On the support varieties of Demazure modules

TL;DR

This paper studies the support varieties of Demazure modules in algebraic group representations, providing explicit calculations for types A1 and A2, and exploring their geometric and order-theoretic properties.

Contribution

It offers explicit computations of support varieties for Demazure modules in low-rank cases and analyzes their orbit structure and relations with Bruhat order.

Findings

Support varieties computed explicitly for types A1 and A2.

Support varieties' inclusion relations align with Bruhat order.

Orbital variety relations established for support varieties.

Abstract

We consider the support varieties of Demazure modules, certain $B$-modules important in the representation theory of reductive groups. In many cases we are able to compute these support varieties over $B_1$, the first Frobenius kernel of a Borel subgroup, and relate them to orbital varieties. We provide a complete calculation of the support varieties when the underlying algebraic group has type $A_1$ or $A_2$. We also consider the saturation ($G$ orbit of) $B_1$ support varieties for Demazure modules and show that the inclusion relations among these varieties are compatible with the Bruhat order.