Some conjectures on the quotients of the tensor products in the category $\mathscr{X}$

Abstract

Let ${\bf G}$ be a connected reductive algebraic group defined over the finite field $\mathbb{F}_q$ with $q$ elements. We propose some conjectures concerning the simple quotients of $M\otimes N$, where $M,N$ are objects in the representation category $\mathscr{X}({\bf G})$ introduced by the author in a previous work to study the complex representations of ${\bf G}$. We provide several pieces of evidence for these conjectures. In particular, we show that these conjectures are valid for ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$.