On q-deformed gl(l+1)-Whittaker function

TL;DR

This paper introduces a new explicit form of q-deformed gl(l+1)-Whittaker functions that generalize classical solutions, connect to characters of algebraic groups, and recover known formulas in specific limits.

Contribution

The paper presents a novel explicit expression for q-deformed gl(l+1)-Whittaker functions, linking them to characters and generalizing the Shintani-Casselman-Shalika formula.

Findings

Explicit q-deformed Whittaker functions reduce to classical functions as q->1.

The functions can be represented as characters of C* x GL(l+1).

In the limit q->0, the functions recover characters of finite-dimensional gl(l+1) representations.

Abstract

We propose new explicit form of q-deformed Whittaker functions solving q-deformed gl(l+1)-Toda chains. In the limit q->1 constructed solutions reduce to classical class one gl(l+1)-Whittaker functions 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 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 Gelfand-Zetlin bases.