MIT Lie Groups Seminar

Abstract: It's an old idea of Harish-Chandra that representations of a real reductive G can be studied by understanding their restrictions to a maximal compact K. Traditionally this was done using the Cartan-Weyl highest weight theory for K representations.

I will recall a (closely related but very different) description of K representations, in which the highest weight is replaced by the Harish-Chandra parameter of a discrete series on a Levi subgroup of G. Using this new parameter, one can define a very natural "height" of a representation of K, taking non-negative integer values. As evidence that height is natural, I offer

Conjecture. Suppose G is split, and pi_1 and pi_2 are two (minimal) principal series representations of G. If pi_1 and pi_2 have the same restriction to the center of G, then the sets of heights of K-types of pi_1 and of pi_2 are the same.

For example, split G_2 has principal series having two different restrictions to K. In each, the first heights appearing are

0, 3, 6, 9, 10,12, 15...

F4 has principal series representations having three different restrictions to K. But in all three, the first heights appearing are

0, 8, 11, 15,16, 21, 24, 26, 27, 29, 30, 32, 33, 36....

The numbers for split E_8 (which also has three types of principal series) are

0, 29, 46, 57, 58, 68, 75, 84, 87, 91, 92...

It seems more or less impossible to find a similar statement using highest weights. I will give some evidence for this conjecture.

Of course there should be an elementary construction of these sequences of heights from the root system of G and the central character; I have no idea how to make such a construction.

If there is time, I will talk about the possibility of finding parallel statements for p-adic groups.

Abstract: We explain a uniform construction of cuspidal character sheaves on Z/mZ-graded Lie algebras. We show that they all arise from the Fourier-Sato transform of nearby cycle sheaves associated to very special supercuspidal data. These data essentially come from Lusztig’s cuspidal character sheaves on (ungraded) reductive Lie algebras. We make use of Lusztig-Yun’s work on graded Lie algebras where all simple perverse sheaves on the nilpotent cone (Fourier-Sato transforms of character sheaves) are produced via spiral induction. This is based on joint work with Wille Liu, Cheng-Chiang Tsai and Kari Vilonen.

Abstract: In this talk, we will explore how the Khovanov-Rozansky homology of the (m,n)-torus knot can be extracted from the finite-dimensional representation of the rational Cherednik algebra at slope m/n, equipped with the Hodge filtration. This result confirms a conjecture of Gorsky, Oblomkov, Rasmussen, and Shende. Our approach involves constructing a family of coherent sheaves on the Hilbert scheme of points on the plane, coming from cuspidal mirabolic D-modules.

Abstract: A very important result of Lusztig asserts that characters of Deligne--Lusztig representations are obtained by the sheaf-to-function correspondence from character sheaves. The goal of my talk is to outline a proof of a generalization of this result using the categorical trace machinery. This is a joint work in progress with Dennis Gaitsgory and Nick Rozenblyum.

Abstract: A microlocal approach to Arthur theory for p-adic groups was revisited by Ciubotaru-Mason-Brown-Okada, suggesting that some endoscopic Arthur packets can be closely approximated by an analogue: The collection of irreducible representations whose Gelfand-Kazhdan dimension is minimal among those that admit a prescribed infinitesimal character.

In joint work with Emile Okada, we refined this heuristic into a precise characterization for split symplectic and odd orthogonal groups. A key added ingredient are the 'weakly spherical' representations - those admitting non-zero vectors invariant under a maximal compact, not necessarily hyperspecial, subgroup.

Roughly, we discovered that the microlocal packets are unions of those endoscopic packets that contain a weakly spherical representation, with the union's structure governed by nilpotent cone geometry. In particular, by relating Arthur parameters to Springer theory, it appears that weak sphericity corresponds to parameters factoring through Lusztig's canonical quotient associated with the relevant nilpotent orbit.

Abstract: I’ll describe an ongoing project, joint with Arnaud Eteve, Alain Genestier and Vincent Lafforgue. We start with an algebro-geometric incarnation of the Fargues-Scholze D(Bun_G), given by the category of sheaves on the loop group equivariant with respect to Frobenius-twisted conjugacy, and our goal is to define on it a ‘spectral’ action of QCoh of the stack of arithmetic local systems. This is achieved by a fusion procedure. In the process one has to resolve difficulties associated with taking nearby cycles along a multi-dimensional base.

Abstract: We will discuss work, joint with Victor Ginzburg, on certain quantizations (or non- commutative deformations) of objects and morphisms of interest in the geometric Lang- lands program. First, we will discuss the quantization of a morphism of group schemes used by Ngô in his proof of the fundamental lemma, which confirms a conjecture of Nadler. Afterwards, we will discuss how this morphism can be used to construct a D-module ana- logue of a recent equivalence proved by Bezrukavnikov–Deshpande in the ℓ-adic setting, identifying a certain braided monoidal subcategory of the category of G-equivariant D- modules on G known as vanishing D-modules with the category of a modules for a ring known as the spherical nil-DAHA. We will also explain the construction of a certain bi- module isomorphism used to construct this braided monoidal equivalence, whose existence was originally conjectured by Ben-Zvi–Gunningham.
Time permitting, we will also discuss another application of our methods: a proof of a conjecture of Braverman and Kazhdan, known as the exactness conjecture, in the D- module setting.

