Real Reductive Groups/atlas Seminar

Records from Spring-Summer 2020



  • 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.