The Hecke-Baxter operators via Heisenberg group extensions

TL;DR

This paper introduces a new perspective on the Hecke-Baxter operator by relating it to generalized Whittaker functions on extended Lie groups, providing a novel geometric interpretation.

Contribution

It proposes a new definition of the Hecke-Baxter operator using generalized Whittaker functions on extended Lie groups, linking representation theory and geometric structures.

Findings

Identifies the Hecke-Baxter operator with a generalized Whittaker function.

Shows how to lift this Whittaker function to a matrix element of an extended symplectic group.

Provides a new geometric interpretation of the Hecke-Baxter operator.

Abstract

The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator was introduced as an element of the $O_{\ell+1}$-spherical Hecke algebra associated with the Gelfand pair $O_{\ell+1}\subset GL_{\ell+1}(\mathbb{R})$. It was specified by the property to act on an $O_{\ell+1}$-fixed vector in a $GL_{\ell+1}(\mathbb{R})$-principal series representation via multiplication by the local Archimedean $L$-factor canonically attached to the representation. In this note we propose another way to define the Hecke-Baxter operator, identifying it with a generalized Whittaker function for an extension of the Lie group $GL_{\ell+1}(\mathbb{R})\times GL_{\ell+1}(\mathbb{R})$ by a Heisenberg Lie group. We also show how this Whittaker function can be lifted to a matrix element of an extension of the Lie group $Sp_{2\ell+2}(\mathbb{R})\times Sp_{2\ell+2}(\mathbb{R})$ by a Heisenberg Lie group.