Grade zero part of forced graded algebras

TL;DR

This paper investigates a specific subcategory of representations for semisimple algebraic groups in characteristic p, using translation functors to analyze irreducible characters when Lusztig's character formula fails.

Contribution

It introduces translation functors in the subcategory derived from quantum groups at p-th roots of unity to study characters beyond Lusztig's formula.

Findings

Provides new tools for understanding irreducible characters

Analyzes the cohomological differences between quantum and algebraic groups

Offers insights into cases where Lusztig's character formula does not hold

Abstract

The paper concerns a certain subcategory of the category of representations for a semisimple algebraic group $G$ in characteristic $p$, which arise from the semisimple modules for the corresponding quantum group at a $p$-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for $G$ when the Lusztig character formula does not hold.