Real Reductive Groups/atlas Seminar

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

NO MEETING December 29, 2020


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.

There is no course credit (I'm not organized enough to have arranged this a year in advance) but I am happy to discuss reading course credit for active participation.


NEXT MEETING:
  • 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.



  • October 2, 2019:
    • David Vogan
    • The mysterious KGB: how atlas writes down a group plus a bit more.
    • Note that this meeting is Wednesday 4-5.
    • video from seminar. There were disasters. The content in the video is at 4:15-8:30 and 39:45-end; but even there the screen shown is not the one being described in the audio. I tried to edit out the empty time from 8:30-39:45 but so far failed; if anyone cares to do that, it would be gratefully received!
  • Oct 28, 2019:
    • Timothy Ngotiaoco
    • Principal series representations and parameters on the split torus
    • video from seminar
  • Nov 18, 2019:
    • David Vogan
    • Parameters for general groups continued: what needs to be entered and what it means
    • video from seminar
    • Regrettably I lost the file of atlas interaction from this session; my computer battery died before I had saved the terminal window.
  • Nov 25, 2019:
    • Lucas Mason-Brown
    • Constructing standard modules by (mostly) real parabolic induction
    • video from seminar
  • Dec 2, 2019:
    • Timothy Ngotiaoco
    • How to compute branching: restriction of (g,K)-modules to K
    • Unforgiveably, I hit "pause" in the recording and never restarted. I hope to post some atlas interactions related to what Timothy presented.
  • Dec 9, 2019:
    • Timothy Ngotiaoco
    • Writing irreducibles in terms of standards: KLV polynomials
    • video from seminar
  • End of seminar for the fall semester.



    • Feb 11, 2020:
      • David Vogan
      • Bottom layer arguments
      • video from seminar
      • Lots of silly mistakes arising from mislabeling of the picture from last week; I'll try to fix the labeling and add some pictures of K-types for standards.
      • Here is a picture of the K-types of a standard representation of G=Sp(4,R) attached to the KGB element 5, along with the signature of the invariant form; and the corresponding information for a principal series for L=Sp(2,R), to see the matching on the bottom layer.
      • Certainly it would be cool to have a corresponding picture for x=4, for example for r=parameter(KGB(G,4),[2,1],[3/2,-3/2]).
      • If you feel ambitious, the Tex files Sp4LKT.tex and Sp4x5.tex are in the directory

        http://www-math.mit.edu/~dav/atlassem/

        (You may want to tell me if you undertake such a project, so that I can tell other people not to do the same thing!)
      • In any case, I STRONGLY recommend that you play with the commands

        set q=parameter(KGB(G,5),[1,2],[0,3])
        print_branch_std_long(q,KGB(G,2),20)
        print_sig_irr_long(q,KGB(G,2),20)

