Real Reductive Groups/atlas Seminar

Spring 2021: Meets on Tuesdays 4:00-5:00 beginning January 5


Meetings for the foreseeable future will be on Zoom. Zoom now requires a password; to get that, send me a note (dav at...) with a few words about your interests. If you don't have Zoom installed on your computer, I think that clicking this link will prompt you to do that installation. Once the installation is done, if you click on the link after 4 on Tuesday you'll be able to see blackboards/shared computer output, and to hear audio if you have a speaker. If you have a microphone and a camera, you are welcome to use those to ask questions.

Background: This is will be a working/learning seminar on (infinite-dimensional) representations of real reductive groups, aimed at grad students having some familiarity with representations of compact Lie groups. We'll use the atlas software; you should follow the directions on the web site to install it on your laptop.

If you're on the MIT math department network, atlas and the related program fokko are installed.

A good general introduction to what the seminar is about can be found at this page from a 2017 workshop. The mathematical subject matter is described in slides from Vogan's lecture. The main ideas about how to realize this mathematics on a computer are described in Adams's lecture. A quick introduction to the syntax for the software is in van Leeuwen's presentation.

First goal is to learn how the software represents real reductive groups (precisely, the group of real points of any complex connected reductive algebraic group) and their representations; making sense of the software will lead to an understanding of the underlying mathematics. Second goal is to use the software to investigate experimentally questions about reductive groups. (To indicate the range of possibilities: one recent project concerned enumerating the even-dimensional homogeneous projective varieties whose primitive cohomology in the middle dimension is zero.)

Plan is to meet weekly in the spring on Tuesdays 4:00-5:00 (but the timing may be negotiable). I hope very much to have more talks by participants. Some of the sessions should be entirely focused on the software: understanding how various scripts work; designing scripts to do new things; and sometimes just trying things.


  • July 27, 2021:
    • Speaker: David Vogan
    • Topic: Using D-modules to understand Springer, coherent continuation....
    • (will be) video from seminar

    • Goal is to outline a connection among Springer's construction of W reps, characteristic sycles of D-modules, and W-reps on W-cells.
    • This is in the OneNote page for July 27, 2021, page labelled "Big Project." Some of the mathematical background is on the first OneNote page for July 27, labelled "Background for Springer correspondence."



This web page was getting ridiculously large, so I've split it into several pieces...


Seminar Fall 2019: introduction to atlas

Seminar Spring-Summer 2020: bottom layer, how to find and classify unitary representations

Seminar Fall 2020: Weyl group representations and atlas

Seminar Spring 2021: Weyl group representations and cells



  • January 5, 2021:
    • Speaker: David Vogan
    • Topic: Lusztig's (O,x,\xi) parametrization of Weyl group representations
    • video from seminar

    • Introduced the important invariants b_sigma ("fake degree," the lowest degree of S(h) where sigma appears) and a_sigma ("formal degree," defined using the corresponding Hecke algebra representation) for a W rep sigma.

    • Stated 0 \le a_sigma \le b_sigma \le #(pos roots), and gave Lusztig's definition that sigma is special when a_sigma=b_sigma.

