# 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

We will mostly hold "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

- 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)

DateSpeakerTitleSlotVideoNotes

Sep 9 | Jay Taylor | Broué's abelian defect group conjecture, I | W3-5pm | IAS/YT | |

Sep 16 | Daniel Juteau | Broué's abelian defect group conjecture, II | W3-5pm | IAS/YT | |

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 | |

Oct 7 | Raphaël Rouquier | Finite groups as algebraic groups in non-defining characteristic | W3-5pm | IAS/YT | |

Oct 14 | Jize Yu | An introduction to affine Grassmanians and the geometric Satake equivalence | W3-5pm | IAS/YT | |

Oct 21 | Linyuan Liu | (Equivariant) Cohomology of the affine Grassmannian and Ginzburg’s picture | W3-5pm | IAS/YT | |

Oct 28 | Anne Dranowski | Derived Equivariant Cohomology of the affine Grassmannian and Bezrukavnikov and Finkelberg’s equivalences | W3-5pm | IAS/YT | |

Nov 4 | Jize Yu | The Derived Geometric Satake Equivalence of Bezrukavnikov and Finkelberg | W3-5pm | IAS/YT | |

Nov 11 | Tony Feng | Iwahori-Whittaker category and geometric Casselman-Shalika | W3-5pm | IAS/YT | |

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 | |

Dec 9 | Peter Fiebig | Lefschetz operators, Hodge-Riemann forms, and representations | W3-5pm | IAS/YT | |

Dec 16 | Dima Arinkin | Hecke category via derived convolution formalism | W3-5pm | IAS/YT |

### Term II (Jan 11 – Apr 9, 2021)

DateSpeakerTitleSlotVideoNotes

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 | |

Jan 27 | Geordie Williamson | The Hecke category action on the principal block via Smith theory | W3-5pm | IAS/YT | |

Feb 3 | Pablo Boixeda Alvarez | The K-ring of Steinberg varieties | W3-5pm | IAS/YT | |

Feb 10 | Shotaro Makisumi | Equivariantization and de-equivariantization | W3-5pm | IAS/YT | |

Feb 17 | Tom Braden | Gaitsgory's central sheaves | W3-5pm | IAS/YT | |

Feb 24 | James Tao | The affine Hecke category is a monoidal colimit | W3-5pm | IAS/YT | |

Mar 3 | Tsao-Hsien Chen | On two geometric realizations of the anti-spherical module | W3-5pm | IAS/YT | |

Mar 10 | Pablo Boixeda Alvarez | Two Geometric Realizations of the Affine Hecke Algebra I | W3-5pm | IAS/YT | |

Mar 17 | Roman Bezrukavnikov | Affine Hecke category and noncommutative Springer resolution | W3-5pm | IAS/YT | |

Mar 24 | Ivan Losev | Parabolic version of the two realizations theorem and applications to modular representation theory | W3-5pm | IAS/YT | |

Mar 31 | No talk: Week of Virtual Workshop on Representation Theory and Geometry | ||||

Apr 7 | Jens Eberhardt | K-Motives and Koszul Duality in Geometric Representation Theory | W3-5pm | IAS/YT |

### Overflow

DateSpeakerTitleSlotVideoNotes

April 14 | Zhiwei Yun | Microlocal sheaves on certain affine Springer fibers | W3-5pm | IAS/YT | |

May 5 | Simon Riche | Towards a modular "2 realizations" equivalence | W7-9am |
IAS/YT | |

May 12 | Pavel Etingof | Frobenius exact symmetric tensor categories | W3-5pm | IAS/YT |

## Abstracts

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

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 $\mathrm{GL}_n(q)$ and $\mathrm{SL}_n(q)$. 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)

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 $\mathrm{SL}(2,q)$.

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

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 $\mathrm{GL}_2$ over $\mathbb{Q}_p$).

