Steinberg quotients, Weyl Characters, and Kazhdan-Lusztig Polynomials

TL;DR

This paper explores the structure of Steinberg quotients in algebraic groups and quantum groups, relating their multiplicities to Weyl characters and Kazhdan-Lusztig polynomials, and provides an algorithm for computing minimal characters.

Contribution

It extends the concept of Steinberg quotients to quantum groups, linking multiplicities to Kazhdan-Lusztig polynomials and offering an explicit computation algorithm.

Findings

Quantum Steinberg quotients relate to algebraic counterparts and Weyl characters.

Minimal characters are close to quantum Steinberg quotients but not always equal.

An explicit algorithm for computing minimal characters is provided.

Abstract

Let $G$ be a reductive group over a field of prime characteristic. An indecomposable tilting module for $G$ whose highest weight lies above the Steinberg weight has a character that is divisible by the Steinberg character. The resulting "Steinberg quotient" carries important information about $G$-modules, and in previous work we studied patterns in the weight multiplicities of these characters. In this paper we broaden our scope to include quantum Steinberg quotients, and show how the multiplicities in these characters relate to algebraic Steinberg quotients, Weyl characters, and evaluations of Kazhdan-Lusztig polynomials. We give an explicit algorithm for computing minimal characters that possess a key attribute of Steinberg quotients. We provide computations which show that these minimal characters are not always equal to quantum Steinberg quotients, but are close in several nontrivial cases.