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 principal series representations of \(GL_{\ell +1}(\mathbb {R})\) is given by multiplication by the Archimedean L-factors associated with these representations. In this note, we propose an extension of the construction to other (non-spherical) \(GL_{\ell +1}(\mathbb {R})\) principal 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 Archimedean L-factor associated with the corresponding principal series representation of \(GL_{\ell +1}(\mathbb {R})\) .
