A note on affine quotients and equivariant double fibrations

Abstract

We consider two linear reductive algebraic groups $ G $ and $ G' $ over $ C $. Take a finite dimensional rational representation $ W $ of $ G \times G' $. Let $ Y = W // G := Spec C[W]^G $ and $ X = W // G' := \Spec C[W]^{G'} $ be the affine quotients. The quotient space $ X $ (respectively $ Y $) naturally inherits the action of $ G $ (respectively $ G' $). In this note, we study the interrelation between the orbit structures of $ X / G $ and $ Y / G' $. In a good situation, we can embed $ Y / G' $ into $ X / G $, and the embedding map preserves important properties such as the closure relation and nilpotency. We give a sufficient condition for the existence of such embedding, and provide many examples arising from the natural representations of classical groups. As an application we consider the geometric problem of unimodular congruence classes of bilinear forms proposed by Djokovic-Sekiguchi-Zhao.