compactifi\'ees, Springer 2006]. (8/09/2011)

100. A UNIPOTENT SUPPORT FOR IRREDUCIBLE REPRESENTATIONS, 1992

Let $G$ be a connected reductive group defined over a finite field F_q of sufficiently large characteristic. For any unipotent element u\in G(F_q) let \Gamma_u be the generalized Gelfand-Graev representation (GGGR) associated by Kawanaka to u; this is a representation of G(F_q) whose character is zero outside the unipotent set. Let \r be an irreducible complex representation of G(F_q); let \r' be the representation of G(F_q) which is dual to \r in the sense of [47]. In [57, 13.4] a unipotent conjugacy class C of G was attached to \r. In this paper the following properties of C are proved (see Theorem 11.2).
(i) The average value of the character of \r on C(F_q) is nonzero and C is characterized by having maximum dimension among unipotent classes with this property.
(ii) If g\in G is such that \tr(g,\r)\ne0 then the unipotent part of g lies in C or in a conjugacy class of dimension <\dim C.
(iii) For some u\in C(F_q), \r' appears with non-zero multiplicity in \Gamma_u; for any u\in C(F_q), \r' appears with small multiplicity in \Gamma_u; if C' is a unipotent class in G such that dim(C')>dim(C) or dim(C')=dim(C), C'\ne C, then \r' does not appear in \Gamma_u for u\in C'(F_q).
Note that something close to (i) has been conjectured in [40]; (ii) has been hinted at in [76,p.177,line 13]; (iii) has been conjectured by Kawanaka. It is natural to call C the unipotent support of \r. One of the keys to the proof of (i)-(iii) is Theorem 7.3 of this paper which gives an explicit decomposition of a GGGR in terms of intersection cohomology complexes of closures of unipotent classes with coefficients in various local systems. A step in the proof of this theorem is a formula for the Fourier transform of a GGGR viewed as a function on Lie(G(F_q)), involving a Slodowy slice. The connection between GGGR and Slodowy slices found in this paper is perhaps related to the observation made several years later by [Premet, Special transversal slices ..., Adv.in Math.2002] that a W-algebra (a characteristic zero analogue of the endomorphism algebra of a GGGR) is a quantized version of the coordinate ring of a Slodowy slice.
In this paper we also give a (provisional) definition (see Theorem 10.7) of the unipotent support of a character sheaf on G. The actual definition (partly conjectural) is given in [212]. (2/22/2012)

104. AFFINE QUIVERS AND CANONICAL BASES, 1992.

In this paper I fix an affine quiver of type A,D or E (but not A_{2n}) with one of the two orientations in which every vertex is a sink or a source. In this case I construct explicitly the perverse sheaves on the space of representations of fixed dimension of the quiver which comprise the canonical basis introduced in [97]. Unlike in the finite type case, these perverse sheaves can be higher dimensional local systems on an open subset of their support (the dimension is that of an irreducible representation of a symmetric group). Also, I describe explicitly (enumerate) the irreducible components of the Lagrangian variety \Lambda attached in [95] to the affine quiver and show that they are in natural bijection with the perverse sheaves in the canonical basis. In this paper, the affine quivers are studied in terms of a finite subgroup of SL_2(C) (MacKay correspondence) and I reprove from this point of view the classification of the indecomposable representations of this quiver, which goes back in various degrees of generality to Weierstrass/Kronecker (affine A_1), Gelfand-Ponomarev (affine D_4), Donovan- Freislich, Nazarova and [Dlab and Ringel, Memoirs AMS, 1976]. Another result of this paper is the construction of a new basis of the algebra U^+ (with v=1) attached to our quiver (later called the semicanonical basis [147]) in which the basis elements appear as constructible functions on the Lagrangian variety \Lambda. (7/11/2011)

110. COXETER GROUPS AND UNIPOTENT REPRESENTATIONS, 1993.

This paper contains things that I did in 1982. One of the results of the classification [57] of unipotent representations of a Chevalley group over F_q was that the set of unipotent representations depends only on the Weyl group W, not on the underlying root system or Chevalley group. Therefore one can asks whether the set of unipotent representations makes sense when W is replaced by a finite Coxeter group when the root system and the Chevalley group are not defined. This question is answered in this paper: the set of unipotent representations is attached to the finite Coxeter group by W heuristic considerations by postulating certain properties that this set should have which are known in the crystallographic case and showing that these postulates have a unique solution in the general case. The degrees of the unipotent representations are computed, the classification of representations in families, the classification of unipotent cuspidal representations are given in each noncrystallographic case. For example if W is of type H_4 there are 104 unipotent representations of which 50 are cuspidal; the largest family contains 74 representations of degree cq^6+higher powers of q where c is an algebraic integer (independent of q) divided by 120. The results of this paper have been a starting point for the investigations of Brou\'e, Malle, Michel on a (heuristic) theory of unipotent representations associated to a finite complex reflection group.(2/19/2012)

111. (with I.Grojnowski) A COMPARISON OF BASES OF QUANTIZED ENVELOPING ALGEBRAS, 1993.

