On extensions of principal series representations

TL;DR

This paper computes extension groups between characters of a Borel subgroup and applies these results to principal series representations of split reductive groups over finite fields, advancing understanding of their structure.

Contribution

It provides explicit calculations of Ext^1 groups between characters and principal series representations, offering new insights into their extensions over finite fields.

Findings

Explicit formulas for Ext^1_B(χ₁, χ₂)

Application to Ext^1_G(π₁, π₂) between principal series

Enhanced understanding of representation extensions over finite fields

Abstract

We compute $\mathrm{Ext}^{1}_B(\chi_1,\chi_2)$ between two characters $\chi_1,\chi_2$ 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 $\mathrm{Ext}^1_G(\pi_1,\pi_2)$ between principal series representations $\pi_1,\pi_2$ of $G(\mathbb{F}_q).$