MIT Lie Groups Seminar

Abstract: A variant of the nearby cycle construction, due initially to Grinberg, is the main geometric ingredient in our construction of character sheaves for graded Lie algebras. In this talk I will explain this construction and how it allows us to associate character sheaves to representations of Hecke algebras at roots of unity. In particular, I will explain how the Hecke relations arise from appropriate b-functions associated to sections of certain D-modules. The b-functions that arise in our applications can be deduced from previous results of Fumihiro Saito on prehomogenous vector spaces. I will also explain the relationship to invariant systems of differential equations. This is joint work with Ting Xue and Misha Grinberg.

Video

Abstract: The Deligne-Simpson problem asks for a criterion of the existence of connections on an algebraic curve with prescribed singularities at punctures. We give a solution to a generalization of this problem to G-connections on P^1 with a regular singularity and an irregular singularity (with a condition called isoclinic). Here G can be any complex reductive group. Perhaps surprisingly, the solution is expressed in terms of rational Cherednik algebras. This is joint work with Konstantin Jakob.

Video

Abstract: Let G be a connected reductive group over an algebraically closed field. There is a natural partition of G into finitely many subsets called strata. Each stratum is a union of conjugacy classes of fixed dimension. The strata of G can be indexed by a set Str(G) which is independent of the ground field. Let CS(G) be the set of unipotent character sheaves (up to isomorphism). These are certain simple perverse sheaves on G. It turns out that there is a natural surjective map from CS(G) to Str(G).

Video

Abstract: In the representation theory of p-adic groups, one way to study irreducible representations is by attaching local factors to them. These local factors are complex valued functions of a complex variable, and they often encode properties of the representations in question. In this talk, I will explain an analog of these local factors for irreducible generic representations of GL(n, F_q). I will talk about two families of gamma factors for the tensor product representation (based on constructions by Jacquet--Piatetski-Shapiro--Shalika and Shahidi) and about the Bessel function of a generic representation (defined by Gelfand--Graev). I will give a conceptual quantitative interpretation for the absolute value of these gamma factors. I will also explain a relation between the Bessel function and Kloosterman sums and sheaves. This talk is based on my two recent preprints, where the most recent one is a joint work with David Soudry.

Slides

Video

Abstract: I will discuss a conjectural categorical form of the (arithmetic) local Langlands correspondence for p-adic groups and establish the unipotent part of such correspondence (for characteristic zero coefficient field). As an application, we show that the coherent Springer sheaf on the moduli of unipotent Langlands parameter is a honest coherent sheaf (i.e. a complex concentrated in degree zero). Joint work with Tamir Hemo.

Video

Abstract: Given the current knowledge of complex representations of finite groups of Lie type, obtaining good upper bounds for their character values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We will report on recent results that produce such character bounds, and discuss some applications of them, in and outside of group theory.

Video

Abstract: In this talk I will explain how a certain construction from the theory of integrable systems (a vertex-IRF transformation) can be interpreted in terms of familiar objects from (geometric) representation theory and how this interpretation gives new results of independent interest. For instance, I will show how to construct Whittaker vectors for representations of \mathfrak{gl}_n using the Gelfand-Tsetlin subalgebra and how it is related to nonstandard quantizations of the group GL_n.

Video

Abstract: Introduced by Lusztig in the 1990s, the braid group symmetries constitute an essential part in the theory of quantum groups. The i-quantum groups are coideal subalgebras of quantum groups arising from quantum symmetric pairs, which are now viewed as natural generalizations of quantum groups. In this talk, I will present our construction of relative braid group symmetries (associated to the relative Weyl group of a symmetric pair) on i-quantum groups. These new symmetries admit many properties similar to Lusztig's symmetries. Our construction has further led to compatible relative braid group actions on modules. This is joint work with Weiqiang Wang.

Video

Abstract: The geometric Langlands conjecture of Beilinson-Drinfeld describes the category of D-modules on the moduli stack of G-bundles on a smooth projective curve in terms of spectral data involving the dual group. It can be thought of as an analogue of Langlands's conjectures in the unramified function field case, but where all of the objects have purely algebro-geometric nature. In this talk, I will describe on-going work with Dennis Gaitsgory settling this problem. I will aim to keep the talk as accessible as possible while emphasizing the new aspects of our work.