At the end of 1991 there were two definitions of a canonical basis of the plus part U^+ of the quantized enveloping algebra of a Kac-Moody Lie algebra with symmetric Cartan matrix: the algebraic one in [Kashiwara, Duke Math.J. 1991] and a topological one in [97]. (But it was already known that, for finite types, both these definitions agree with the original definition [92], see [95],[97].) In this paper it is shown that these two bases agree in the general case. The new idea of this paper is a geometric interpretation of the symmetric bilinear form (,) on U^+. Namely for b,b' in the basis [97], it is shown that the rational function (b,b') expanded in a power series in v^{-1} has coefficients given by the dimensions of the equivariant Ext groups between the equivariant simple perverse sheaves which represent b,b'. (These Ext groups can be defined along the same lines as the equivariant homology spaces in [81].) In particular these coefficients are natural numbers.
The direct sum of the equivariant Ext groups above (for various degrees and various b,b') is naturally an algebra which, by [Varagnolo,Vasserot, arxiv:0901.3992], coincides with the KLR-algebra introduced combinatorially in [Khovanov,Lauda, arxiv:0803.4121] and [Rouquier, arxiv:0812.5023]. (11/07/2010)

112. TIGHT MONOMIALS IN QUANTIZED ENVELOPING ALGEBRAS, 1993.

In this paper I show that the construction [97] in terms of quivers of the canonical basis of the plus part U^+ of a quantized enveloping algebra can be generalized to the case where the quiver is allowed to have loops (this was not allowed in [97]). The resulting class of algebras includes the usual U^+ but also the classical Hall algebra with their canonical bases. Moreover, the plus part of a quantized (Borcherds) generalized Kac-Moody Lie algebra as described in [Kang,Schiffmann, Adv.Math.2006]) is in fact a subalgebra of one of our U^+, and the canonical basis described in [loc.cit.] is closely connected with the canonical basis of U^+ introduced in this paper, see [Kang,Schiffmann, arxiv:0711.1948].
In the U^+ of this paper there are elements F_i^{(a)} of the canonical basis indexed by a vertex i of the quiver and a natural number a. In the case without loops these elements are divided powers of a single element F_i but in the general case this is not so. It seems that the elements F_i^{(a)} generate the algebra U^+. This is known from [87] in the case without loops and is proved in the paper in the case where there is only one vertex and any number of loops.
Consider now a monomial m=F_{i_1}^{(a_1)}F_{i_2}^{(a_2)}...F_{i_n}^{(a_n)} in the F_i^{(a)}. We say that m is tight if it belongs to the canonical basis. In this paper I give a criterion to determine whether m is tight. The criterion is in terms of a certain positivity property of a quadratic form. Using this criterion I show that m is always tight if there is exactly one vertex and at least two loops. I also investigate the existence of tight monomials in the loop free case of small rank. It was already known from [92] that in type A_2 all elements of the canonical basis are tight monomials. In the paper I show that in type A_3 there is an abundance of tight monomials. In some sense (explained in the paper), 80% of the canonical basis consists of tight monomials; they fall into 8 families indexed by the various reduced expressions of the longest Weyl group element). Later, in [N.Xi, Comm.Alg.1999], the remaining elements of the canonical basis were described explicitly in this case; they are not tight monomials. The tight monomials in type A_4 are described in [Y.Hu,J.Ye,X.Yue, J.Alg.2003]. But in higher rank there are fewer and fewer tight monomials.(10/24/2010)

122. QUANTUM GROUPS AT v=INFINITY, 1995.

The main conjecture of this paper has now been proved for type A in: [K.McGerty, Int.Math.Res.Not. 2003] and in general in: [J.Beck and H.Nakajima, Duke Math.J. 2004].

