On parabolic Whittaker functions II

TL;DR

This paper derives a new integral representation for a specific Grassmannian Whittaker function, linking it to Gromov-Witten invariants, representation theory, and toric degenerations, advancing the understanding of these mathematical structures.

Contribution

It introduces a Givental-type integral representation for the $ ext{Gr}_{m,N}$-Whittaker function, connecting it to toric degenerations and total positivity in a novel way.

Findings

Derived a stationary phase integral representation for the Grassmannian Whittaker function.

Connected the toric degeneration of Grassmannians with total positivity.

Provided a representation theory interpretation of the toric degeneration.

Abstract

We derive a Givental-type stationary phase integral representation for the specified $\Gr_{m,N}$-Whittaker function introduced in \cite{GLO2}, which presumably describes the $S^1\times U_N$-equivariant Gromov-Witten invariants of Grassmann variety $\Gr_{m,N}$. Our main tool is a generalization of Whittaker model for principal series $\CU(\frak{gl}_N)$-modules. In particular, our construction includes a representation theory interpretation of the Batyrev--Ciocan-Fontanine--Kim--van Straten toric degeneration of Grassmannian, providing a direct connection between this toric degeneration of $\Gr_{m,N}$ and total positivity for unipotent matrices.