Equivariant deformations of the affine multicone over a flag variety

Abstract

We prove that the invariant Hilbert scheme parameterising the equivariant deformations of the affine multicone over a flag variety is, under certain hypotheses, an affine space. More specifically, we obtain that the isomorphism classes of equivariant deformations of such a multicone are in correspondence with the orbits of a well-determined wonderful variety. (C) 2007 Elsevier Inc. All rights reserved

Equivariant degenerations of spherical modules for groups of type \mathsf {A}

“…In Section 2 we present known results, mostly from [1] and [3], in the form we need them. In Section 3, which may be of independent interest, we formulate a criterion about non-extension of invariant sections of the TOME 62 (2012), FASCICULE 5 normal sheaf.…”

“…In this section we gather known results, mostly from [1] and [3], together with immediate consequences relevant to our purposes. In particular we explain that to prove Theorem 1.1 it is sufficient to show that M S is smooth when S is the weight monoid of a spherical module W for G of type A.…”

“…In Section 4 we review the known classification of spherical modules [17,2,21] as presented in [19] and reduce the proof of Theorem 1.1 to a case-by-case verification. We perform this case-by-case analysis in Section 5, using results from [3] mentioned in Section 2 and, for the most involved cases, also the exclusion criterion of Section 3.…”

“…Bravi and S. Cupit-Foutou [3] generalized Jansou's result as follows. Given a free submonoid S of Λ + such that coming from the action of T on Y induces an action of T on M S , and they proved that M S is (isomorphic to) a multiplicity-free representation of T whose weight set is the set of spherical roots of a wonderful variety associated to S. The connections between the moduli schemes M Y and wonderful varieties have been studied further in [10,11].…”

Classification of strict wonderful varieties

“…Consider the adjoint action of G: G acts by conjugation on its Lie algebra g. Panyushev characterises in [21] (see also [22] for a classification-free proof) the adjoint nilpotent orbits in g which are G-spherical and provides the list of spherical nilpotent orbits. More specifically, he proves that an adjoint nilpotent orbit G.e is spherical if and only if (ad e) 4 = 0. In Appendix B we reproduce the list obtained in [21] of nilpotent spherical orbits of height 3, i.e., with (ad e) 3 = 0.…”