# Geometric and Modular Representation Theory Seminar (Special Year Seminar 2020-21)

This is the seminar website for the 2020-2021 IAS Special Year. You can jump to the schedule and abstracts.

## Information

At least for the first semester, we will be mostly holding "hybrid seminars," with talks on Zoom but some members participating locally.

• Time: Wednesdays 3-5pm or 7-9am (Princeton time)
• Location: Zoom and (starting 9/30) Simonyi 101

We will hold the Princeton time constant. Please note that daylight savings may affect the meeting time in your location.

### Speaking slots

We have reserved three slots for this seminar: Wednesdays 7-9am, Wednesdays 3-5pm, and Thursdays 7-9am. The first two are speaking slots, and the speaker may decide (based on their timezone) which slot they prefer. Then those that cannot make it meet in the next slot to watch a recording of the seminar and discuss.

So, on any given week, we will do either of the following:

• The seminar takes place on Wednesday 3-5pm. Those who can't attend it watch the recording together on Thursday 7-9am.
• The seminar takes place on Wednesday 7-9am. Those who can't attend it watch the recording together on Wednesday 3-5pm.

The schedule of talks below shows the speaking slot.

### For speakers

IAS announces the talks for the upcoming week each Friday. Please send your title and abstract to Shotaro Makisumi by Tuesday before that, e.g. for the talk on Sep 9th, we would like your title/abstract by Sep 1st. Speakers should also send the talk notes so that they can be posted here before the talk.

### Seminar format

Our seminar is reserved for 2 hours each week, but we expect each meeting to run closer to one hour than to two. We will adopt the following structure:

• The first hour will be the talk itself, broken down as follows: 25 minutes, 5 min break, 25 minutes, 5 min break.
• The second hour starts with questions, followed by discussion, and finally a truly optional extra 25 min of talk if necessary.

### Participation guidelines

We may change these depending on how things go.

• You will be muted by default.
• All members are encouraged if possible to have their video on. This makes it much easier for the speaker to see what is being followed, and encourages audience participation.
• To ask questions, you can do either of the following:
• First type your question in the chat. Anyone in the audience is welcome to answer these questions. If questions are unresolved after some time, the organizer will stop the speaker, and then you can unmute yourself and ask the question.
• Unmute yourself and ask the question directly.

## Schedule of talks and notes

Abstracts can be found further down the page. The videos are also collected here.

### Term I (Sep 21 – Dec 18, 2020)

Date Speaker Title Slot Video Notes
Sep 9 Jay Taylor Broué's abelian defect group conjecture, I W3-5pm IAS/YT .pdf
Sep 16 Daniel Juteau Broué's abelian defect group conjecture, II W3-5pm IAS/YT .pdf
Sep 23 No talk: Week of short postdoc talks
Sep 30 Raphaël Rouquier Finite groups as algebraic groups in defining characteristic W3-5pm IAS/YT .pdf
Oct 7 Raphaël Rouquier Finite groups as algebraic groups in non-defining characteristic W3-5pm IAS/YT .pdf
Oct 14 Jize Yu An introduction to affine Grassmanians and the geometric Satake equivalence W3-5pm IAS/YT .pdf
Oct 21 Linyuan Liu (Equivariant) Cohomology of the affine Grassmannian and Ginzburg’s picture W3-5pm IAS/YT .pdf
Oct 28 Anne Dranowski Derived Equivariant Cohomology of the affine Grassmannian and Bezrukavnikov and Finkelberg’s equivalences W3-5pm IAS/YT .pdf
Nov 4 Jize Yu The Derived Geometric Satake Equivalence of Bezrukavnikov and Finkelberg W3-5pm IAS/YT .pdf
Nov 11 Tony Feng Iwahori-Whittaker category and geometric Casselman-Shalika W3-5pm IAS/YT .pdf
Nov 18 No talk: Week of Virtual Workshop on Recent Developments in Geometric Representation Theory
Nov 25 No talk: IAS's Thanksgiving break starts Wednesday 2pm
Dec 2 George Lusztig Geometric Satake equivalence: a historical survey W3-5pm IAS/YT .pdf
Dec 9 Peter Fiebig Lefschetz operators, Hodge-Riemann forms, and representations W3-5pm IAS/YT .pdf
Dec 16 Dima Arinkin Hecke category via derived convolution formalism W3-5pm IAS/YT .pdf

