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

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.