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Seminar by Dimitri Lebedev (ITEP) I

Title: Topological field theories and local Archimedean Langlands correspondence (1/3)

2013.10.21 | Christine Dilling

Date Wed 15 Jun
Time 11:00 12:00
Location Aud. D2


Archimedean local L-factors were introduced to simplify functional equations of global L-functions. From the point of view of arithmetic geometry these factors complete the Euler product representation of global L-factors. A known construction of non-Archimedean local L-factors is rather transparent and uses characteristic polynomials of the image of the Frobenius homomorphism in finite-dimensional representations of the local Weil-Deligne group closely related to the local Galois group. On the other hand, Archimedean L-factors are expressed through products of Gamma-functions and thus are analytic objects avoiding simple algebraic interpretation. In the series of lectures we approach the problem of proper interpretation of Archimedean L-factors in terms of topological field theory.

In this first Lecture we introduce certain universal Baxter operators for classical Lie groups. The universal Baxter operator is an element of the Archimedean spherical Hecke algebra H(G;K), K be a maximal compact subgroup of a Lie group G. It has a defining property to act in spherical principle series representations of G via multiplication on the corresponding local Archimedean L-factors. Being restricted to the maximal Cartan torus it coincides with the Baxter operators for different quantum integrable systems. We consider in details the case of the Lie group G = GLn (R) and show the explicit relation with the Baxter operators arising in the theory of quantum Toda chains. We use the Baxter operator to construct the Toda chain eigenfunctions in the form of stationary phase integral proposed by Givental. We provide also
universal Baxter operators for classical groups SO2n , Sp2n using the results of Piatetski-Shapiro and Rallis on integral representations of local Archimedean L-factors.

This lecture is based on common papers with A.Gerasimov and S.Oblezin:

1. Baxter operator and Archimedean Hecke algebra, Commun. Math. Phys. 284:3 (2008) 867-896; [math.RT/0706.347]

2. Archimedean L-factors and Topological Field Theories I in Communications in Number Theory and Physics, v 5, no 1, 2011; [math.NT/0906.1065]

3. On universal Baxter operators for classical groups [math. RT /1104.0420]