Real Reductive Groups/atlas Seminar

Records from Fall 2020



  • 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.