On the Satake correspondence for the equivariant quantum differential equations and qKZ difference equations of Grassmannians

TL;DR

This paper explores the relationship between equivariant quantum differential and qKZ equations for Grassmannians and projective spaces, establishing a Satake correspondence that leads to new formulas and insights into solution behaviors.

Contribution

It demonstrates a Satake correspondence linking Grassmannian and projective space systems, extending the B-theorem, and analyzing Stokes phenomena with K-theoretic interpretations.

Findings

Joint system for G(k,n) is gauge equivalent to a system on the exterior power of P^{n-1}

B-theorem for Grassmannians extends from projective spaces via Satake identification

Stokes bases correspond to K-theoretic classes of exceptional collections

Abstract

We consider the joint system of equivariant quantum differential equations (qDE) and qKZ difference equations for the Grassmannian $G(k,n)$, which parametrizes $k$-dimensional subspaces of $\mathbb{C}^n$. First, we establish a connection between this joint system for $G(k,n)$ and the corresponding system for the projective space $\mathbb{P}^{n-1}$. Specifically, we show that, under suitable \textit{Satake identifications} of the equivariant cohomologies of $G(k,n)$ and $\mathbb{P}^{n-1}$, the joint system for $G(k,n)$ is gauge equivalent to a differential-difference system on the $k$-th exterior power of the cohomology of $\mathbb{P}^{n-1}$. Secondly, we demonstrate that the \textcyr{B}-theorem for Grassmannians, as stated in arXiv:1909.06582, arXiv:2203.03039, is compatible with the Satake identification. This implies that the \textcyr{B}-theorem for $\mathbb{P}^{n-1}$ extends to $G(k,n)$ through the Satake identification. As a consequence, we derive determinantal formulas and new integral representations for multi-dimensional hypergeometric solutions of the joint qDE and qKZ system for $G(k,n)$. Finally, we analyze the Stokes phenomenon for the joint system of qDE and qKZ equations associated with $G(k,n)$. We prove that the Stokes bases of solutions correspond to explicit $K$-theoretical classes of full exceptional collections in the derived category of equivariant coherent sheaves on $G(k,n)$. Furthermore, we show that the Stokes matrices equal the Gram matrices of the equivariant Euler-Poincar\'e-Grothendieck pairing with respect to these exceptional $K$-theoretical bases.