Category ${\mathscr{C}}_{k}$ of multi-loop algebra representations versus modular representations: Questions of G. Lusztig

TL;DR

This paper investigates the relationship between Lusztig's category of multi-loop algebra representations and modular representations, providing counterexamples to some of Lusztig's posed questions.

Contribution

It demonstrates that certain questions posed by Lusztig about the connection between these representation categories have negative answers.

Findings

Some of Lusztig's questions about the category ${\\mathscr{C}}_{k}$ are negatively answered.

Counterexamples show the disconnect between multi-loop algebra representations and modular representations.

The results clarify limitations in the correspondence between these two types of representations.

Abstract

Lusztig defined an abelian category ${\mathscr{C}}_{k}$ of a class of representations of a multi-loop algebra and asked various questions connecting it to the modular representation theory of simple algebraic groups in char. p>0. The aim of this paper is to show that some of these questions have negative answer.