Fall 2021: Meets on Thursdays 10:30-12:00 beginning October 7
Meetings for the foreseeable future will be
on Zoom. If
you click after 10:30 a.m. (Eastern time) Thursday you'll be able to
see notes/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.
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.
Last Tuesday 4:00 meeting is 10/5/21. Beginning 10/7/21, will meet Thursdays 10:30-12:00.
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
(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.)
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
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
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
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.
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.
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.
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.
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.
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.
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).
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).
Goal is to understand central characters of Arthur's unipotent reps, with the goal of understanding which unipotent reps of Spin groups fail to factor to SO. Today was introduction to central characters, how to compute them "by hand" in atlas, and how to compute unipotent reps in atlas.
This is in the OneNote page for July 27, 2021, pages labelled "Blocks and central characters" and "Central characters."
August 24, 2021:
Speaker: David Vogan
Topic: Computing central characters of (unipotent) representations B
Recalled duality for blocks, and how it's related to central characters; use that to prove that unipotent reps attached to distinguished nilpotents in ^\vee Spin(n) must factor to SO..
This is also be in the OneNote page for July 27, 2021, pages labelled "central character," "intermission," (which talks more explicitly about duality between U(p,q) (p+q=3) and GL(3,R)) and "duality."
Annegret Paul and I recently used atlas to verify Theorem. If a special nilpotent orbit O meets a Levi subgroup L in an L-orbit OL, then OL must be special in L.
Discussed what this has to do with computing the Springer W rep attached to O, and stated a conjectural characterization of special in this setting.
All this is on the OneNote section labelled September 7, 2021.
Goal is to formulate precisely (and prove) the algorithm for going from a G nilpotent to the special W rep in its Lusztig family. Hiding in here is a criterion for the orbit to be special..
Last week I talked about how to go from a nilpotent orbit to the associated special orbit. This week I'll use similar ideas to go from an arbitrary irreducible W rep to the unique special in its (Lusztig) family.
Brief introduction to the new script to_ht.at, which is meant to be an outline of how to use a desired new library function full_deform_to_ht(Param,int),
November 4, 2021, 10:30-12:00:
Speaker: David Vogan
Topic: W reps, cells, and local structure of characters I
David finally agreed to stop talking about this, after a discussion of its relation to the script lusztig_cells.at and of what mathematical results might be needed to complete these ideas.
November 25, 2021, 10:30-12:00:
Speaker:
Topic: NO MEETING (Thanksgiving)
December 2, 2021, 10:30-12:00:
Speaker: David Vogan
Topic: Counting real forms of nilpotents using W reps