MIT Lie Groups Seminar

Abstract: For an affine symplectic variety in characteristic $0$ Kontsevich classified its deformation quantizations. Later, Bezrukavnikov and Kaledin gave a classification of deformation quantizations of an arbitrary symplectic variety $X$ (under some mild assumptions), and generalized the picture to positive characteristic. In the case when $H^i(X, O_X)=0$ for $i=1,2,3,$ they constructed a distinguished quantization $O_h$.

However, this construction is not functorial: the algebra $O_h$ is not equivariant with respect to automorphisms of $X$. Moreover, the distinguished quantization may not exist without the cohomology vanishing assumption.

I will explain a construction of a functorial in $X$ quantization $QCoh_h$ of the category of quasi-coherent sheaves on $X$.

If time permits, given a Lagrangian fibration $X \rightarrow Y$, I will discuss an explicit description of $QCoh_h$.

The talk is based on the joint work with D. Kubrak, R. Travkin, and V. Vologodsky.

Abstract: Smith-Treumann localization to the fixed points of a cyclic group action has proven to be a powerful tool in representation theory (e.g. Leslie-Lonergan, Riche-Wiliamson,...). However, the scope of its applications is limited by strict conditions, in particular on the characteristic of the field of coefficients. In the talk I'll propose a different variant of the localization to the fixed points of a cyclic group action, which on the one hand maintains some of the nice properties of Smith-Treumann localization, while on the other hand is applicable in a different range of settings. 

Abstract: In a letter to Howe in 1975, Langlands speculated that the theta correspondence could be an instance of functoriality. This later turned out to be false as the theta correspondence does not always preserve $L$-packets. In 1989, Adams proposed a remedy: instead of $L$-packets, the theta correspondence should preserve $A$-packets. Mœglin showed that Adams was mostly correct, but also that there are examples where Adams' conjecture fails: there exists a representation lying in an $A$-packet whose theta lift does not lie in any $A$-packet. The goal of this talk is to generalize Adams' conjecture to $ABV$-packets (also called micro packets). We will discuss how the generalization explains the failure of Adams' conjecture, along with providing some evidence for the generalization.

Abstract: The view from 10,000 meters is that the unitary dual of a reductive algebraic group over a local field should consist of the representations whose existence Arthur conjectured in the 1980s, together with others arising by deformation. Here is a precise conjecture in that direction:

CONJECTURE. Suppose $G$ is a real reductive algebraic group, and $\pi$ is a unitary representation of $G$ having integral infinitesimal character. Then $\pi$ is an Arthur representation.

The conjecture is true for many classical groups up to rank 6. My guess is that it is true for ALL classical groups (and that the technology of the experts is sufficient to prove that).

I have checked that it is true for all representations of split $G_2$; for all but two representations of split $F_4$; for all but six representations of split $E_7$; and all but 27 representations of split $E_8$. It is true for the complex forms of $G_2$, $F_4$, and $E_6$.

I will say a little about why the conjecture is plausible, and where the counterexamples come from.

Related elementary problem: suppose $H \subset G$ is complex reductive, and $\gamma$ in $X^*(H)$ is a weight. Give a simple algorithm to calculate all the maximal proper Levi subgroups $L \supset H$ so that $\gamma$ is in the $\mathbb{Q}$-span of the roots of $L$.

Abstract: Let $G$ be a connected reductive group defined and split over a finite field $F_q$ and such that the longest element in the Weyl group $W$ of $G$ is central in $W$. Let $U$ be the set of isomorphism classes of unipotent representations of the finite group $G(F_q)$. We define an involution $\xi\mapsto\xi^!$ of $U$ such that, for any $\xi\in U$, the dimension of $\xi^!$ (a polynomial in $q$ with rational coefficients) is obtained (up to sign) by changing $q$ to $-q$ in the dimension of $\xi$ (also a polynomial in $q$ with rational coefficients). This is a recent joint work with P.Deligne.

Abstract: Let $G$ and $G^{\vee}$ be Langlands dual semisimple complex Lie groups, and $\mathcal{N}$ and $\mathcal{N}^\vee$ be the respective nilpotent cones. It is expected that $\mathcal{N}$ and $\mathcal{N}^\vee$ form a pair, satisfying the properties of the conjectural symplectic duality. For such symplectic dual pair there are natural questions concerning the relation between symplectic leaves of the dual varieties. In the example of nilpotent cones, the answers are connected with certain duality maps, the best known one is the order reversing duality map (BVLS duality) between the sets of special orbits, studied by Barbasch and Vogan, and by Lusztig and Spaltenstein. Extending the image to non-special orbits, there are duality maps by Sommers and Achar, generalizing BVLS duality. In this talk I will explain how these duality maps (or their refined versions) fit into the context of symplectic duality between $\mathcal{N}$ and $\mathcal{N}^\vee$. It is based on the joint works with Ivan Losev, Lucas Mason-Brown, Shilin Yu and the ongoing project with Shilin Yu.

Abstract: Let $G$ be a connected reductive group over a field of positive characteristic. In two recent papers, P. Sobaje proved some remarkable properties of expressions of the form "(character of a not-to-small tilting $G$-module)/(character of the Steinberg module)." In this talk, I will review Sobaje's results, and I will explain a geometric incarnation of these results in terms of parity sheaves and "Smith-Treumann localization" on the dual affine Grassmannian. If time permits, I will also discuss connections to some homological phenomena in the derived category of G-modules. This is joint work with P. Sobaje.

