Total positivity, Schubert positivity, and Geometric Satake

TL;DR

This paper unifies three notions of positivity on a subgroup related to a complex algebraic group and confirms a conjecture about parametrizing its nonnegative part, leveraging the geometric Satake correspondence.

Contribution

It proves the equivalence of Schubert, total, and MV-positivity on the subgroup and parametrizes its nonnegative part, connecting these concepts via affine Grassmannian geometry.

Findings

The three notions of positivity coincide.

The nonnegative part of the subgroup is explicitly parametrized.

Connections established through the geometric Satake correspondence.

Abstract

Let G be a simple and simply-connected complex algebraic group, and let X \subset G^\vee be the centralizer subgroup of a principal nilpotent element. Ginzburg and Peterson independently related the ring of functions on X with the homology ring of the affine Grassmannian Gr_G. Peterson furthermore connected this ring to the quantum cohomology rings of partial flag varieties G/P. The first aim of this paper is to study three different notions of positivity on X: (1) Schubert positivity arising via Peterson's work, (2) total positivity in the sense of Lusztig, and (3) Mirkovic-Vilonen positivity obtained from the MV-cycles in Gr_G. Our first main theorem establishes that these three notions of positivity coincide. The second aim of this paper is to parametrize the totally nonnegative part of X, confirming a conjecture of the second author. In type A a substantial part of our results were previously established by the second author. The crucial new component of this paper is the connection with the affine Grassmannian and the geometric Satake correspondence.