北京雁栖湖应用数学研究院

I will talk about (possible) conjectures on constructions of automorphic kernel functions, with which the integral transforms may yield information about automorphic $L$-functions as the Langlands conjecture asserts or about the Langlands functorial transfers. 

If $G$ is a finite reductive group, Deligne-Lusztig theory aims to construct representations of $G$ on the $\ell$-adic cohomology of certain $G$-varieties (now called Deligne-Lusztig varieties). Knowing the endomorphism algebra of such representations would be a first step in a complete understanding of their decomposition into irreducible components. We present here a way to construct many elements of these endomorphism algebras, involving a categorical action of the braid group on the derived category of constructible sheaves on the flag variety.

The Gan-Gross-Prasad conjecture is one of the most important problems in the field of automorphic forms, relating automorphic periods for classical groups to $L$-functions. It has a wide spectrum of applications to number theory, arithmetic geometry, and representation theory. In this talk, we will survey the main progress of this conjecture (for which the unitary case has been completely solved in characteristic zero) together with its major applications observed in recent years.

Modular kernels are two-variable meromorphic functions defined on the upper half-plane that satisfy specific modularity conditions and additional properties for congruence subgroups. The modular kernels for $SL(2, Z)$ were studied by Duke and Jenkins in 2008. Those for the theta group were introduced by Radchenko and Viazovska in their proof of the Fourier interpolation formula. More sophisticated modular kernels were introduced by Cohn, Kumar, Miller, Radchenko, and Viazovska in their recent work on the universal optimality of the $E(8)$ and Leech lattices. In contrast, Beta-Gamma systems represent some of the simplest examples of non-rational conformal field theories. In this talk, we will introduce Beta-Gamma systems twisted by finite dimensional representations of the modular group $SL(2, Z)$ and show that their 2-point correlation functions are exactly the modular kernels. This connection raises numerous intriguing questions relevant to both fields, which we aim to explore.

The Gelfand-Kirillov dimension is an important notion in the study of modules over non-commutative noetherian rings. In this talk, I will explain this notion for mod p or (locally analytic) $p$-adic representations of $p$-adic groups, and discuss some recent progress and open questions. 

We will discuss the structure of representations for (quantum) affine Lie algebras obtained by functors of parabolic induction and twisted localization.

We generalize S. Gelfand's classical construction of a Novikov algebra from a commutative differential algebra to get a deformation family $(A,\circ_q)$ of Novikov algebras by an admissible commutative differential algebra, which ensures a bialgebra theory of commutative differential algebras, enriching the antisymmetric infinitesimal bialgebra. Moreover, a deformation family of Novikov bialgebras is obtained, under certain further condition. In particular, we obtain a bialgebra variation of S. Gelfand's construction with an interesting twist: every commutative and cocommutative differential antisymmetric infinitesimal bialgebra gives rise to a Novikov bialgebra whose underlying Novikov algebra is $(A,\circ_{-\frac{1}{2}})$ instead of $(A,\circ_0)$ which recovers the construction of S. Gelfand. This is the joint work with Yanyong Hong and Li Guo.

Deformed W-algebras are two parameter algebras associated to a simple Lie algebra g, obtained from fields commuting with screening operators. We discuss some remarkable specializations of deformed W-algebras : The first classical limit is known to be isomorphic to the center of the quantum affine affine algebra at the critical level. The second classical limit is not well understood. We interpret this limit as a “folding” of the Grothendieck ring of finite-dimensional representations of a quantum affine algebra associted to the simply-laced Lie algebra g′ (corresponding to g). We construct a corresponding novel quantum integrable model. We conjecture, and verify in a number of cases, that its spaces of states of can be identified with finite-dimensional representations of the Langlands dual quantum affine algebra (joint work with E. Frenkel and N. Reshetikhin). Third (mixed) limit : we use this limit to state a general conjecture on the parametrization of simple modules of non simply-laced quantized Coulomb branches (and of truncated shifted quantum affine algebras). We have several evidences, including a general result for simple finite-dimensional representations.

In 1981, Drinfeld enumerated the number of irreducible l-adic local systems of rank two on a projective smooth curve fixed by the Frobenius endomorphism. Interestingly, this number looks like the number of points on a variety over a finite field. Deligne proposed conjectures to extend and comprehend Drinfeld's result. By the Langlands correspondence, it is equivalent to count certain cuspidal automorphic representations over a function field. I will present some counting results where we connect counting to the number of stable Higgs bundles using Arthur's non-invariant trace formula.

The Gaiotto conjecture aims to realize the category of representations of a basic classical quantum supergroup as a certain twisted D-module category on the affine Grassmannian. It is the quantum analog of a class of examples of the local conjecture of Ben-Zvi-Sakellaridis-Venkatesh in the relative Langlands program. In this talk, we will introduce the precise statement of the Gaiotto conjecture and the recent progress. This talk is based on joint works with Michael Finkelberg and Roman Travkin.

Representation theory is an important branch of mathematics that seeks to understand the structure of groups by representing their elements as linear maps. With the emergence of the Local Langlands program, the representation theory of $p$-adic groups has become one of the most prominent topics in recent almost 50 years. At the end of the last century, Vignéras proposed studying characteristic $l$ representations (where $l$ different from $p$) via complex representation theory, also known as modular $l$-representations. Although these two kinds of representations share many fundamental properties, recent research has shown that their categorical structures exhibit significant differences. In particular, the $l$-modular block decomposition is only known for a few groups. In this talk, we will introduce the l-modular block decomposition for $GL_n​$ and $SL_n$​, compare it with complex representations, and discuss predictions for general groups