        that were used to make Sp4x5.tex; see what changes when you change lambda and nu!
    • Feb 25, 2020:
    • Mar 24, 2020:
      • Timothy Ngotiaoco
      • Cohomological induction, unitarity, and bottom layers
      • video from seminar. The recording failed to capture Timothy's camera, so the video is kind of useless until 43:00. Timothy's audio explains nicely his notes that we could see during the talk, but those are not visible on the recording. Timothy has provided pictures of his notes, posted below.
      • First page of notes from the seminar.
      • Second page of notes from the seminar.
      • Third page of notes from the seminar.
      • Fourth page of notes from the seminar.
      • some atlas code related to the Stein complementary series Timothy talked about at the end.
    • Mar 31, 2020:
      • David Vogan
      • How Marc van Leeuwen has made atlas much faster at computing unitary representations
      • video from seminar
      • some atlas code run during the seminar: concerns how to time an atlas session, and the (very crude) structure of the "is_unitary" algorithm.
      • "Blackboard notes" from the seminar.
    • May 19, 2020:
      • David Vogan
      • How to enumerate the unitary representations from last week's conjecture.
      • video from seminar
      • This turned out to be all about a question from Roger about how to make the unitarity conjecture at integral infinitesimal character into a finite question. This can be done crudely with the Dirac inequality, but a nicer conjectural answer is formulared in the notes. Statement is that if the infinitesimal character lambda is strictly larger than 1 on a simple coroot alpha-vee, then the unitary reps of infinitesimal character lambda are each cohomologically induced in the good range from unitary on a unique maximal theta-stable parabolic attached to that alpha. We verified this for Sp(8,R) and a couple of lambda (4,2,2,0) and (4,2,1,1).
      • atlas code from seminar, and extensions
    • June 2, 2020:
      • Timothy Ngotiaoco
      • Salamanca/Vogan paper "Classification of unitary representations..." continued. Description of the set of lowest K-types for which the conjecture in the paper gives no reduction of unitarity. Possibly more atlas demonstration of these ideas.
      • Timothy's notes from his talk.
      • David Vogan began to talk about a way of describing K-types for classical groups that interpolates between usual highest weights and atlas. This description is meant both to help with using the software, and to support the proofs of theorems. (After all, the atlas point of view is built for studying infinite-dimensional representations.)
      • atlas code run during the seminar.
      • OneNote notebook used to begin describing K-types for classical G.
      • video from seminar
    • June 9, 2020:
      • David Vogan
      • K-types for classical groups.
      • Continuation of the discussion growing from Timothy's talks about how to organize representations of K in a way close to unitarity.
      • atlas code run during the seminar.
      • video from seminar
    • June 16, 2020:
      • David Vogan
      • K-types and unitary duals for classical groups.
      • Hope to describe in some detail the unitary duals of U(1,1) and U(2,1) using the "K-data" defined last week, as a model of what might work for more G.
      • video from seminar
    • June 23, 2020:
      • David Vogan
      • Finding nonunitarity certificates for spherical reps of U(2,2).
      • Since the general scheme requires knowing nonunitarity certificates for "small" L\cap K-types for all Levis of theta-stable parabolics, I decided to pause to look at how one might find such certificates. Had atlas run through lots of spherical representations of U(2,2), for each non-unitary one noting the lowest height K-types contributing to the negative signature. The code was very crude, really just allowing human inspection to do the work; but it would be easy and useful to write a script that automates this entirely: instead of calling "is_unitary(p)" and "print_sig_irr_long," one could call "hermitian_form_irreducible(p)," pick out the first term with non-integer coefficient, and add that to a list of proposed nonunitarity certificates. (There are similar things done in the script "hermitian.at") The tricky part of the script would be to run over an appropriately large collection of nu's; but what I did in the OneNotes pages for U(2,2) would be easy to generalize to spherical reps of U(p,q), and it would give at least a good start toward finding certificates. I highly recommend this exercise!
      • video from seminar
    • June 30, 2020:
      • David Vogan
      • K-types and unitary duals for classical groups (continued).
      • Hope to describe in some detail the unitary duals of U(n,1) using the "K-data" defined June 9, as a model of what might work for more G. For U(p,q), the approximate picture one might hope for is that the unitary dual of U(p,q) can be written as a finite union of pieces. Each piece should be identified by cohomological induction with the unitary representations of some product of U(p_j,q_j) that are (nearly?) "spherical": containing a K-type which on U(p_j,q_j) is just det^{m_j}: highest weight (m_j,...,m_j)(m_j,...,m_j).
      • video from seminar
    • July 7, 2020:
      • David Vogan
      • K-types and unitary duals for classical groups (fizzling out).
      • Looked again at the "K-data" defined June 9, to see how that shows which bottom layer arguments can succeed. But I decided to abandon even the small goal of proving the results of Baldoni, Knapp, and Speh about U(n,2). (They proved many things about unitary representations of U(n,2), but, contrary to what I said in the talk, I'm not sure they reached a complete classification.)
      • video from seminar
    • July 14, 2020:
      • David Vogan
      • is_unitary: what makes the software large and slow, and how to make it smaller and faster
      • The atlas software can do more or less any character-theoretic calculation for representations of exceptional groups, with the exception of complex E8 (for which results of Lusztig and others tell us a great deal). But the "is_unitary" operation for complicated representations even of E7 is recursive, combining results from hundreds of millions of character calculations. The present software cannot complete these calculations even with a terabyte of RAM.
      • We have made some changes recently that shrink the memory requirements of these calculations by large factors (a factor of ten for E6). The result is to extend significantly what can be done on desktop machines. It has already allowed us to complete some is_unitary calculations for E8 using a few hundred gigabytes of RAM (where a terabyte did not suffice before). I'll explain the (extremely simple) idea, and run some examples.
      • video from seminar
    • July 21, 2020:
      • David Vogan
      • is_unitary: what makes the software large and slow, and how to make it smaller and faster (continued)
      • Last week I outlined the structure of the is_unitary algorithm, and sketched the nature of the large number of large (signature) formulas needed to carry it out in a group of rank seven or eight.
      • This time I defined precisely the "alcoves" on which the signature formula s are guaranteed to be constant. More or less these are alcoves for the affine Weyl group, which are analogues of Weyl chambers and well understood. But as is often the case, the real group brings some order 2 dirt into the game, so that interesting mathematical questions remain. I didn't really make those questions explicit; I'll do that July 28
      • The atlas software modified to store only one deformation formula per alcove is available by some sequence of commands like

        git fetch origin
        git checkout davidfast
        make veryclean
        make optimize=true

