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

Abstract

Let $\mathbf{G}$ 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 $\mathcal{X}(\mathbf{G})$ introduced by the author in a previous work to study the complex representations of $\mathbf{G}$. We provide several pieces of evidence for these conjectures. In particular, we show that these conjectures are valid for $\mathbf{G}=S{L}_2({\bar{\mathbb{F}}}_q)$.

19 pages, accepted by Journal of Lie Theory