IRREDUCIBILITY OF COMMUTING VARIETY ASSOCIATED WITH (son+m, son⊕som)

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 [5], [6] and SabourinYu [8], [9]. 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, [5]; • if the rank of (g, g0) equals 1, then C(g1) is irreducible only in one case, namely, (som+1, som), [6], [8]; • for (sl2n, sp2n) and (E6, F4) the corresponding commuting variety is irreducible, [6]; • if (g, g0) = (so2+m, so2⊕som), then C(g1) is irreducible, [9]. For all other symmetric pairs the problem is open. Here we extend the result of [9] to all symmetric pairs (son+m, son ⊕ som). The scheme of the proof is similar to that of [9]. But as it often happens, the argument in a general situation is shorter and simpler, than in a particular case.