MIT Lie Groups Seminar

Abstract: Unipotent representations of semisimple Lie groups is a very important and somewhat conjectural class of unitary representations. Some of these representations for complex groups (equivalently, Harish-Chandra bimodules) were defined in the seminal paper of Barbasch and Vogan from 1985 based on ideas of Arthur. From the beginning it was clear that the Barbasch-Vogan construction doesn't cover all unipotent representations. The main construction of this talk is a geometric construction of Harish-Chandra bimodules that should exhaust all unipotent bimodules. A nontrivial result is that all unipotent bimodules in the sense of Barbasch and Vogan are also unipotent in our sense. The proof of this claim is based on the so called symplectic duality that in our case upgrades a classical duality for nilpotent orbits in the version of Barbasch and Vogan. Time permitting I will explain how this works. The talk is based on a joint work with Lucas Mason-Brown and Dmytro Matvieievskyi.

Abstract: When W is the Weyl group of a reductive group G, we can categorify its Hecke algebra by means of equivariant sheaves on the double flag variety of G. We will define a functor from the resulting category to a certain category of modules over a polynomial extension of C[W]. We will prove that, on objects called Rouquier complexes, our functor yields the equivariant Borel-Moore homology of a generalized Steinberg variety attached to a positive element in the braid group of W. Some reasons this may be interesting: (1) In type A, the triply-graded Khovanov-Rozansky homology of the link closure of the braid is a summand of the weight-graded equivariant homology of this variety. This extends previously-known results for the top and bottom "a-degrees" of KR homology. (2) The "Serre duality" of KR homology under insertion of full twists leads us to conjecture a mysterious homeomorphism between pieces of different Steinbergs. (3) We find evidence for a rational-DAHA action on the (modified) homology of the Steinbergs of periodic braids. It seems related to conjectures of Broué-Michel and Oblomkov-Yun in rather different settings.

Abstract: We develop a new type of harmonic analysis on an extended symplectic group $G=\BG_m\times \Sp_2n$ over $p$-adic fields. It is associated with the Langlands $\gamma$-functions attached to irreducible admissible representations of $G(F)$ and the standard representation of the dual group. Our work can be viewed as an extension of the work of Godement-Jacquet (which is a generalization of Tate's thesis). We confirm a series of conjectures in the local theory of the Braverman-Kazhdan proposal in this setting. This is a joint work with D. Jiang and L. Zhang.

Abstract: We describe a strategy for classifying character sheaves in the setting of graded Lie algebras. Via a nearby cycle construction we show that irreducible representations of Hecke algebras of complex reflection groups at roots of unity enter the description of character sheaves. We will explain connection to the work of Lusztig and Yun where (Fourier transforms of) character sheaves are parametrized by irreducible representations of trigonometric double affine Hecke algebras (DAHA). We will discuss some conjectures arising from this connection, which relate finite dimensional irreducible representations of trigonometric DAHAs to irreducible representations of Hecke algebras. This is based on joint work with Kari Vilonen and partly with Misha Grinberg.

Abstract: It is expected that the Betti form of the geometric Langlands equivalence will ultimately fit into an equivalence of four dimensional topological field theories. In this talk I will give an overview of ongoing work in the theory of sheaves of higher categories in derived algebraic geometry, and explain how it can be used to define a candidate four dimensional theory for the spectral side.

Abstract: We report on recent progress on the computation of the Green functions of a reductive group over a finite field, as introduced by Deligne and Lusztig in the 1970s. By work of Lusztig and Shoji, it is known that these Green functions coincide with another type of Green functions defined in terms of character sheaves. And there is a purely combinatorial algorithm for computing the values of these functions, up to certain signs. These signs have been explicitly determined in almost all cases. We show how the missing cases, which occur in groups of exceptional type in bad characteristics, can be solved by a purely group-theoretical computation.