Video

Abstract: Vincent Lafforgue gave a construction of a semisimple Langlands parameter from an automorphic form for any reductive group over a function field by using excursion operators. Our aim is to give a general approach for proving certain local-global compatibilities satisfied by these Langlands parameters. The main consequence for the Langlands correspondence is to show that Lafforgue's construction is compatible with Lusztig's theory of character sheaves at a given point of a smooth curve over a finite field. Namely, using the theory of character sheaves, one attaches a torus character and a two-sided cell to an irreducible representation of a reductive group over a finite field. If our automorphic form lives in an isotypic component determined by this irreducible representation, we show that the torus character and two-sided cell determine the semisimple and unipotent parts of the image of the tame generator under the Langlands correspondence, respectively.
The main theorem in the general approach to local-global compatibility is that nearby cycles commute with pushforward of certain perverse sheaves from the stack of global shtukas to a power of a curve. To prove this result, the main technical ingredient is the notion of what we call $\Psi$-factorizability, where nearby cycles over a general base are independent of the composition of specializations chosen, and the $\Psi$-factorizability statements we make give some answers to a question raised by Genestier-Lafforgue. Using this theorem, to compute the action of framed excursion operators, one may instead compute the monodromy of certain nearby cycles sheaves on certain restricted shtukas. In our case of interest, perverse sheaves over restricted shtukas are related to the monodromic affine Hecke category by a horocycle correspondence. That is, one may reduce certain questions in the global function field Langlands program to questions in local geometric Langlands.

Video

Abstract: Given a tensor triangulated category T one can try to study it by establishing an “exhaustive” family of fiber functors $F_\lambda: T \to T_\lambda$ where the categories $T_\lambda$ are already “understood”. I’ll focus on two specific examples where this strategy leads to some (partial) information about the Balmer spectrum for the category T. One example deals with the stable category of lattices for a finite group scheme defined over a ring unifying the classical results of Hopkins, Neeman for a derived category of a commutative ring with more recent theorems in modular representation theory. The other considers the spectrum of a stable category of a small quantum group. Joint work with T. Barthel, D. Benson, S. Iyengar, H. Krause; and C. Negron.

Video

Abstract: Pre-Tannakian categories are a purely categorical axiomatization of the additional structure the category of finite dimensional representations of a group has. If a Pre-Tannakian category C admits a structure preserving "fiber functor" to vector spaces then Tannakian reconstruction says that necessarily C = Rep(G) for some group (scheme) G. However there are a number of interesting examples of pre-Tannakian categories which do not admit a fiber functor, and classifying them is an important open problem. We introduce a notion of a discrete pre-Tannakian category, with the property that Rep(G) is discrete if and only if G is a profinite group, and show that all discrete pre-Tannakian categories come from oligomorphic groups via a previous construction of ours.

Video

Abstract: We describe a new realization of Lusztig's asymptotic affine Hecke algebra $J$ in terms of coherent sheaves on a moduli stack of Deligne–Langlands parameters. More precisely, we show that $J$ arises from a certain restriction of the "coherent Springer sheaf," which is associated to the (usual) affine Hecke algebra via the categorical local Langlands correspondence. We will then explain how to "upgrade" this sheaf-theoretic realization of $J$ to a categorification using certain coherent sheaves on Springer fibers, following a conjecture of Qiu–Xi. Finally, we will use the asymptotic picture to motivate the result, independently announced by Hemo–Zhu, that the coherent Springer sheaf lies in cohomological degree $0$ (i.e., is a sheaf rather than a complex).

Video

Abstract: I will report on a joint project with Dan Ciubotaru, David Kazhdan and Yakov Varshavsky, in which we describe unipotent invariant distributions on a p-adic group in terms of the Langlands dual group. Some of our constructions are related to the results by Li, Nadler, Yun presented by Zhiwei Yun in his recent seminar talk.

Video

Abstract: We will exposit some tenets of the local geometric Langlands philosophy with concrete representation-theoretic consequences. In particular, we will focus on the problem of affine Beilinson-Bernstein localization at the critical level. Then, we will explain how to actually prove some of these statements. Some of this work is joint with Sam Raskin.

Video

Abstract: For characters of p-adic reductive groups there is the notion of wave-front set, which is a set of nilpotent orbits that describes the asymptotic behavior of a character near the identity. (Maybe one can think of the closure of wave-front set as singular support.) There is a long-standing conjecture that any wave-front set is contained in a single geometric orbit, as worked out by many authors for several types of depth-0 representations. In this talk, we explain how the above conjecture cannot hold in general, because an analogous assertion does not hold for graded Lie algebras. We will discuss this last statement in the context of graded Springer theory.

Slides

Video

Abstract: An important invariant for admissible representations of reductive p-adic groups is the wavefront set, the collection of the maximal nilpotent orbits in the support of the orbital integrals that occur in the Harish-Chandra-Howe local character expansion. We compute the geometric and Okada's canonical unramified wavefront sets for representations in Lusztig's category of unipotent reduction for a split group in terms of the Kazhdan-Lusztig parameters. We use this calculation to give a new characterisation of the anti-tempered unipotent Arthur packets. Another interesting consequence is that the geometric wavefront set of a unipotent supercuspidal representation uniquely determines the nilpotent part of the Langlands parameter; this is an extension to p-adic groups of Lusztig's result for unipotent representations of finite groups of Lie type. The talk is based on joint work with Lucas Mason-Brown and Emile Okada.