        This will make the version of atlas that I used during the session except that it's faster than what I used during the session. Here's my laptop with master:
        atlas> is_unitary(F4_s.trivial)
        #def_forms = 100 max res size = 19MB CPU time = 1 secs
        ...
        #def_forms = 15000 max res size = 164MB CPU time = 15 secs
        Value: true
        atlas> quit
        Bye.
        17.791u 0.168s 1:16.67 23.4%

        So 18 seconds, 164 megabytes, something more than 15,000 formulas stored. Same calculation in davidfast:
        atlas> is_unitary(F4_s.trivial)
        #alcv_forms = 101 max res size = 19MB CPU time = 1 secs
        ...
        #alcv_forms = 5000 max res size = 110MB CPU time = 10 secs
        Value: true
        atlas> quit
        Bye.
        11.930u 0.083s 0:29.95 40.1%

        Now it's 12 seconds, 110 megabytes, storing 5000 formulas.
        For is_unitary(E6_q.trivial) the differences are more dramatic:
        In master:
        atlas> is_unitary(E6_q.trivial)
        #def_forms = 100 max res size = 23MB CPU time = 1 secs
        ...
        #def_forms = 700000 max res size = 7244MB CPU time = 17 mins
        Value: true
        atlas> quit
        Bye.
        1177.941u 9.530s 33:43.20

        In davidfast:
        atlas> is_unitary(E6_q.trivial)
        #alcv_forms = 100 max res size = 24MB CPU time = 1 secs
        ...
        #alcv_forms = 90000 max res size = 2747MB CPU time = 11 mins
        Value: true
        atlas> quit
        Bye.
        708.535u 2.654s 2:41:22.44

        2.7 gigs of memory in 11 minutes, storing 90,000 formulas. So a bit more than half the time and a third of the memory use.
      • I am sorry to say that I accidentally failed to record this sesssion, so the only record is the written notes in same OneNote notebook as before.
    • July 28, 2020:
      • David Vogan
      • Mathematical questions about alcoves
      • Formulated some more or less simple mathematical questions about alcoves as they appear in the unitarity calculation; and ran through the steps needed to get the "smaller faster" version of the atlas software that they allow.
      • video from seminar
    • August 25, 2020:
      • Speaker: David Vogan
      • Getting atlas to compute the Lusztig-Bezrukavnikov bijection
      • video from seminar
      • atlas script to be used in the seminar.
      • Lusztig paper "Cells in affine Weyl groups IV," source of his conjecture.
      • Suppose G is complex connected reductive, with B and H as usual. Write X*(H)^+ for the dominant weights for H; of course these index also irr algebraic representations of G, or irreducible representations of a maximal compact K of G. They are also in bijection (by taking lowest K-type) with the irreducible tempered representations of G as a real group, of real infinitesimal character: atlas parameters with nu=0.

        Lusztig conjectured in the early 90s that the set X^*(H)^+ was in one-to-one correspondence with G-conjugacy classes of pairs (\xi, \tau), with \xi\in g^* a nilpotent linear functional, and \xi an irreducible algebraic representation of the isotropy group G^\xi. This conjecture was proven by Bezrukavnikov and Ostrik in the 2000s, but their proof seems not to be constructive. The bijection was computed explicitly for GL(n,C) by Pramod Achar in his MIT thesis. For other G there are are only partial results.

