MIT Lie Groups Seminar

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: I will discuss joint work with Gaitsgory and V. Lafforgue on the structure and spectral theory of unramified automorphic forms over function fields. In particular, we will explain how the Arthur-Ramanujan conjecture can be proved using the general theory.

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.