Commuting involutions and degenerations of isotropy representations

Abstract

Let σ 1 and σ 2 be commuting involutions of a semisimple algebraic group G. This yields a $$ {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} $$ -grading of $$ \mathfrak{g} $$ = Lie(G), $$ \mathfrak{g}={\oplus_{i,j=0,1 }}{{\mathfrak{g}}_{ij }} $$ , and we study invariant-theoretic aspects of this decomposition. Let $$ \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle $$ be the $$ {{\mathbb{Z}}_2} $$ -contraction of $$ \mathfrak{g} $$ determined by σ 1. Then both σ 2 and σ 3 := σ 1 σ 2 remain involutions of the non-reductive Lie algebra $$ \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle $$ . The isotropy representations related to $$ \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_2}} \right) $$ and $$ \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_3}} \right) $$ are degenerations of the isotropy representations related to $$ \left( {\mathfrak{g},{\sigma_2}} \right) $$ and $$ \left( {\mathfrak{g},{\sigma_3}} \right) $$ , respectively. These degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various $$ {{\mathbb{Z}}_2} $$ -gradings associated with the $$ {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} $$ -grading of $$ \mathfrak{g} $$ and study the special case in which σ 1 and σ 2 are conjugate.