        I'll talk about Achar's concrete description of the LB bijection, and demonstrate how atlas can help in computing it.
      • Promised link to information about Beijing conference on associated varieties, unipotent representations, and Dirac cohomology.
    • September 1, 2020:
    • September 8, 2020:
    • September 15, 2020:
      • Speaker: David Vogan
      • Topic: Understanding signatures as coherent sheaves II
      • video from seminar
      • Jeff's explanation of how to make atlas compute K-multiplicities in the ring of regular functions on the normalization of a K-orbit closure.
    • September 22, 2020:
      • Speaker: David Vogan
      • Topic: Dirac inequality and computing the unitary dual
      • video from seminar
      • I'll review how to use the Casimir operator to make computing the unitary dual a finite problem; how Parthasarathy's Dirac inequality improves things (making a smaller finite problem); and Chaoping Dong's idea for getting an improvement on the old Helgason-Johnson "nu is in the convex hull of rho" bound. (This seems likely to spill into next week; we'll see!)
    • September 29, 2020:
      • Speaker: David Vogan
      • Topic, 2020: Finding the sharpest Dirac inequality
      • video from seminar
      • Here is a short atlas file that computes the Dirac inequality bound on the infinitesimal character for a K-type. The file is still very incomplete, but I'll try to use it to look at Chaoping Dong's conjecture about bounding the continuous parameter for nontrivial unitary representations.
    • October 6, 2020:
      • Speaker: David Vogan
      • Topic: Weyl group representations and nilpotent orbits
      • video from seminar
    • October 27, 2020:
      • Speaker: David Vogan
      • Topic: Examples of associated varieties.
      • video from seminar
      • Talked about examples of the last theorem Timothy explained, relating the K-types of a Harish-Chandra module to its associated cycle. Concentrated almost entirely on the case of Sp(8,R) and the complex nilpotent corresponding to the partion 2+2+2+2. Showed how to use atlas to make a list of modules with that associated variety of annihilator, and how to compute the actual associated variety.

      • atlas interaction from seminar
    • November 3, 2020:
      • Speaker: David Vogan
      • Topic: More examples of associated varieties, and some theory
      • video from seminar
      • More about the algebraic geometry and combinatorics of this situation (there is a natural graph with vertices the finite set of K-orbits in O \cap (g/k)^*), and state a bunch of open problems. Atlas examples with _reducible_ associated varieties.
    • November 10, 2020:
      • Speaker: David Vogan
      • Topic: How to count representations with given associated variety of annihilator
      • video from seminar
      • Given a complex nilpotent orbit O for G and an infinitesimal character gamma, I have explained in the last two weeks how to make atlas LIST the parameters with infinitesimal character gamma and associated variety of annihilator equal to O-bar. This calculation is tractable but slow (matter of many minutes?) in rank four.
      • This time I will try to explain (work of Barbasch-V from early 1980s) how to determine the NUMBER of parameters that will be in the LIST, without actually calculating the list. This is a structure theory/Weyl group calculation, and ought to be accessible to atlas in higher rank (but I haven't yet tried this!)
      • Ultimate goal (research problem that I don't know how to solve) is to find a clear and precise relationship between real forms of nilpotent orbits in G and ^\vee G, and cells of representations.
      • So what I ACTUALLY did was describe some results about induction in classical Weyl groups, which McGovern used in the 1990s to calculate cells as W representations for classical groups.
    • November 17, 2020:
      • Speaker: David Vogan
      • Topic: Lusztig's families and cells in Sp(p,q)
      • video from seminar
      • Last week I outlined (sloppily and a bit incorrectly) how to calculate the coherent continuation representation of W in the case of Sp(p,q).
      • This week I will translate that answer using Lusztig's notion of families and special representations into a description of cells for Sp(p,q), and at the same time of the real forms of nilpotent orbits for this group.
      • Ultimate goal (research problem that I don't know how to solve) is to find a clear and precise relationship between real forms of nilpotent orbits in G and ^\vee G, and cells of representations.
    • November 24, 2020:
      • Speaker: David Vogan
      • Topic: Coherent continuation computation in atlas
      • video from seminar
      • Focused on details of how the Weyl group action on atlas parameters is computed, printed, and understood.
      • Jeff's wonderful explanation of how to see this in atlas (near the beginning of video) appears as the second page of the OneNote notebook for November 24, 2020.
    • December 1, 2020:
    • December 8, 2020:
      • Speaker: Jeffrey Adams
      • Topic: Computing Harish-Chandra's character formulas
      • video from seminar
      • Jeff's notes for his lecture; go to the "Computing global characters" page.
    • December 15, 2020:
      • Speaker: Jeffrey Adams
      • Topic: Computing Harish-Chandra's character formulas (continued).
      • video from seminar
    • December 22, 2020:
      • Speaker: Jeffrey Adam
      • Topic: Computing Harish-Chandra's character formulas: worked examples in atlas./li>
      • Jeff's notes for his lecture; go to the "Computing global characters" page.
    • January 5, 2021:
      • Speaker: David Vogan
      • Topic: Lusztig's (O,x,\xi) parametrization of Weyl group representations
      • (will be) 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.