On q-Deformed gll+1-Whittaker Function I

We propose a new explicit form of q-deformed Whittaker functions solving q-deformed gl(l+1)-Toda chains. In the limit q -> 1 the constructed solutions reduce to the classical gl(l+1)-Whittaker functions of class one in the form proposed by Givental. An important property of the proposed expression for the q-deformed gl(l+1)-Whittaker function is that it can be represented as a character of C* x GL(l+1). This provides a q-version of the Shintani-Casselman-Shalika formula for the p-adic Whittaker function. The Shintani-Casselman-Shalika formula is recovered in the limit q -> 0 when the q-deformed Whittaker function is reduced to a character of a finite-dimensional representation of gl(l+1) expressed through the Gelfand-Zetlin basis.