(ir)reducibility of Some Commuting Varieties Associated with Involutions

The ground field k is algebraically closed and of characteristic zero. Let g be a reductive algebraic Lie algebra over k and σ an involutory automorphism of g. Then g = g0 ⊕ g1 is the direct sum of σ-eigenspaces. Here g0 is a reductive subalgebra and g1 is a g0-module. Let G be the adjoint group of g and G0 ⊂ G a connected subgroup with LieG0 = g0. The commuting variety of (g, g0) is the following set: C(g1) = {(x, y) ∈ g1 × g1 | [x, y] = 0}. The problem whether C(g1) is irreducible was considered by Panyushev [6], [7] and SabourinYu [9], [10]. Suppose g is simple. Then the known results are • if the rank of the symmetric pair (g, g0) is equal to the semisimple rank of g (called the maximal rank case), then the corresponding commuting variety is irreducible, [6]; • if the rank of (g, g0) equals 1, then C(g1) is irreducible only in one case, namely, (som+1, som), [7], [9]; • for (sl2n, sp2n) and (E6, F4) the corresponding commuting variety is irreducible, [7]; • if (g, g0) = (so2+m, so2⊕som), then C(g1) is irreducible, [10]. For all other symmetric pairs the problem is open. In sections 1–3, we extend the result of [10] to all symmetric pairs (son+m, son ⊕ som). The scheme of the proof is similar to that of [10]. But as it often happens, the argument in a general situation is shorter and simpler, than in a particular case. In [7], it was conjectured that C(g1) is irreducible if the rank of the symmetric pair is greater than 1. This conjecture is not true. In section 4, we prove that C(g1) is reducible for symmetric pairs (gln+m, gln⊕glm) with n 6= m, (so2n, gln) with odd n, and (E6, so10⊕k).