Abstract: It is well known that Hall-Littlewood polynomials naturally arise from the problem of counting partial flags preserved by a nilpotent matrix over a finite field. I give an explicit interpretation of the modified Macdonald polynomials in a similar spirit, via counting parabolic bundles with nilpotent endomorphism over a curve over finite field. The result can also be interpreted as a formula for a certain truncated weighted counting of points in the affine Springer fiber over a constant nilpotent matrix. This leads to a confirmation of a conjecture of Hausel, Letellier and Rodriguez-Villegas about Poincare polynomials of character varieties. On the other hand, it naturally leads to interesting expansions of Macdonald polynomials and related generating functions that appear in the shuffle conjecture and its generalizations.

Abstract: This talk is based on the paper (joint with M. Finkelberg and I. Mirković).

Let G be a reductive complex algebraic group. Recall that a geometric Satake isomorphism is an equivalence between the category of G(O)-equivariant perverse sheaves on the affine Grassmannian for G and the category of finite dimensional representations of the Langlands dual group \hat{G}. It follows that for any G(O)-equivariant perverse sheaf P there exists an action of the dual Lie algebra \hat{\mathfrak{g}} on the global cohomology of P.

We will explain one possible approach to constructing this action. To do so, we will describe a new geometric construction of the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra \hat{\mathfrak{g}} based on certain one-parametric deformation of zastava spaces. We will introduce the so-called Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian that is a one-parametric deformation of the affine Grassmannian Gr_G. We will discuss the case G=SL_2 as an example.

Abstract: The commuting scheme has always been of great interest in invariant theory but it was only recent that it appears as a primordial object in the study of the Hitchin fibration for higher dimensional varieties. I will explain how the invariant theory for the commuting scheme, in particular the Chevalley restriction theorem for the commuting scheme, is used in the study of Hitchin fibration and the proof of the Chevalley restriction theorem in the case of symplectic Lie algebras. The talk is based on joint work with Ngo Bao Chau.

Abstract: The concept of a unipotent representation has its origins in the representation theory of finite Chevalley groups. Let G(Fq) be the group of Fq-rational points of a connected reductive algebraic group G. In 1984, Lusztig completed the classification of irreducible representations of G(Fq). He showed:

1. All irreducible representations of G(Fq) can be constructed from a finite set of building blocks -- called 'unipotent representations.'
2. Unipotent representations can be classified by certain geometric parameters related to nilpotent orbits for a complex group associated to G(Fq).

Now, replace Fq with C, the field of complex numbers, and replace G(Fq) with G(C). There is a striking analogy between the finite-dimensional representation theory of G(Fq) and the unitary representation theory of G(C). This analogy suggests that all unitary representations of G(C) can be constructed from a finite set of building blocks -- called `unipotent representations' -- and that these building blocks are classified by geometric parameters related to nilpotent orbits. In this talk I will propose a definition of unipotent representations, generalizing the Barbasch-Vogan notion of `special unipotent'. The definition I propose is geometric and case-free. After giving some examples, I will state a geometric classification of unipotent representations, generalizing the well-known result of Barbasch-Vogan for special unipotents.

This talk is based on forthcoming joint work with Ivan Loseu and Dmitryo Matvieievskyi.

Abstract: Let $(G,G')$ be a reductive dual pair inside a finite symplectic group. By restricting the Weil representation to the dual pair, there exists a relation (called the finite Howe correspondence) between the irreducible representations of the two groups $G,G'$. In this talk, we would like to discuss some progress on the understanding of the correspondence by using Lusztig's classification on the representations of finite classical groups. In particular, we will focus on the following three subjects:

1. the decomposition of the uniform projection of the Weil character
2. the commutativity between the Howe correspondence and the Lusztig correspondence
3. the description of the Howe correspondence on unipotent characters in terms of the symbols by Lusztig.

Abstract: I will present joint work with Yiannis Sakellaridis and Akshay Venkatesh, in which we apply a perspective from topological field theory to the relative Langlands program. To a spherical variety one can assign two quantization problems, automorphic and spectral, both resulting in structures borrowed from QFT. The automorphic quantization (or A-side) organizes objects such as periods, Plancherel measure, theta series and relative trace formula, while the spectral quantization (or B-side) organizes L-functions and Langlands parameters. Our conjectures describe a duality operation on spherical varieties, which exchanges automorphic and spectral quantizations (and may be seen as Langlands duality for boundary conditions in 4d TFT, a refined form of symplectic duality / 3d mirror symmetry).

