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.
