Whittaker functions, geometric crystals, and quantum Schubert calculus

TL;DR

This paper explores the connections between Whittaker functions, geometric crystals, and quantum Schubert calculus, providing an integral formula for Whittaker functions and linking it to mirror symmetry and representation theory.

Contribution

It proves a new integral formula for Whittaker functions using geometric crystals and establishes their relation to mirror symmetry and representation theory.

Findings

Derived an integral formula for Whittaker functions over geometric crystals.

Connected Whittaker functions to mirror symmetry of flag varieties.

Suggested Whittaker functions as geometric analogues of irreducible characters.

Abstract

This mostly expository article explores recent developments in the relations between the three objects in the title from an algebro-combinatorial perspective. We prove a formula for Whittaker functions of a real semisimple group as an integral over a geometric crystal in the sense of Berenstein-Kazhdan. We explain the connections of this formula to the program of mirror symmetry of flag varieties developed by Givental and Rietsch; in particular, the integral formula proves the equivariant version of Rietsch's mirror symmetry conjecture. We also explain the idea that Whittaker functions should be thought of as geometric analogues of irreducible characters of finite-dimensional representations.