The Hecke-Baxter operators via Heisenberg group extensions

The $G{L}_{\mathrm{\ell }+1}(\mathbb{R})$ Hecke-Baxter operator was introduced as an element of the $O_{\mathrm{\ell }+1}$-spherical Hecke algebra associated with the Gelfand pair $O_{\mathrm{\ell }+1}\subset G{L}_{\mathrm{\ell }+1}(\mathbb{R})$. It was specified by the property to act on an $O_{\mathrm{\ell }+1}$-fixed vector in a $G{L}_{\mathrm{\ell }+1}(\mathbb{R})$-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 $G{L}_{\mathrm{\ell }+1}(\mathbb{R})\times G{L}_{\mathrm{\ell }+1}(\mathbb{R})$ 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 $S{p}_{2\mathrm{\ell }+2}(\mathbb{R})\times S{p}_{2\mathrm{\ell }+2}(\mathbb{R})$ by a Heisenberg Lie group.