Differential operators on quantized flag manifolds at roots of unity III

TL;DR

This paper investigates the cohomology of sheaves of twisted differential operators on quantized flag manifolds at prime power roots of unity, linking representation theory and geometric structures.

Contribution

It provides a detailed description of cohomology in this setting and relates the number of irreducible modules to Springer fiber cohomology, supporting Lusztig's conjecture.

Findings

Cohomology of twisted differential operators is explicitly described at roots of unity.

Number of irreducible modules matches the dimension of Springer fiber cohomology.

Supports a weak version of Lusztig's conjecture on non-restricted representations.

Abstract

We describe the cohomology of the sheaf of twisted differential operators on the quantized flag manifold at a root of unity whose order is a prime power. It follows from this and our previous results that for the De Concini-Kac type quantized enveloping algebra, where the parameter $q$ is specialized to a root of unity whose order is a prime power, the number of irreducible modules with a certain specified central character coincides with the dimension of the total cohomology group of the corresponding Springer fiber. This gives a weak version of a conjecture of Lusztig concerning non-restricted representations of the quantized enveloping algebra.