### Term II (Jan 11 – Apr 9, 2021)

Date Speaker Title Slot Video Notes
Jan 13 Tony Feng Introduction to Smith theory W3-5pm IAS/YT
Jan 20 Daniel Juteau New Age Linkage or: The linkage principle and the tilting character formula via Smith-Treumann theory W3-5pm IAS/YT .pdf
Jan 27 Geordie Williamson The Hecke category action on the principal block via Smith theory W3-5pm IAS/YT
Feb 3 Pablo Boixeda Alvarez TBD W3-5 IAS/YT
Feb 10 TBD TBD TBD IAS/YT
Feb 17 TBD TBD TBD IAS/YT
Feb 24 TBD TBD TBD IAS/YT
Mar 3 TBD TBD TBD IAS/YT
Mar 10 TBD TBD TBD IAS/YT
Mar 17 TBD TBD TBD IAS/YT
Mar 24 TBD TBD TBD IAS/YT
Mar 31 No talk: Week of spring special year conference
Apr 7 TBD TBD TBD IAS/YT

## Abstracts

September 9: Broué's abelian defect group conjecture, I by Jay Taylor (University of Southern California / Institute for Advanced Study)

Abstract: This talk will form part of a series of three talks focusing on Broué’s Abelian Defect Group Conjecture, which concerns the modular representation theory of finite groups. We will pay particular attention here to the ‘geometric’ form of the conjecture which concerns finite reductive groups such as $GL_n(q)$ and $SL_n(q)$. Broué’s conjecture gives a strong structural reason for many numerical coincidences one sees amongst characters and is part of a general ‘local/global phenomena’ that is abundant in the theory.

In this first talk we will briefly recall the necessary background material, get to the point where we can state the conjecture, and discuss some important examples.

September 16: Broué's abelian defect group conjecture, II by Daniel Juteau (CNRS / Université de Paris / Institute for Advanced Study)

Abstract: In this second talk about Broué’s Abelian Defect Group Conjecture, we will explain its geometric version in the case of finite groups of Lie type: the equivalence should be induced by the cohomology complex of Deligne-Lusztig varieties. This was actually the main motivation for the conjecture in the first place. We will illustrate those ideas with the case of SL(2,q).

September 30: Finite groups as algebraic groups in defining characteristic by Raphaël Rouquier (UCLA / Institute for Advanced Study)

Abstract: We will discuss p-local representation theory and conjectures of Broue and Alperin, the latter been inspired by finite groups of Lie type in characteristic p. The issue is the extent to which the category of representations can be reconstructed from that of proper parabolic subgroups. In different settings, a Borel suffices (Arkhipov-Bezrukavnikov-Ginzburg for rational representations, Colmez for GL_2 over Q_p).

October 7: Finite groups as algebraic groups in non-defining characteristic by Raphaël Rouquier (UCLA / Institute for Advanced Study)

Abstract: We will discuss the conjecture of Broue relating modular representations of finite groups of Lie type in non-defining characteristic to those of normalizers of Levi subgroups, with a focus on GL_n. The categories of representations can be related to (quantized) Hilbert schemes of points on surfaces and one obtains two variable versions of decomposition numbers.

October 14: An introduction to affine Grassmanians and the geometric Satake equivalence by Jize Yu (Institute for Advanced Study)

Abstract: This is the first talk in a series of three talks towards understanding Bezrukavnikov-Finkelberg's derived geometric Satake equivalence. In this talk, we recall the geometry of equal characteristic affine Grassmannians and some of the ingredients of proving the geometric Satake equivalence in this setting.

October 21: (Equivariant) Cohomology of the affine Grassmannian and Ginzburg’s picture by Linyuan Liu (Institute for Advanced Study)

Abstract: This is the second talk in a series of three talks on the derived Satake. I will give an overview of an article by Ginzburg which laid the foundational ideas for this equivalence.

October 28: Derived Equivariant Cohomology of the affine Grassmannian and Bezrukavnikov and Finkelberg’s equivalences by Anne Dranowski (Institute for Advanced Study)

Abstract: This is the third talk in a series of three talks on the derived Satake equivalence. I will give an overview of the article of Bezrukavnikov and Finkelberg which explains how the equivariant derived category of the affine Grassmannian can be described in terms of coherent sheaves on the Langlands dual Lie algebra.

