On a Classical Limit of q-Deformed Whittaker Functions

Abstract

Previously, we derive a representation of q-deformed $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker function as a sum over Gelfand–Zetlin patterns. This representation provides an analog of the Shintani–Casselman–Shalika formula for q-deformed $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker functions. In this note, we provide a derivation of the Givental integral representation of the classical $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker function as a limit q → 1 of the sum over the Gelfand–Zetlin patterns representation of the q-deformed $${\mathfrak{gl}_{\ell+1}}$$ -Whittaker function. Thus, Givental representation provides an analog the Shintani–Casselman–Shalika formula for classical Whittaker functions.