Abstract: Let $F$ be a local field of characteristic zero. The general linear groups $GL_n(F)$ and $GL_m(F)$ act naturally on the space $M_{m,n}(F)$ of $m\times n$ matrices, with associated representation $\omega = C_c^\infty(M_{m,n}(F))$, the space of compactly supported functions on $M_{m,n}(F)$. Given an irreducible representation $\pi$ of $GL_n(F)$, a basic problem is to describe the structure of the big theta lift $\Theta(\pi)$, the maximal $\pi$-isotypic quotient of $\omega$, as a representation of $GL_m(F)$. This may be viewed as a kind of transcendental invariant theory, and the case $m=n=1$ already appears in Tate’s thesis. The problem has been studied extensively (by Howe, Mínguez, Fang-Sun-Xue, among others), yet it is still not fully understood.

Following ideas of Adams–Prasad–Savin, one can enrich the picture by considering derived theta lifting. In this talk I will discuss some recent progress in this direction, and highlight the phenomenon relating the vanishing of higher theta lifts and the holomorphic of the Godement–Jacquet $L$-function of $\pi$ and $\pi^\vee$ at certain critical points. This is based on joint work with Rui Chen, Yufeng Li, Xiaohuan Long, and Chenhao Tang.

Abstract: The microlocal point of view was introduced by M. Sato in the 1960s for studying partial differential equations. It was then adopted by M. Kashiwara and P. Schapira and developed into a systematic theory in the context of sheaves on manifolds. The theory has since had applications in many fields, including partial differential equations, symplectic geometry, geometric Langlands, and exponential sums. In this talk, I will explain the basic ingredients of this theory, and discuss recent development of its analogues in the positive characteristic as well as non-Archimedean contexts.

Abstract: The Rankin—Selberg integral formula (for G=GL_n * GL_{n-1} and H=GL_{n-1}) is a formula relating integrals of Hecke eigenforms of G along H to the L-function of the corresponding Galois representation (called the Rankin—Selberg L-function). In this talk, I will introduce a generalization of this formula over function fields in the everywhere unramified setting, relating self-intersection numbers of Rankin—Selberg cycles on the moduli of G-Shtukas to higher derivatives of the Rankin—Selberg L-function. This generalized formula can be regarded as a higher-dimensional analogue of Yun—Zhang’s higher Gross—Zagier formula.

Abstract: Kazhdan-Lusztig-Vogan polynomials provide fairly complete information about the singularities of K-orbit closures in flag varieties; but in view of their complexity, it is often convenient to have weaker information available in a more explicit form. The combinatorial notion of pattern avoidance provides a tool for doing this. I will characterize smoothness and rational smoothness of K-orbit closures via pattern avoidance in all classical cases. The conditions exhibit many similarities to but also intriguing differences from their counterparts for Schubert varieties.

Abstract: G-shtukas (classical and p-adic) for a reductive group G and their moduli spaces play an important role in the Langlands program (equi-characteristic and p-adic). Similarly, moduli spaces of G-displays can be viewed as period spaces for Shimura varieties in mixed characteristic. In this talk I will explain a general formalism how to construct moduli spaces of G-bundles on certain algebraic stacks. I will apply this formalism to construct and study moduli spaces of (truncated) G-shtukas, (truncated) G-displays, and (truncated) prismatic Breuil-Kisin-Fargues modules, focussing on the case of local G-shtukas.

Abstract: A surprising prediction of the geometric Langlands conjecture is that the category of automorphic sheaves with nilpotent singular support is independent of the complex structure of the underlying curve. In this talk, we give a gentle introduction to the main objects appearing in the Betti geometric Langlands correspondence and then present (the idea behind) a proof of the Ben-Zvi-Nadler conjecture not relying on the geometric Langlands conjecture.

Abstract:A natural enlargement of the BGG Category O for a semisimple Lie algebra is the category of weight modules with trivial central character and finite-dimensional weight spaces supported on the root lattice. We will present a new geometric realization of this category in terms of gluing sheaves on the flag variety; this realization is Koszul dual to a well-studied gluing construction of Kazhdan and Laumon. We will explain its relationship to a proposed Koszul duality relating the small quantum group and the semi-infinite flag variety to the geometry of affine Springer fibers.

Abstract:Just as local arithmetic Langlands studies vector spaces with an action of a group over a local field, local geometric Langlands studies categories with an action of a group over $k((t))$, where $k$ is an algebraically closed field. We explain a partial classification, joint with Gurbir Dhillon and Yakov Varshavsky, which gives an explicit "restricted" local geometric Langlands correspondence for $k$ not of small positive characteristic. This can be used (joint additionally with Arnaud Eteve, Dennis Gaitsgory, and Sam Raskin) to give an explicit equivalence of Fargues-Scholze type over function fields of sufficiently large characteristic.

Abstract: Shifted Yangians are a family of algebras which can be used to quantize slices in the affine Grassmannian. They admit a family of coproducts which make the direct sum of their category $\mathcal{O}$s into a monoidal category. I will give a description of the Grothendieck ring of this monoidal category. In particular, I will explain that it is isomorphic to the Cox ring of an infinite Bott-Samelson variety.

Abstract: An influential conjecture of Finkelberg--Mirkovic identifies the regular block of representations of a reductive group in positive characteristic with Iwahori constructible perverse sheaves on the affine Grassmannian. We will review the statement and history of this conjecture, including its recent resolution by Bezrukavnikov--Riche. We will then discuss a second conjecture, made at the same time by Finkelberg--Mirkovic, which identifies the regular block of representations of a Frobenius kernel with Iwahori constructible perverse sheaves on the semi-infinite flag variety, as well as its proof, which is work in progress with Achar, Riche, and Taylor.