Algebraic approach to contraction families

TL;DR

This paper introduces an algebraic framework for contraction group schemes, compares related quotient schemes, and explores their actions in symmetric and theta-stable parabolic subgroup contexts.

Contribution

It provides a purely algebraic approach to contraction group schemes and analyzes their quotient schemes and actions in specific subgroup cases.

Findings

Algebraic construction of contraction group schemes

Comparison of quotient schemes with related schemes

Analysis of actions in symmetric and theta-stable parabolic subgroups

Abstract

In this paper, we give a purely algebraic approach to the contraction group scheme predicted by Bernstein--Higson--Subag and constructed by Barbasch--Higson--Subag. We also compare quotient schemes of contraction group schemes with other related schemes, equipped with actions of contraction group schemes in the cases of symmetric and $\theta$-stable parabolic subgroups.