Abstract: The central characters of the finite-dimensional Kac-Moody superalgebras can be described by their "cores"; this notion can be nicely interpreted in terms of the Duflo-Serganova functors. I will discuss an extension of these results to affine Lie superalgebras.

Abstract: One of the fundamental contributions of Kazhdan and Lusztig's 1979 Inventiones paper was the notion of "cells" in Weyl groups. They gave a decomposition of the left regular representation of W as a direct sum of "left cell" representations, which encode deep and powerful information about group representations. In the case of the symmetric group S_n=W, the left cells are irreducible representations. In all other cases they are not. Lusztig in his 1984 book gave a beautiful description of all left cells in terms of the geometry of a nilpotent orbit.

There is a parallel notion of "Harish-Chandra cells" in the representation theory of a real reductive group G(R). Again each cell is a representation of W, encoding deep information about the G(R) representations. I will formulate a conjecture extending Lusztig's calculation of left cell representations to this case, and explain its connection with Arthur's theory of unipotent representations.

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Abstract: In 1992, Henniart proved that supercuspidal representations for p-adic GLn are determined by their character on so-called very regular elements. This has been useful in many ways as it allows for convenient comparison between various constructions of supercuspidal representations for GLn. We describe a version of this type of result which holds for (some) representations of reductive groups over finite fields. This is joint work with Masao Oi.

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Abstract:The program of Sakellaridis and Venkatesh proposes a unified framework to study integral representations of L-functions through the lens of spherical varieties. For X an affine spherical variety, the (hypothetical) IC complex of the infinite-dimensional formal arc space of X is conjecturally related to special values of local unramified L-functions. We formulate this relation precisely using a new conjectural geometric construction of the crystal basis of a finite-dimensional representation (determined by X) of the dual group. We prove these conjectures for a large class of spherical varieties. This is joint work with Yiannis Sakellaridis.

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Abstract:(work in progress with R. Rouquier) In this talk I will present a computational (yet conjectural) method to determine some decomposition matrices for finite groups of Lie type. I will first explain how one can produce a "natural" self-equivalence in the case of $\mathrm{GL}_n(q)$ coming from the topology of the Hilbert scheme of $\mathbb{C}^2$. The combinatorial part of this equivalence is related to Macdonald's theory of symmetric functions and gives $(q,t)$-decomposition numbers. The evidence suggests that the case of finite unitary groups is obtained by taking a suitable square root of that equivalence.

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Abstract:This talk is based on joint work with M.Khovanov and Y.Kononov. By evaluating a topological field theory in dimension 2 on surfaces of genus 0,1,2 etc we get a sequence. We investigate which sequences occur in this way depending on the assumptions on the target category.

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Abstract:Suppose that one has a supercuspidal representation of a Levi subgroup of some reductive $p$-adic group $G$. Bernstein associated to this a block Rep$(G)^s$ in the category of smooth $G$-representations. We address the question: what does Rep$(G)^s$ look like?

Usually this is investigated with Bushnell--Kutzko types, but these are not always available. Instead, we approach it via the endomorphism algebra of a progenerator of Rep$(G)^s$. We will show that Rep$(G)^s$ is "almost" equivalent with the module category of an affine Hecke algebra -- a statement that will be made precise in several ways.

In the end, this leads to a classification of the irreducible representations in Rep$(G)^s$ in terms of the complex torus and the finite groups that are canonically associated to this Bernstein component.

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Let X be a projective curve, G a reductive group. Let Bun be the stack of G-bundles over X, and consider the category of D-modules on Bun. (This category appears on the “automorphic” side of the geometric Langlands correspondence.) Drinfeld and Gaitsgory prove that, despite the “unbounded” (non-quasi compact) nature of Bun, the category of D-modules is well-behaved (compactly generated).

In this talk, we will “compactify” this category in a stronger sense; this can be viewed as compactifying the quantized cotangent bundle to Bun. While the basic idea of such compactification goes back to ideas of Deligne and Simpson, its construction relies on non-trivial properties of the geometry of Bun (similar to the Drinfeld-Gaitsgory Theorem).

