The Hecke-Baxter operators via Heisenberg group extensions

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.