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 ${{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2}$-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.