The $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator: principal series representations

TL;DR

This paper extends the $GL_{ell+1}(bR)$ Hecke-Baxter operator to non-spherical principal series representations, generalizing key algebraic structures and showing its action corresponds to Archimedean $L$-factors.

Contribution

It introduces a generalization of the Hecke-Baxter operator to non-spherical principal series, expanding the algebraic framework for $GL_{ell+1}(bR)$ representations.

Findings

The generalized Hecke-Baxter operator acts by multiplication with Archimedean $L$-factors.

Extension of spherical concepts to non-spherical principal series representations.

Provides a broader algebraic structure for analyzing $GL_{ell+1}(bR)$ representations.

Abstract

Previously introduced the $GL_{\ell+1}(\mathbb{R})$ Hecke-Baxter operator is a one-parameter family of elements in the commutative spherical Hecke algebra $\mathcal{H}(GL_{\ell+1}(\mathbb{R}),O_{\ell+1})$. Its action on spherical vectors in spherical principle series representations of $GL_{\ell+1}(\mathbb{R})$ is given by multiplication by the Archimedean $L$-factors associated to these representations. In this note we propose an extension of the construction to other (non-spherical) $GL_{\ell+1}(\mathbb{R})$ principle series representations providing a relevant generalization of the notions of spherical vector, commutative spherical Hecke algebra and the Hecke-Baxter operator to the general case. Action of the introduced Hecke-Baxter operator on the generalized spherical vectors is given by multiplication by the Archiemdean $L$-factor associated to the corresponding principle series representation of $GL_{\ell+1}(\mathbb{R})$.