Quotients of commuting schemes associated to Symmetric Pairs

Abstract

Let g = g0 ⊕ g1 be a Z2-grading of a classical Lie algebra such that (g, g0) is a classical symmetric pair. Let G be a classical group with Lie algebra g and let G0 be the connected subgroup of G with Lie(G0) = g0. For d ≥ 2, let C d (g1) be the d-th commuting scheme associated with the symmetric pair (g, g0). In this article, we study the categorical quotient C d (g1)//G0 via the Chevalley restriction map. As a consequence we show that the categorical quotient scheme C d (g1)//G0 is normal and reduced. As a part of the proof, we describe a generating set for the algebra k[g d 1 ] G 0 , which are of independent interest.