Baxter operator and Archimedean Hecke algebra

TL;DR

This paper introduces Baxter Q-operators for Lie algebras, linking Whittaker functions, integral representations, and Hecke algebras, and proves related conjectures with explicit eigenvalues connected to L-factors.

Contribution

It constructs Baxter Q-operators for specific Lie algebras and connects them to integral representations and Hecke algebra elements, providing new proofs of conjectures.

Findings

Baxter Q-operators for gl(n+1) and so(2n+1] introduced.

Eigenvalues expressed via Gamma-functions and related to L-factors.

Provided a short proof of Friedberg-Bump and Bump conjectures.

Abstract

In this paper we introduce Baxter integral Q-operators for finite-dimensional Lie algebras gl(n+1) and so(2n+1). Whittaker functions corresponding to these algebras are eigenfunctions of the Q-operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G=GL(n+1) proved earlier by Stade. We also identify eigenvalues of the Baxter Q-operator acting on Whittaker functions with local Archimedean L-factors. The Baxter Q-operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra H(G(R),K), K being a maximal compact subgroup of G. Finally we stress an analogy between Q-operators and certain elements of the non-Archimedean Hecke algebra H(G(Q_p),G(Z_p)).