November 4: The Derived Geometric Satake Equivalence of Bezrukavnikov and Finkelberg by Jize Yu (Institute for Advanced Study)

Abstract: This is the last talk towards understanding Bezrukavnikov-Finkelberg's derived geometric Satake equivalence. With the preparations from previous talks, we will introduce two filtrations: a topological filtration on the equivariant cohomology and an algebraic filtration on the Kostant functor. The comparison of these two functors will lead us to the main statement of Bezrukavnikov-Finkelberg's derived geometric Satake equivalence.

November 11: Iwahori-Whittaker category and geometric Casselman-Shalika by Tony Feng (MIT / Institute for Advanced Study)

This talk will be an exposition of a recent paper of Bezrukavnikov-Gaitsgory-Mirkovic-Riche-Rider giving an Iwahori-Whittaker model for the Satake category. The main point is that their argument works for modular coefficients. I will give some motivation via the Casselman-Shalika formula in the theory of p-adic groups, introduce the Iwahori-Whittaker model, and sketch the proof of the equivalence.

December 2: Geometric Satake equivalence: a historical survey by George Lusztig (MIT / Institute for Advanced Study)

Before the "geometric Satake equivalence" there was a decategorified version of it which however contained most of its essential features. In my talk I will talk about some of the ideas which have led to this theory. In particular I will explain the connection with modular representations.

December 9: Lefschetz operators, Hodge-Riemann forms, and representations by Peter Fiebig (Friedrich-Alexander-Universität Erlangen-Nürnberg / Institute for Advanced Study)

Motivated by a formal similarity between the Hard Lefschetz theorem and the geometric Satake equivalence we study vector spaces that are graded by a weight lattice and are endowed with linear operators in simple root directions. We allow field coefficients in characteristics different from 2. In the case that a “Hodge-Riemann form” exists, the operators (and the grading) yield a semisimple representation of the associated Lie algebra. We then explore the analogous theory with the field replaced by the ring of p-adic integers. In this setup we obtain tilting modules for the associated algebraic group.

December 16: Hecke category via derived convolution formalism by Dima Arinkin (University of Wisconsin–Madison)

The talk is about convolution in the setting of geometric representation theory. What are its formal properties? As a starting point, let G be a group and let D(G) be the derived category of constructible sheaves on it. Convolution turns D(G) into a monoidal category, which is rigid (every object is dualizable) if and only if G is proper (this statement is due to Boyarchenko and Drinfeld).

In this talk, I develop the formalism of convolution using the language of derived algebraic geometry, and then apply these techniques to the (spherical) Hecke category and related objects.

January 13: Introduction to Smith theory by Tony Feng (MIT / Institute for Advanced Study)

Smith theory is a type of equivariant localization with respect to a cyclic group of prime order p, with coefficients in a field of the same characteristic p. It has been the source of various recent advances in modular representation theory and mod p Langlands functoriality, some of which will be discussed later in the seminar. I will give an introduction to the basic formalism.

January 20: The linkage principle and the tilting character formula via Smith-Treumann theory by Daniel Juteau (CNRS / Université de Paris / Institute for Advanced Study)

The linkage principle says that the category of representations of a reductive group G in positive characteristic decomposes into "blocks" controlled by the affine Weyl group. We will discuss the beautiful geometric proof of this result that Simon Riche and Geordie Williamson obtained by applying Smith-Treumann theory to the Iwahori-Whittaker model of the Satake category... and, as a bonus, a tilting character formula valid for all dominant weights and all characteristics!

January 27: The Hecke category action on the principal block via Smith theory by Geordie Williamson (University of Sydney / Sydney Mathematical Research Institute / Institute for Advanced Study)

Wall-crossing functors on the principal block of category O give an action of the (finite) Hecke category. If one knows enough about the Hecke category, one can deduce the Kazhdan-Lusztig conjectures from the existence of this action. This is a simple example of the power of categorification. In 2013, Riche and I conjectured that something similar is true for the principal block of reductive algebraic groups: namely that wall-crossing functors give an action of the (affine) Hecke category. We showed that the conjecture implies several rather deep statements in representation theory (mod p analogues of the Kazhdan-Lusztig conjectures). Recently, this conjecture has been proved in two different ways: the first (by Bezrukavnikov and Riche) via mod p localization and the second (by my student Josh Ciappara) via Smith theory. I will give an outline of Josh's proof. Bezrukavnikov and Riche's proof will be discussed later in the seminar.

February 3: TBD by Pablo Boixeda Alvarez

TBD