Representation theory seminar 2022 Semester 2

Organisers: Dougal Davis, Kari Vilonen, Ting Xue

Topics: Real groups, affine Springer fibers.

Time: Thursdays 3:15pm-5pm.

Location: Peter Hall 107

Please contact one of the organisers to be added to the mailing list.

Schedule

Dec 2: Joint Number Theory/Representation Theory seminar, 2 – 3pm, Peter Hall 107.

Simon Marshall (Wisconsin-Madison) Large values of eigenfunctions on hyperbolic manifolds

It is a folklore conjecture that the sup norm of a Laplace eigenfunction on a compact hyperbolic surface grows more slowly than any positive power of the eigenvalue. In dimensions three and higher, this was shown to be false by Iwaniec-Sarnak and Donnelly. I will present joint work with Farrell Brumley that strengthens these results, and extends them to locally symmetric spaces associated to SO(p,q).

Nov 17 Gufang Zhao (Melbourne) Rotations on loop spaces, continued

Nov 10 Arun Ram (Melbourne) Introduction to Hessenberg varieties

It useful, and sensible, to view Hessenberg varieties as ‘the baby case’
of affine Springer fibres. In this talk I will review results of Martha Precup
and Eric Sommers which analyse the affine paving and the equivariant cohomology
of Hessenberg varieties. The primary references are arxiv:1205.3976 and
arxiv:2201.arxiv:2201.13346

Notes

Nov 3 Gufang Zhao (Melbourne) Rotations on loop spaces

This talk follows the paper https://arxiv.org/abs/1002.3636 of Nadler and Ben-Zvi, and tries to explain loop spaces in derived stacks, and rotational symmetries on the loop spaces. The goal is to express the loop spaces and rotation in terms of de Rham complexes of the space (and equivalently D-modules). This talk is example focused and does not attempt to discuss everything in loc. cit..

Oct 27 Davood Nejaty (Melbourne) Moduli scheme of Higgs triples

Hitchin studied the solutions of self-dual Yang-Mills equations over an algebraic curve which yields an integrable system. Then Hausel-Thaddeus conjectured the topological mirror symmetry of $SL_r$ and $PGL_r$ Hitchin spaces by providing evidence such as equality of their Hodge-Deligne polynomials for $r=2,3$ . It was proved for arbitrary $r$ and genus $g>1$ curve by Maulik-Shen. Now we want to have some observations for the moduli space of stable Higgs triples with $L_1,L_2$ -valued fields when $L_1\otimes L_2$ is the canonical bundle of the curve.

Oct 20 Dougal Davis (Melbourne) Adams-Barbasch-Vogan explicitly (continued)

Oct 13 Qixian Zhao (Utah) Around Adams-Barbasch-Vogan (continued)

Dougal Davis (Melbourne) Adams-Barbasch-Vogan explicitly

I will make another attempt at processing the local Langlands correspondence of Adams-Barabsch-Vogan, with the aim of making the bijection between the two sides explicit. I will explain how to parametrise both sides completely combinatorially in terms of the root datum, and write down my best guess at the bijection using these parametrisations. I will revisit the example of $SL_2$ and $PGL_2$ from this perspective and, time permitting, explain what one needs to do to check that my bijection is the correct one.

Notes Blackboard

Oct 6 Yaping Yang (Melbourne) Affine Springer fibers and Hitchin fibers, continued

Notes for all 3 talks

Sep 29 Qixian Zhao (Utah) Around Adams-Barbasch-Vogan

I will first explain what strong real forms and extended groups are. Then, I will explain the statement of local Langlands correspondence over $\mathbb{R}$ , how it interacts with the localization picture, and how this interaction leads to Vogan duality. I will use the $SL_2$ example to demonstrate. Definitions and statements will (hopefully) be given precisely, but there will be no proofs.

Notes

Sep 22 No seminar

Sep 15 Davood Nejaty and Linfeng Wei (Melbourne) Recap of real reductive groups and Beilinson-Bernstein localisation

Sep 8 Yaping Yang (Melbourne) Affine Springer fibers and Hitchin fibers, continued

Sep 1 Yaping Yang (Melbourne) Affine Springer fibers and Hitchin fibers,

In my talk, I will discuss a relation between the homogeneous affine Springer fibers and Hitchin fibers over the weighted projective line. I will mainly follow Section 6.6 of the paper “Geometric representations of graded and rational Cherednik algebras” by Oblomkov and Yun.

Masoud’s notes on affine Springer fibres

Aug 25 Dougal Davis (Melbourne) The local Langlands correspondence for real groups and other stories

In this talk, I will give a rough introduction to the main themes of the real groups strand of our learning seminar, with the intention that the details will be fleshed out in future talks. I will sketch the Adams-Barbasch-Vogan approach to the local Langlands correspondence for real groups, which gives a duality between Grothendieck groups of representations of a real group on one side and of perverse sheaves on a stack of Langlands parameters on the dual side. I will also explain how this category of perverse sheaves is related to coherent sheaves on a different stack of Langlands parameters by the $S^1$ -equivariance story of Ben-Zvi and Nadler, resolving the apparent clash with Matt Emerton’s talk two weeks ago. If time permits, I may also say a few words about Arthur representations and conjectures concerning them.

Aug 18 Volker Heiermann (Université d’Aix-Marseille) Affine Hecke Algebras, Intertwining Operators and Types

To put into relation a Bernstein component of the category of complex smooth representations of a p-adic reductive group and representations of Affine Hecke Algebras, one can in principle use two methods: the theory of types of Bushnell-Kutzko or the intertwining algebra of a certain projective generator of this category. From the first method, it is rather clear that unitarity is preserved, which is less immediate for the second method. The aim of the talk is to explain the situation, as time permits.

Aug 11 Matthew Emerton (Chicago) Categorical perspectives on the arithmetic Langlands program

A categorical perspective, which has long been the norm in the geometric Langlands program, has recently emerged in the arithmetic Langlands program as well. I will explain some of the ideas related to this perspective, including the Fargues–Scholze conjecture; conjectures and results (due variously to Ben-Zvi–Chen–Helm–Nadler, Hellmann, Zhu, as well as the speaker and Toby Gee) regarding fully faithful functors from categories of representations of p-adic reductive groups to categories of coherent sheaves on stacks of Langlands parameters; and conjectural descriptions of the cohomology of locally symmetric congruence quotients (e.g. Shimura varieties) in terms of these categorical ideas.

Aug 4 joint Representation Theory/Number theory seminar

Chen Wan (Rutgers) A multiplicity formula of K-types

In this talk, by using the trace formula method, I will prove a multiplicity formula of K-types for all representations of real reductive groups in terms of the Harish-Chandra character.

Jul 28 David Ridout (Melbourne) Inverse quantum hamiltonian reduction

Let g be a (complex, finite-dimensional) simple Lie algebra. Quantum hamiltonian reduction is a cohomological functor from category O for an affine vertex algebra $V_k(g)$ to the corresponding category for a W-algebra $W_k(g;f)$ . This functor depends on the orbit of a nilpotent f in g, as does the W-algebra. Unfortunately, it is only well understood for regular and minimal orbits.

A new approach to W-algebra module categories starts with Adamovic’s recent observation that there exist functors going the other way: from the W-algebra to the affine vertex algebra (or more typically, to another W-algebra). Interestingly, these functors do not naturally land in category O but in a much larger category. More interestingly, it appears that this approach will work for general nilpotent orbits.

We currently only understand these “inverse functors” in a few examples. I will attempt to explain some of what we know (and maybe some of what we believe) when $g=sl_2$ .