Abstract: We formulate a new version of the local Langlands correspondence for discrete L-parameters which involves (Weyl orbits of) packets of representations of all twisted Levi subgroups of a connected reductive group G through which a given parameter factors and prove that this version of the correspondence is true if one assumes the pre-existing local Langlands conjectures. Twisted Levi subgroups are crucial objects in the study of supercuspidal representations; this work is a step towards deepening the relationship between the representation theory of p-adic groups and the Langlands correspondence. This talk will also serve as a rough introduction to the refined local Langlands correspondence (and its generalizations). This is joint work with David Schwein.

Abstract: A symplectic reflection group $W$ is a finite subgroup of $\mathrm{Sp}(V)$ the group of automorphisms of a symplectic vector space which is generated by symplectic reflections (\emph{i.e.} a non-trivial element of $\mathrm{Sp}(V)$ whose fixed-point locus has maximal dimension). To such a group, Etingof and Ginzburg attached an associative algebra, $\mathcal S(W,V)$, the symplectic reflection algebra. We will describe a natural class of isomorphisms between such algebras which we (unimaginatively) call admissible.

In the case where $W$ is a complex reflection group and $V = \mathfrak h \oplus \mathfrak h^*$ is the cotangent bundle of a reflection representation of $W$, the symplectic reflection algebra is a rational Cherednik algebra $\mathcal H(W, \mathfrak h)$, and the notion of admissibility is a rigid enough that we can completely classify its admissible star operations. When $W$ is a real reflection group such operations are a torsor for $\mathrm{SL}_2(\mathbb C)$ but when $W$ is non-real, it turns out that there are essentially only two such star operations. Time permitting we will discuss how these interact with the "trigonometric" presentation of $\mathcal H(W,\mathfrak h)$ and conjectures of Etingof and Shelley-Abrahamson on unitary representations of rational Cherednik algebras.

Abstract: Kottwitz introduced certain compact ''fake'' unitary group Shimura varieties and determined their local Hasse-Weil zeta functions at primes of good reduction. For primes $p$ where the level is arbitrarily deep, the local Hasse-Weil zeta functions were further studied in the case of signature $(1,n-1)$ and $p$-adic uniformization (by Xu Shen), and in the case of arbitrary signature but under the assumption that the group at $p$ is a product of Weil restrictions of general linear groups (by Scholze and Shin). In this talk, I will explain joint work with Jingren Chi in which we generalize the above works by allowing the group at $p$ to be any inner form of a product of Weil restrictions of general linear groups. New phenomena arise when the group at $p$ is not quasi-split, both in geometry and in harmonic analysis. A key result is a crucial vanishing property of twisted orbital integrals of Scholze's test functions at $p$. This talk will try to highlight the difficulties which arise and explain how they are resolved for these special examples of Shimura varieties.

Abstract: Trigonometric Dunkl-Cherednik operators provide a representation of Cherednik’s trigonometric double affine Hecke algebra. We present a the- orem of existence and uniqueness of ”nonsymmetric shift operators” for these Dunkl-Cherednik operators. These are trigonometric differential- reflection operators which shift the parameters of the Dunkl-Cherednik operators by integers, and which restrict on the space of W -invariant poly- nomials to the well known hypergeometric shift operators. The latter are instrumental in the theory of symmetric Macdonald polynomials.

Joint work with Valerio Toledano Laredo

Abstract: I will discuss certain global deformations of Hecke categories, constructed using sheaves with infinite dimensional stalks. The tame local Betti Langlands equivalence is an extension of Bezrukavnikov's equivalence, that provides a coherent realization of the universal monodromic affine Hecke category. This is joint work with Gurbir Dhillon.

Abstract: The Connes-Kasparov isomorphism is a statement in $C^*$-algebra K-theory about the reduced $C^*$-algebra of an almost-connected Lie group. Work of Vincent Lafforgue in the 1990’s led to a proof of the isomorphism using index theoretic techniques, so it is worthwhile to ask what, if anything, the isomorphism has to say about representation theory? Because $C^*$-algebra $K$-theory is about projective, rather than irreducible, modules, the answer isn’t evident. But Lafforgue himself recovered Harish-Chandra’s classification of the discrete series of a real reductive group from the isomorphism. In this talk I shall present a different answer, which involves David Vogan’s tempiric representations of a real reductive group (that is, tempered irreducible representations with real infinitesimal character). The new assertion is in fact equivalent to the Connes-Kasparov isomorphism for real reductive groups. This is joint work with Bob Yuncken and Jacob Bradd.