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

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 $\mathrm{GL}_n$. 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)

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)

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)

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)

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 $\mathcal{O}$ 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:** *The K-ring of Steinberg varieties* by Pablo Boixeda Alvarez (Institute for Advanced Study)

In the upcoming series of talks we discuss some equivalences by Bezrukavnikov between categories of coherent sheaves on Steinberg varieties and of perverse sheaves on the affine flag variety.

In this talk I discuss the decategorified isomorphism preceeding Bezrukavnikov’s equivalence of categories. This is a ring isomorphism of the K-ring of the Steinberg variety under convolution and the affine Hecke algebra originally due to Kazhdan, Lusztig and Ginzburg.

**February 10:** *Equivariantization and de-equivariantization* by Shotaro Makisumi (Columbia University / Institute for Advanced Study)

This is the second in a series of talks on "two realizations." We discuss equivariantization and de-equivariantization for a group $G$, which relate categories with $G$-actions and categories with $\operatorname{Rep}G$-actions, and how this relates to the notion of category over a stack. We will deduce a "de-equivariantization principle" which will be helpful in understanding the proof strategy of Arkhipov-Bezrukavnikov and the full "two realizations," to be explained in future talks.

**February 17:** *Gaitsgory's central sheaves* by Tom Braden (University of Massachusetts, Amherst / Institute for Advanced Study)

A theorem of Bernstein identifies the center of the affine Hecke algebra of a reductive group $G$ with the Grothendieck ring of the tensor category of representations of the dual group $G^\vee$. Gaitsgory constructed a functor which categorifies this result. This functor sends Satake sheaves on the affine Grassmannian of $G$ to Iwahori-equivariant perverse sheaves on the affine flag variety, and the sheaves in the image lie in the center of this category. The functor is given as nearby cycles for a family over a curve whose general fiber is the affine Grassmannian times the finite flag variety and whose special fiber is the affine flag variety. As a result, the functor carries an important additional structure, an endomorphism coming from monodromy of nearby cycles.

**February 24:** *The affine Hecke category is a monoidal colimit* by James Tao (MIT)

We will discuss the following theorems concerning colimits taken in the infinity-category of monoidal DG-categories. (No familiarity with infinity-categories will be required or assumed.) The affine Hecke category is the monoidal colimit of its finite type Hecke subcategories. The category of D-modules on the loop group is the monoidal colimit of the categories of D-modules on standard parahoric subgroups. These theorems, and many analogous ones, are consequences of a general inductive characterization of colimits indexed by a category of `words' in a Coxeter group. We will also discuss two applications-in-progress: the construction of a functor from the affine to the finite Hecke category in type A, and the construction of new `deformations' of the affine Hecke category.

**March 3:** *On two geometric realizations of the anti-spherical module* by Tsao-Hsien Chen (University of Minnesota, Twin Cities / Institute for Advanced Study)

The anti-spherical modules over the affine Hecke algebras admit two different realizations: one realization is in terms of the space of Whittaker functions on the affine flag manifolds and the other realization, due to Kazhdan-Lusztig, is in terms of the equivariant K-theory of the Springer resolution of the nilpotent cone for the dual group. I will explain the work of Arkhipov-Bezrukavnikov on the equivalence between the Iwahoric-Whittaker category and the equivariant derived category for the Springer resolution, which provides a geometric lift (or categorification) of the above two realizations of the anti-spherical modules.

**March 10:** *Two Geometric Realizations of the Affine Hecke Algebra I* by Pablo Boixeda Alvarez (Institute for Advanced Study)

In this talk we will begin the discussion of the results in Bezrukavnikov's ``On Two Geometric Realizations of the Affine Hecke algebra''. We will put all the previous tools described in this series of talks together to construct the equivalence of monoidal categories categorifying the realization of the affine Hecke algebra as the K-ring of the Steinberg variety.

