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.
Categories: math.RT