126. BRAID GROUP ACTIONS AND CANONICAL BASES, 1996.

Let U be the quantized enveloping algebra corresponding to a given root datum. Let U^+ be the plus part of U. Let E_i be the standard generators of U^+. Let T_i be the symmetries of U defined in [107, Part VI] and let B be the canonical basis of U^+ defined in [107, 14.4]. In this paper I show that T_i respects B as much as possible. More precisely, we have U^+=(U^+\cap T_i^{-1}U^+)\oplus U^+E_i and I show that the associated projection U^+\mapsto(U^+\cap T_i^{-1}U^+) applies B to a basis of U^+\cap T_i^{-1}U^+ (union with 0). Similarly we have U^+=(T_iU^+\cap U^+)\oplus E_iU^+ and I show that the associated projection U^+\mapsto(T_iU^+\cap U^+) applies B to a basis of T_iU^+\cap U^+ (union with 0). I then show that these bases of U^+\cap T_i^{-1}U^+, T_iU^+\cap U^+ correspond to each other under T_i.
According to [Baumann, arxiv:1104.0907], an analogus result holds when the canonical basis B is replaced by the semicanonical basis [147] assuming that the root datum is simply laced and v=1).
The results of this paper have been used in [Beck,Chari,Pressley, Duke Math.J. 1999] to give a characterization of the canonical basis B of U^+ (in the affine case) in terms of a basis B' of U^+ of PBW type, constructed using (in part) iterations of symmetries T_i; the results of this paper are used to show that any element of B' is congruent to a unique element of B modulo v^{-1} times the Z[v^{-1}]lattice generated by B. (This extends the results of [92] in the finite type case.) (7/11/2011)

132. CELLS IN AFFINE WEYL GROUPS AND TENSOR CATEGORIES, 1997.

The main conjecture of this paper is proved in [Bezrukavnikov, Adv.Studies Pure Math.40, Mat.Soc.Japan 2004].

138. ON QUIVER VARIETIES, 1998.

Theorem 5.5 has been strengthened in [Malkin,Ostrik,Vybornov, Adv.in Math.2006] where it is shown that the morphism in that Theorem is in fact an isomorphism of algebraic varieties.(8/17/2010)

148. FERMIONIC FORM AND BETTI NUMBERS, 2000

This paper contains a conjecture which expresses the Betti numbers of the Nakajima quiver varieties in terms of a certain complicated but in principle computable fermionic form. This conjecture has now been proved in [Kodera,Naoi, arxiv:1103.4207]. (3/24/2011)

157. RATIONALITY PROPERTIES OF UNIPOTENT REPRESENTATIONS, 1982.

Let G be a split connected reductive group over F_q. For each w in the Weyl group W of G let R_w be the virtual representation of G(F_q) associated to w in [22]. Let r be a unipotent representation of G(F_q) that is, an irreducible representation appearing in R_w for some w. Let A(r) be the set of w in W such that r appears in R_w. Let A'(r) be the elements of minimal length of A(r). One of the main observations of this paper is that if r is cuspidal, A'(r) is contained in a single conjugacy class C(r) of W and that for w in C(r), the multiplicity of r in R_w is equal to (-1)^{semisimple rank of G}. From this it is deduced that a unipotent representations of G(F_q) whose character has values in rational numbers is actually defined over the rational numbers; in particular if G is of classical type any unipotent representation is defined over the rational numbers. (This is not true for unitary groups over F_q). The proof is not constructive since it uses the Hasse principle for division algebras. It is also observed that an analogue A\mapsto C(A) of the correspondence r\mapsto C(r) holds when r is replaced by a unipotent cuspidal character sheaf A. For example if G is of type E_8/F_4/G_2 and A is the unique unipotent cuspidal character sheaf with unipotent support (the closure of the conjugacy class \gamma of a unipotent element whose centralizer has group of components S_5/S_4/S_3) then C(A) contains an element which is "regular" of order 6/4/3 (=largest order of an element of S_5/S_4/S_3). In the paper it is noted that in these three cases C(A) consists of elements of a single length (40/12/4). It is interesting that C(A) also corresponds to \gamma under a quite different correspondence described in [197]. The rationality property of unipotent representations described in this paper was known to me (with a different proof, also explained in the paper) since 1982 when it was the object of a lecture that I gave at a US-France Conference on Representation Theory in Paris. The results of this paper were presented at a conference in Rome (June 2001) and one in Isle de Berder (Bretagne) in September 2001. (7/08/2011)

167. AN INDUCTION THEOREM FOR SPRINGER'S REPRESENTATIONS, 2004.

The theorem in the title was stated without proof in [48] for reductive groups in characteristic zero and it was one of the main tools in the computation in [48] of the Springer correspondence for groups of type E_n. This paper (written in 2001) contains a proof of that theorem, valid in arbitrary characteristic. It uses the connection between Green functions of a reductive group over a finite field and character sheaves [89] and also some arithmetic considerations.(2/24/2012)