**March 17:** *Affine Hecke category and noncommutative Springer resolution* by Roman Bezrukavnikov (MIT / Institute for Advanced Study)

I will start with a few comments on the proof of the equivalence presented in the previous talks. Then I will focus on the description of the *abelian* category of perverse sheaves on the affine flag variety on the coherent side, where the answer is given in terms of the so called noncommutative Springer resolution. This is a key ingredient in applications to representations of semisimple Lie algebras, leading to the proof (with Mirkovic) of Lusztig's 1999 conjecture for large p, as well as applications to cells in affine Weyl groups and asymptotic Hecke algebras and potential applications to characters of modules over an affine Lie algebra at the critical level and (Kac - De Concini) quantum groups at a root of unity. The story can be viewed as upgrading a (very) special case of geometric Langlands duality to a statement involving abelian, rather than triangulated, categories.

**March 24:** *Parabolic version of the two realizations theorem and applications to modular representation theory* by Ivan Losev (Yale University / Institute for Advanced Study)

The goal of this talk is two-fold. We state a parabolic version of the two realizations theorem and sketch a proof. This version relates Iwahori-constructible sheaves on parabolic affine flag variety to coherent sheaves on a parabolic version of the Steinberg. We also explain an application of this result to the representation theory of semisimple Lie algebras in positive characteristic. Our result here is an explicit character formula for appropriately equivariant simple modules with distinguished p-character. The talk is based on 2005.10030, joint with Bezrukavnikov.

**April 7:** *K-Motives and Koszul Duality in Geometric Representation Theory* by Jens Eberhardt (Mathematical Institute of the University of Bonn)

Perverse sheaves and intersection cohomology are central objects in geometric representation theory. This talk is about their long-lost K-theoretic cousins, called K-motives. We will discuss definitions and basic properties of K-motives and explore potential applications to geometric representation theory. For example, K-motives shed a new light on Beilinson-Ginzburg-Soergel's Koszul duality — a remarkable symmetry in the representation theory and geometry of two Langlands dual reductive groups. We will see that this new form of Koszul duality does not involve any gradings or mixed geometry which are as essential as mysterious in the classical approaches.

**April 14:** *Microlocal sheaves on certain affine Springer fibers* by Zhiwei Yun (MIT)

For homogeneous affine Springer fibers (those with $\mathbb{G}_m$ symmetry), we realize them as Lagrangian cycles inside ambient symplectic varieties, and make sense of microlocal sheaves supported on these affine Springer fibers. We also propose a conjectural equivalence between the microlocal sheaf category and a coherent sheaf category in terms of Springer fibers for the Langlands dual group, and are able to prove the conjecture in some cases. This is joint work with Roman Bezrukavnikov, Pablo Boixeda Alvarez and Michael McBreen.

**May 5:** *Towards a modular "2 realizations" equivalence* by Simon Riche (Université Clermont Auvergne, IAS)

I will report on a project joint with Roman Bezrukavnikov (and partly with Laura Rider) aiming at constructing a variant for positive-characteristic coefficients of the equivalence constructed by Bezrukavnikov in "On two geometric realizations of an affine Hecke algebra". In a first paper we have obtained a description of the "regular quotient" of the category of Iwahori-equivariant perverse sheaves on the affine flag variety in terms of representations of the centralizer of a regular unipotent element in the dual group. In a second paper (in preparation) we obtain a version of this equivalence "in families" over the adjoint quotient, which allows to obtain the desired equivalence "over the regular locus". In the talk I will explain these constructions, and how we plan to use them to construct the full equivalence.

**May 12:** *Frobenius exact symmetric tensor categories* by Pavel Etingof (MIT)

I will report on a joint work in progress with K. Coulembier and V. Ostrik. We show that a symmetric tensor category in characteristic p>0 admits a fiber functor to the Verlinde category (semisimplification of Rep(Z/p)) if and only if it has moderate growth and its Frobenius functor (an analog of the classical Frobenius in the representation theory of algebraic group) is exact. For example, for p=2 and 3 this implies that any such category is (super)-Tannakian. We also give a characterization of super-Tannakian categories for p>3. This generalizes Deligne's theorem that any symmetric tensor category over C of moderate growth is super-Tannakian to characteristic p. At the end I'll discuss applications of this result to modular representation theory.