Rationality of quotients by linear actions of affine groups

TL;DR

This paper investigates the rationality of quotients by the special affine group acting on vector spaces, establishing positive results for large, indecomposable representations and providing detailed conditions for certain extensions.

Contribution

It advances understanding of the rationality problem for quotients by affine groups, especially for large and indecomposable representations, with new theorems for specific extension cases.

Findings

Rationality holds for sufficiently large, indecomposable representations.

Explicit results for two-step extensions with completely reducible components.

Provides conditions under which the quotient V/G is rational.

Abstract

Let G be the (special) affine group, semidirect product of SL_n and C^n. In this paper we study the representation theory of G and in particular the question of rationality for V/G where V is a generically free G-representation. We show that the answer to this question is positive if the dimension of V is sufficiently large and V is indecomposable. We have a more precise theorem if V is a two-step extension 0 -> S -> V -> Q -> 0 with S, Q completely reducible.