Video

Abstract: A conjecture of Hiraga, Ichino and Ikeda expresses the formal degree of a discrete series of a (algebraic) reductive group over a local field in terms of the adjoint gamma factor of its Langlands parameter. It can be checked for real reductive groups using work of Harish Chandra and the explicit form of the Langlands correspondence in this case. For classical groups over a p-adic field, the conjecture was established for odd orthogonal groups as well as unitary groups by two completely different methods. In this talk, I will explain a proof for symplectic and even orthogonal p-adic groups based on properties of twisted endoscopy and ideas originating from Shahidi relating residue of intertwining operators to twisted orbital integrals. This method can actually be readily adapted to treat odd orthogonal and unitary groups as well.

Slides

Video

Abstract: My talk is based on the joint work with E. Gorsky and O. Kivinen. I will explain a construction that associates a coherent sheaf on the Hilbert scheme of points on the plane to plane curve singularity. The global sections of the sheaf are equal to cohomology of the corresponding Affine (type A) Springer fiber. The construction categorifies HOMFLYPT homology/cohomogy of compactified Jacobian conjecture if combined with Soergel bimodule/ Sheaves of Hilbert scheme theorem of Oblomkov-Rozansky. I will also discuss generalizations outside of type A.

Video

Abstract: I'll start by summarizing the main results of the series [AGKRRV], where it is shown that the trace of Frobenius on the category of automorphic sheaves with nilpotent singular support identifies with the space of unramified automorphic functions. We'll then discuss conjectural counterparts of this statement in the local and global ramified settings.

Video

Abstract: Given a reductive group G and some auxiliary data, one has the Hitchin fibration associated with the adjoint action of G on Lie(G), which is successfully used by B. C. Ngô to prove the endoscopic fundamental lemma for Lie algebras. Following the same idea, there is a group analogue called the multiplicative Hitchin fibration by replacing the Lie algebra with reductive monoids, and one can hope to directly prove the fundamental lemma at group level. This project is close to completion and we report the results so far. There are many new features that are not present in the additive case, among which is a pleasant surprise that there might be some strata in the support theorem that are not explained by endoscopy.

Video

Abstract: This talk is based on the joint work with Jiajun Ma and Congling Qiu on theta correspondence of type I dual pairs over a finite field F_q. We study the Hecke algebra modules arising from theta correspondence between certain Harish-Chandra series for these dual pairs. We first show that the normalization of the corresponding Hecke algebra is related to the first occurrence index, which leads to a proof of the conservation relation. We then study the deformation of this Hecke algebra module at q=1 and generalize the results of Aubert-Michel-Rouquier and Pan on theta correspondence between unipotent representations along this way.

Video

Abstract: I will speak on joint work with Linus Hamann and Kieu Hieu Nguyen. For odd unramified unitary groups over a p-adic field, there is a Langlands correspondence using trace formula techniques due to Mok and Kaletha--Minguez--Shin--White. Another correspondence was constructed recently by Fargues--Scholze using p-adic geometry. We show these correspondences are compatible by first proving an analogous result for unitary similitude groups by studying the cohomology of local Shimura varieties. If time permits, we will discuss applications to the Kottwitz conjecture and eigensheaf conjecture of Fargues.

Video

Abstract: In joint work with Tom Braden we give a purely algebraic description of the category of perverse sheaves (with coefficients in any field) on $S^n(C^2)$, the n-fold symmetric product of the plane. In particular, using the geometry of the Hilbert scheme of points, we relate this category to the symmetric group and its representation ring. Our work is motivated by analogous structure appearing in the Springer resolution and Hilbert-Chow morphism..

Video

Abstract: We discuss new categorical interpretations of two distinct q-deformations of the rational numbers. The first one was introduced in a different context by Morier-Genoud and Ovsienko, and enjoys fascinating combinatorial, topological, and algebraic properties. The second one is a natural partner to the first, and is new. We obtain these deformations via boundary points of a compactification of the space of Bridgeland stability conditions on the 2-Calabi--Yau category of the A2 quiver. The talk is based on joint work with Louis Becker, Anand Deopurkar, and Anthony Licata.

Slides

Video

Abstract: In joint work with Ben Elias, David Rose, and Logan Tatham, we give a generators and relations presentation of the monoidal category generated by fundamental representations of the symplectic group (and its q-analogue). I will discuss our work, which generalizes both the Temperely-Lieb algebra description of representations of SL_2 and Kuperberg’s ``spider" description of representations of Sp_4. I will then comment on joint work in progress with Haihan Wu which gives an analogous presentation of the monoidal category generated by exterior powers of the vector representation of the orthogonal group (and its q-analogue).

Video

Abstract: We define an analogue of Category O in the context of Kazhdan and Laumon's 1988 gluing construction for perverse sheaves on the basic affine space G/U. We explicitly classify its simple objects, and then use our understanding of the structure of this category to discuss its connections to some other interesting objects in representation theory, namely Braverman-Kazhdan's Schwartz space on G/U and perverse sheaves on the semi-infinite flag variety.

Video