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

TL;DR

The paper proposes conjectures about the simple quotients of tensor products in a specific representation category of a reductive algebraic group over finite fields, supported by evidence including the case of SL_2.

Contribution

It introduces new conjectures on the structure of tensor product quotients in a specialized representation category, extending understanding in algebraic group representations.

Findings

Conjectures are supported by evidence.

Conjectures hold for G=SL_2.

Provides insights into simple quotients of tensor products.

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 ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$.