On universal Baxter operator for classical groups

TL;DR

This paper constructs universal Baxter operators for classical groups SO_{2 ext{ extmu}l} and Sp_{2 ext{ extmu}l} within the Archimedean spherical Hecke algebra, extending previous work on GL_{l+1}(R).

Contribution

It introduces universal Baxter operators for classical groups, expanding the framework beyond GL_{l+1}(R) using integral representations of local L-factors.

Findings

Constructed Baxter operators for SO_{2 ext{ extmu}l} and Sp_{2 ext{ extmu}l}.

Connected Baxter operators with local Archimedean L-factors.

Extended the theory of Baxter operators to classical groups.

Abstract

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. Recently such operators were introduced for G=GL_{\ell+1}(R) as generalizations of the Baxter operators arising in the theory of quantum Toda chains. In this note we provide universal Baxter operators for classical groups SO_{2\ell}, Sp_{2\ell} using the results of Piatetski-Shapiro and Rallis on integral representations of local Archimedean L-factors.