Time: 14:00-15:30, Thursday, July 31, 2025
Venue: E10-212, Yungu Campus
Speaker: Professor Sergei Oblezin, BIMSA, Tsinghua University
Title: Automorphic L-functions and Hecke-Baxter operators
Abstract: It is well-known that for an admissible representation of the split real group GL(n,R), there is a canonically attached local Archimedean L-factor, a function in a single complex variable capturing data of the original representation. The Archimedean L-function is essentially given by a product of Gamma-functions. This form is motivated by deep connections with the theory of zeta-functions of the global number field. In particular, the local L-factors of representations of non-compact groups play an essential role in formulation of the Langlands correspondence for these groups. However, the standard construction of the Archimedean L-factors follows a round-about strategy.
In my talk, I first introduce the Hecke-Baxter operator, a one-parameter family of elements of spherical Hecke algebra associated with the Gelfand pair (GL(n,R), O(n)). This operator is specified by the property that its action on a O(n)-fixed vector in a principal series GL(n,\mathbb{R})-representation reproduces the Archimedean L-function attached to this representation as the eigenvalue. I review representation theoretic properties of the Hecke-Baxter operator and discuss generalizations to some other classes of GL(n,R)-representations. Finally, I introduce a global version of the Hecke-Baxter operator acting on automorphic forms with eigenvalues given by (products of) zeta-functions.
This talk is a report on an ongoing joint project with A.Gerasimov and D. Lebedev devoted to Archimedean ($\infty$-adic) arithmetic geometry and representation theory.