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Given a representation V of a reductive group G, Braverman-Finkelberg-Nakajima defined a Poisson variety called the Coulomb branch, using a convolution algebra construction. This variety comes with a natural deformation quantization, called a Coulomb branch algebra. Important cases of these Coulomb branches are (generalized) affine Grassmannian slices, and their quantizations are truncated shifted Yangians.

Motivated by the geometric Satake correspondence and the theory of symplectic duality/3d mirror symmetry, we expect a categorical g-action on modules for these truncated shifted Yangians. I will explain three results in this direction. First, we have an indirect realization of this action, using equivalences with KLRW-modules. Second, we have a geometric relation between these generalized slices by Hamiltonian reduction. Finally, we have an algebraic version of this Hamiltonian reduction which we are able to relate to the first realization.

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Khovanov-Rozansky homology is a knot invariant which, by the result of Khovanov, can be computed as the Hochschild cohomology functor applied to Rouquier complexes of Soergel bimodules. I will describe a new geometric model for the Hochschild cohomology of Soergel bimodules, living in the monodromic Hecke category. I will also explain how it allows to identify objects representing individual Hochsсhild cohomology groups as images of explicit character sheaves. Based on the joint work with Roman Bezrukavnikov.

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Abstract: Kazhdan and Lusztig proved the Deligne-Langlands conjecture, a bijection between irreducible representations of unipotent principal block representations of a p-adic group with certain unipotent Langlands parameters in the Langlands dual group (plus the data of certain representations). We lift this bijection to a statement on the level of categories. Namely, we define a stack of unipotent Langlands parameters and a coherent sheaf on it, which we call the coherent Springer sheaf, which generates a subcategory of the derived category equivalent to modules for the affine Hecke algebra (or specializing at q, unipotent principal block representations of a p-adic group). Our approach involves categorical traces, Hochschild homology, and Bezrukavnikov's Langlands dual realizations of the affine Hecke category. This is a joint work with David Ben-Zvi, David Helm and David Nadler.

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Abstract: The action of quantum groups on the K-theory of Nakajima varieties takes the simplest form in the stable bases, invented by D.Maulik and A.Okounkov, and in their most advanced (elliptic) version by M.Aganagic and A.Okounkov. In collaboration with A.Smirnov we discovered and proved the factorization property of elliptic stable envelopes. As a consequence, we proved the conjectures of E.Gorsky and A.Negut. Also it gives a new interesting description of the operators of quantum difference equations, shift operators and other quantities in enumerative geometry. The talk is based on joint works with A.Smirnov.

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Abstract: Recently Kaletha constructed the local Langlands correspondence (i.e., L-packets and their L-parameters) for a wide class of supercuspidal representations. In this talk, I would like to discuss my ongoing work on the twisted endoscopic character relation for Kaletha's supercuspidal L-packets.

The strategy is to imitate Kaletha's proof of the standard endoscopic character relation in the setting of twisted endoscopy. Thus first I am going to review Kaletha's construction of supercuspidal L-packets and his proof of the standard endoscopic character relation. Then I will explain a few key points in the twisting process with an emphasis on Waldspurger's philosophy "l'endoscopie tordue n'est pas si tordue".

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Abstract: In geometric representation theory one tries to understand group representations using geometry. But sometimes one can try to go in the opposite direction. In this talk we will illustrate this by showing that a number of features in geometry (such as Springer correspondence attached to unipotent classes) can be recovered from pure algebra (such as the generic degrees of representations of Weyl groups).

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Abstract: It is well known that (complex, smooth) irreducible representations of GL_n(F) are closely related to Lusztig's (dual) canonical basis of type A. This suggests studying irreducibility of parabolic induction in terms of geometry of preprojective varieties. Motivated by works of Geiss-Leclerc-Schroer, Kang-Kashiwara-Kim-Oh and others, I will recall some old and new conjectures and indicate some modest progress towards them. I will also explain a construction of standard modules via RSK correspondence.

Based on joint works with Alberto Minguez and Max Gurevich.