    • Recalled Springer's parametrization of W reps using nilpotent orbits, and started formulating Lusztig's parametrization of W reps using _special_ nilpotent orbits.
  • January 12, 2021:
    • Speaker: David Vogan
    • Topic: HC characters, Weyl group representations, and nilpotent orbits
    • video from seminar
    • Jeff Adams notes about three Weyl group representations closely related to the ones discussed today.
    • link to OneNote notebook with slides from talks 1/5 and 1/12/21.
    • (Unfortunately there are many handwritten notes expanding on the typed part of the notes. Apparently it never occurred to the geniuses at Microsoft that anyone might use both drawing and typing on the same document: the version that you can see on the web has the text and the handwritten notes placed independently, rendering the notes useless or worse. Sigh.)
  • January 19, 2021:
    • Speaker: David Vogan
    • Topic: Duality and the Langlands classification
    • video from seminar
    • link to OneNote notebook with slides from talk 1/19
    • The _second_ page for 1/19/21 is the handwritten notes about duality in the case of integral infinitesimal character that I actually did in the video. The _first_ page is typed notes about the possibly non-integral case. The first page is important, but the main ideas are mostly present on the second page.
  • January 26, 2021:
    • Speaker: David Vogan
    • Topic: Computing Lusztig's (O,x,triv) Weyl group representations using Jeff's sigma_L.at
    • video from seminar
    • Started with a review on OneNote of Lusztig's parametrization of W reps in families. Then did a session using Jeff's script sigma_L.at for computing sigma_L(special orbit,x,triv).
  • February 2, 2021:
    • Speaker: David Vogan
    • Topic: duality and character expansions at non-identity points
    • video from seminar
    • More about duality for cells. Reviewed relation of special W rep sigma_L(1,triv) in each cell to leading term of character expansion at 1.
    • Introduced big idea: occurrence of sigma(x,triv) in cell should be related to leading term of character expansion at x~ (some preimage of class x from A-bar in K(R)). Stated that script sigma_L.at is computing W rep attached to such a leading term.
  • February 9, 2021:
    • Speaker: David Vogan
    • Topic: more about character expansions at non-identity points
    • video from seminar
  • February 23, 2021:
    • Speaker: David Vogan
    • Topic: Calculating the representation I with basis the set of involutions for type C.
    • video from seminar
    • In fact I talked in general terms about why attaching W reps to G(R) reps is worthwhile. The promised calculation of I and the other nice W-reps for Sp(2n,R) is deferred another week.
  • March 2, 2021:
    • Speaker: David Vogan
    • Topic: Calculating the representation I with basis the set of involutions for type C.
    • video from seminar
    • There is a general theorem of Kottwitz calculating I, which for classical W says that I is the sum of all special representations, each with multiplicity equal to the order of Lusztig's associated finite group. I will try to give a fairly detailed proof of this statement in type C, with a sketch of how to extend it to calculate the representations X (basis KGB elements), Y (basis dual KGB elements), and B (basis the parameters in the block of the trivial).
    • Representations of W(C_n) are parametrized by pairs (pi,rho) of partitions of sizes adding to n. In the proof I'll give, the pairs (pi,pi) play a distinguished role. All of the corresponding W(C_n) representations are special in Lusztig's sense. They exist only when n=2m is even; in that case the sum of their dimensions is (2m)!/m!. It would be interesting to understand whether there is a natural "general" way (encompassing the exceptional groups) to view them as distinguished. (They make sense for types B and D as well; it is perhaps worth noting that the isomorphism B_2 \simeq C_2 respects this (unique) distinguished W-representation.)
    • I set up the argument but did not carry it out; so same topic again March 9!
  • March 9, 2021:
    • Speaker: David Vogan
    • Topic: Calculating the representation I with basis the set of involutions for type C.
    • video from seminar
    • More or less completed the outline of proving that for type C, the W rep I is a sum of only special pieces, each with multiplicity |A-bar|.
  • March 16, 2021:
    • Speaker: David Vogan
    • Topic: Calculating the representation B, X, and Y (with basis a block, or KGB(G), or KGB(G.dual)) for type C.
    • video from seminar
    • How to extend the calculation from last week.
  • March 30, 2021:
    • Speaker: Jeffrey Adams
    • Topic: How atlas computes the W reps I, X, Y, and B, CONTINUED.
    • video from seminar
  • April 6, 2021:
    • Speaker: Timothy Ngotiaoco
    • Topic: Galois cohomology, theta cohomology, and atlas.
    • video from seminar

    • Timothy encountered network glitches, and we did not see or hear any of the lecture that he delivered. While we waited in hope, Jeff Adams kindly delievered an introduction to the topic, starting with Cartan's description of real forms for reductive groups.

