The q-Whittaker function Wλ​(x;q) associated to a
partition λ is a q-analogue of the Schur function
sλ​(x), and is defined as the t=0 specialization of the
Macdonald polynomial Pλ​(x;q,t). We show combinatorially how
to expand Wλ​(x;q) in terms of partial flags compatible with a
nilpotent endomorphism over the finite field of size 1/q. This yields an
expression analogous to a well-known formula for the Hall-Littlewood functions.
We show that considering pairs of partial flags and taking Jordan forms leads
to a probabilistic bijection between nonnegative-integer matrices and pairs of
semistandard tableaux of the same shape, proving the Cauchy identity for
q-Whittaker functions. We call our probabilistic bijection the q-Burge
correspondence, and prove that in the limit as q→0, we recover a
description of the classical Burge correspondence (also known as column RSK)
due to Rosso (2012). A key step in the proof is the enumeration of an arbitrary
double coset of GLn​ modulo two parabolic subgroups, which we find to
be of independent interest. As an application, we use the q-Burge
correspondence to count isomorphism classes of certain modules over the
preprojective algebra of a type A quiver (i.e. a path), refined according to
their socle filtrations. This develops a connection between the combinatorics
of symmetric functions and the representation theory of preprojective algebras.Comment: 69 pages. v2: Added Remark 5.1