The q-Whittaker function \(W_\lambda (\textbf{x};q)\) associated to a partition \(\lambda \) is a q-analogue of the Schur function \(s_\lambda (\textbf{x})\) , and is defined as the \(t=0\) specialization of the Macdonald polynomial \(P_\lambda (\textbf{x};q,t)\) . We show combinatorially how to expand \(W_\lambda (\textbf{x};q)\) 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 \(q\rightarrow 0\) , 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 \(\text {GL}_n\) 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.
