On the irreducibility of the commuting variety of a symmetric pair associated to a parabolic subalgebra with abelian unipotent radical
In this paper, we study the commuting variety of symmetric pairs associated to parabolic subalgebras with abelian unipotent radical in a simple complex Lie algebra. By using the ``cascade'' construction of Kostant, we construct a Cartan subspace which in turn provides, in certain cases, useful information on the centralizers of non $\mathfrak{p}$-regular semisimple elements. In the case of the rank 2 symmetric pair $(\mathrm{so}_{p+2},\mathrm{so}_{p}\times \mathrm{so}_{2})$, $p\geq 2$, this allows us to apply induction, in view of previous results of the authors, and reduce the problem of the irreducibility of the commuting variety to the consideration of evenness of $\mathfrak{p}$-distinguished elements. Finally, via the correspondence of Kostant-Sekiguchi, we check that in this case, $\mathfrak{p}$-distinguished elements are indeed even, and consequently, the commuting variety is irreducible.