    • link to Jeff's OneNote page about the seminar.

    • Timothy's _actual_ lecture will take place April 13.
  • April 13, 2021:
    • Speaker: Timothy Ngotiaoco
    • Topic: Galois cohomology, theta cohomology, and atlas.
    • video from seminar
    • Timothy's notes from the seminar. (Zoom in a lot to read!)

    • Suppose P is a profinite group acting continuously on a set X. Timothy recalled the definition of Galois cohomology: H^0(P,X) is the set of P-invariants, and (if X is a group) then H^1(P,X) is 1-cocycles Z^1(P,X) modulo coboundaries.
    • He recalled that if C is an algebraic group defined over a finite field k, P = Z^ is the Galois group, and F in P is the Frobenius element, then H^1(P,C) is isomorphic to H^1(P,C/C_0). Deduced that if G is reductive over k, and T is a rational maximal torus in G, then the set of all G(k)-conjugacy classes of rational tori may be identified with F-twisted conjugacy classes in W.
    • Finally he began to look at P=Gal(C/R), showing that if H is a torus defined over R, so that H(R) is isomorphic to (R^x)^a x (C^x)^b x (S^1)^c, then H^1(P,H) is isomorphic to order 2 elements in (S^1)^c.
    • If T(R) is a connected maximal torus in a maximal compact K(R) of a real reductive G(R), so that H_f = G^{T(R)} is a fundamental Cartan, he defined W_f = [N_G(T)/H_f]^P, which amounts to the part of the Weyl group commuting with the distinguished involution. Next time he'll prove that H^1(P,G) is isomorphic to W_f orbits on H^1(P,H_f) (which recall is a small elementary abeliean 2-group).
  • April 20, 2021:
    • Speaker: Timothy Ngotiaoco
    • Topic: Galois cohomology, theta cohomology, and atlas.
    • video from seminar
    • Timothy's notes from the seminar. (Zoom in a lot to read!)

    • Timothy presented more examples of real Galois cohomology and classification of real forms of stuff.
  • April 27, 2021:
    • Speaker: Jeffrey Adams (UMD)
    • Topic: More about the W representations I, X, and B.
    • video from seminar

    • link to Jeff's OneNote page about the seminar.

    • Jeff Adams resumed his discussion of the W-representations I, X, Y, and B.
    • We promised a transcript of the atlas interaction at 28:00-1:14:00. A small part of it is here.
    • The seminar took place in the branch "jeff". As of May 10, I think that everything mentioned is on "master.".
    • Many of the commands require in addition to G the "CharacterTable" of G. This you can get by "set ct=G.character_table".

  • May 11, 2021:
    • Speaker: Jeffrey Adams
    • Topic: More about W and its representations.
    • Subtitle: How to distinguish φ'2,4 from φ''2,4.
    • I believe that almost all of Jeff's relevant commands now live on the master branch; so something like
      • git checkout master
      • git pull origin master
      • this step might not be needed make verbose=true optimize=true

      should leave you well placed to play along at home!
    • video from seminar

    • The session was largely an atlas demonstration of how to see and understand character tables for (especially classical) reductive groups.

    • An annotated transcript of most of Jeff's interaction with atlas is here

  • May 18, 2021:
    • Speaker: David Vogan
    • Topic: Unproven theorems to be found.
    • video from seminar

    • Link to Microsoft OneNote notebook with notes from the talk (section May 21, 2021, page "Understanding cells..."). Also included are notes about the precise definition of Lusztig's map from F(O) to pairs (x,xi) for type C_n (page "Calculating (x(sigma),xi(sigma))").
    • Notes also include a page "Conjectures" with hints about what ought to be atlas-testable ideas/conjectures. I'll say a bit more about some of those next week.
    • Here is some of the atlas interaction I did, computing cells as W reps.

    • Jeff showed us last week how to write down the character table of any Weyl group, and in particular to make a numbered list of all irreducibles of W.

