Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Junbin Dong

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

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

Abstract

Let be a connected reductive algebraic group defined over the finite field with elements. We propose some conjectures concerning the simple quotients of , where are objects in the representation category introduced by the author in a previous work to study the complex representations of . We provide several pieces of evidence for these conjectures. In particular, we show that these conjectures are valid for .

Paper Structure

This paper contains 4 sections, 11 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Quotients of the tensor products in $\mathscr{X}({\bf G})$
  4. The case ${\bf G}=SL_2(\bar{\mathbb{F}}_q)$

Key Result

Proposition 2.1

For any $J\subset I(\theta)$, the $\Bbbk {\bf G}$-module $\mathbb{M}(\theta)_J$ has the form and the set $\{u\dot{w}\eta(\theta)_J \mid w\in W^J, u\in {\bf U}_{w_Jw^{-1}} \}$ is a basis of $\mathbb{M}(\theta)_J$.

Theorems & Definitions (25)

  • Proposition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Conjecture 3.1
  • Conjecture 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • Conjecture 3.5
  • ...and 15 more