Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On extensions of principal series representations

Gautam H. Borisagar, Asfak Soneji

TL;DR

This work determines when nontrivial extensions occur between $B$-characters of a split reductive group over a finite field and translates these findings to extensions between principal series representations of the finite group of Lie type $G(\mathbb{F}_q)$. The key criterion is that $\mathrm{Ext}^1_B(\chi_1,\chi_2) \neq 0$ precisely when the ratio $\chi_1^{-1}\chi_2$ equals a Frobenius twist of a simple root $\alpha$ on the unipotent radical $N$, i.e., $\chi_1^{-1}\chi_2 = \phi\circ\alpha$, with $\phi$ a Frobenius automorphism; the corresponding Ext space is one-dimensional. Extending to principal series, the paper shows nonzero $\mathrm{Ext}^1_G(\Pi_{\chi_1},\Pi_{\chi_2})$ occurs iff $\chi_1^{-1}\chi_2^{w}=\phi\circ\alpha$ for some Weyl group element $w$, and provides a one-dimensional instance $\mathrm{Ext}^1_G(\Pi_{\chi},\Pi_{\chi'})$ when $\chi^{-1}\chi' = \phi\circ\alpha$. The results rely on Frobenius reciprocity for Ext, Bruhat decomposition, and the $T$-action on $N$, offering concrete criteria for extensions in modular representations of finite groups of Lie type and connecting to known GL$_2$ cases.

Abstract

We compute $\mathrm{Ext}^{1}_B(χ_1,χ_2)$ between two characters $χ_1,χ_2$ of a Borel subgroup $B$ of a split reductive group $G$ over a finite field $\mathbb{F}_q,$ and make an application to the calculation of $\mathrm{Ext}^1_G(π_1,π_2)$ between principal series representations $π_1,π_2$ of $G(\mathbb{F}_q).$

On extensions of principal series representations

TL;DR

This work determines when nontrivial extensions occur between -characters of a split reductive group over a finite field and translates these findings to extensions between principal series representations of the finite group of Lie type . The key criterion is that precisely when the ratio equals a Frobenius twist of a simple root on the unipotent radical , i.e., , with a Frobenius automorphism; the corresponding Ext space is one-dimensional. Extending to principal series, the paper shows nonzero occurs iff for some Weyl group element , and provides a one-dimensional instance when . The results rely on Frobenius reciprocity for Ext, Bruhat decomposition, and the -action on , offering concrete criteria for extensions in modular representations of finite groups of Lie type and connecting to known GL cases.

Abstract

We compute between two characters of a Borel subgroup of a split reductive group over a finite field and make an application to the calculation of between principal series representations of
Paper Structure (3 sections, 6 theorems, 24 equations)

This paper contains 3 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Extensions of characters of a Borel subgroup
  3. On Extensions of Principal series representations

Key Result

Proposition 1

Let $\chi_1,\chi_2$ be two characters $\chi_1,\chi_2: T \rightarrow \mathbb{F}_p^\times$. Then, for $\alpha$ a simple root of $T$ on $N.$ Further, if $\chi_1^{-1}\chi_2 = \alpha$, for $\alpha$ a simple root of $T$ on $N,$ treated as a character on $B,$

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Proposition 4