Commuting involutions and degenerations of isotropy representations

Let $σ_1$ and $σ_2$ 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 $\g<σ_1>$ be the $Z_2$-contraction of $\g$ determined by $σ_1$. Then both $σ_2$ and $σ_3:=σ_1σ_2$ remain involutions of the non-reductive Lie algebra $\g<σ_1>$. The isotropy representations related to $(\g<σ_1>, σ_2)$ and $(\g<σ_1>, σ_3)$ are degenerations of the isotropy representations related to $(\g, {σ_2})$ and $(\g, {σ_3})$, 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 $\g$.

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