    • First thing I'll show is how to make atlas compute the decomposition of any block into cells, and then of cells into irreducibles of W.

    • First question is which representations of W show up in blocks, and with which multiplicities?

    • I hoped that if G was split adjoint the answer was all of them, but that's not right.

    • Suppose C is a cell (set of parameters for G) of regular integral infinitesimal character, and C^\vee is the dual cell for G^\vee. Is it true that EITHER C contains a parameter cohomologically induced from a proper Levi, OR C^\vee does (or both)?

    • Suppose p is good range cohomologically induced from p_L on a theta-stable Levi L in G. Write C and C_L for the corresponding cells. As W representations, it should be true (but it isn't!) that
      C is isomorphic to j_{W_L}^W (C_L)
      (Lusztig's truncated induction). (I think this formula is precisely true when
      GK_dim(p) = GK_dim(p_L) + 1/2(dim g/l)
      but I have not written a proof. What's true in this formula in general is that the = has to be replaced by \le. Is there a substitute for this formula that _is_ true?

    • Comment: as Weyl group representations, C is isomorphic to C^\vee \otimes sgn. These problems are more or less outlining an inductive procedure for computing all cell representations of W.

  • May 25, 2021:
    • Speaker: David Vogan
    • Topic: More on W structure of cells .
    • video from seminar

    • Tried to formulate a problem: by experimentation, determine which subgroups S of A-bar(O) can arise as attached to a cell.

    • Some of the atlas interaction in the seminar.


  • June 8, 2021:
    • Speaker: David Vogan
    • Topic: Exactly what associated varieties look like for 2^{2m+1} orbit in Sp(4m+2,R)>.
    • video from seminar

    • atlas interaction from the seminar.

    • Most of the ideas are sketched in the usual Microsoft OneNote notebook from the seminar.

    • Second topic was calculating harmonic occurrences of W reps.


  • June 15, 2021:
    • Speaker: Roger Zierau
    • Topic: Computing all possible cells for all possible G locally a product of three Sp(4,R)'s
    • video from seminar


  • June 29, 2021:
    • Speaker: Jeffrey Adams
    • Topic: Computing A-bar(O) and Lusztig's W-reps sigma(x,triv) (x in A-bar)
    • video from seminar

    • Jeff introduced the new command show_lusztig_cell_no_dual, now playing on jeff, soon to appear in branches everywhere.


  • July 6, 2021:
    • Speaker: David Vogan
    • Topic: Intro to Springer correspondence
    • video from seminar

    • I talked in a little detail about how the Springer correspondence (between nilpotent orbits and W reps) is defined, since Lusztig's map builds on this; and any hope of justifying the proposed algorithm for computing Lusztig's map relies on understanding the relationship between the Springer correspondence for G and that for subgroups of G.
    • Microsoft OneNote pages from the seminar.


  • July 13, 2021:
    • Speaker: David Vogan
    • Topic: More about Springer correspondence
    • video from seminar

    • Detailed geometric description of how Lusztig-Spaltenstein induction relates to the Springer correspondence.
    • Suggested exercise: given n=p+q, there is a Levi subgroup L(p,q) = GL(p) x Sp(2q) of Sp(2n). Problem is to identify the nilpotent orbit Ind_{L(p,q)}^G (zero orbit) and the corresponding W representation (some pair of partitions).
    • Jeff's hints for making atlas do the exercise.


  • July 20, 2021:
    • Speaker: David Vogan
    • Topic: Lusztig-Spaltenstein induction in atlas
    • video from seminar

    • Talked again about the Lusztig-Spaltenstein result calculating the Springer correspondence for induced nilpotent orbits, and calculated it for the Levi L(p,q) = GL(p) x Sp(2q) of Sp(2n) Ind_{L(p,q)}^G (zero orbit).
    • Jeff's explanation of how to calculate this using atlas.