Commuting involutions and degenerations of isotropy representations

Abstract

Let $\sigma_1$ and $\sigma_2$ be commuting involutions of a semisimple algebraic group $G$. This yields a $Z_2\times Z_2$-grading of $\g=\Lie(G)$, $\g=\bigoplus_{i,j=0,1}\g_{ij}$, and we study invariant-theoretic aspects of this decomposition. Let $\g<\sigma_1>$ be the $Z_2$-contraction of $\g$ determined by $\sigma_1$. Then both $\sigma_2$ and $\sigma_3:=\sigma_1\sigma_2$ remain involutions of the non-reductive Lie algebra $\g<\sigma_1>$. The isotropy representations related to $(\g<\sigma_1>, \sigma_2)$ and $(\g<\sigma_1>, \sigma_3)$ are degenerations of the isotropy representations related to $(\g, {\sigma_2})$ and $(\g, {\sigma_3})$, respectively. We show that 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 $Z_2$-gradings associated with the $Z_2\times Z_2$-grading of $\g$.