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

Abstract

Previously introduced the $G{L}_{\ell +1}(\mathbb{R})$ Hecke-Baxter operator is a one-parameter family of elements in the commutative spherical Hecke algebra $\mathcal{H}(G{L}_{\ell +1}(\mathbb{R}),O_{\ell +1})$. Its action on spherical vectors in spherical principle series representations of $G{L}_{\ell +1}(\mathbb{R})$ 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) $G{L}_{\ell +1}(\mathbb{R})$ 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 $G{L}_{\ell +1}(\mathbb{R})$.

25 pages; minor typos are fixed and important references added