Existence of Affine Pavings for Varieties of Partial Flags Associated to Nilpotent Elements

Abstract

The flag variety of a complex reductive linear algebraic group |$G$| is by definition the quotient |$G/B$| by a Borel subgroup. It can be regarded as the set of Borel subalgebras of |${\rm Lie}(G)$|⁠. Given a nilpotent element |$e$| in |${\rm Lie}(G)$|⁠, one calls Springer fiber the subvariety formed by the Borel subalgebras which |$e$| belongs to. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by De Concini, Lusztig, and Procesi asserts that when |$G$| is classical, a Springer fiber can always be paved by finitely many subvarieties isomorphic to affine spaces. In this paper, we study varieties generalizing the Springer fibers in two ways: being contained in a partial flag variety |$G/P$| (the quotient by a parabolic subgroup, instead of a Borel subgroup) and defined by a more general belonging condition (in terms of an ideal of |${\rm Lie}(P))$|⁠. These varieties arise, for instance, as fibers of resolutions of nilpotent orbit closures. The main result of the paper is a generalization of De Concini, Lusztig, and Procesi's theorem to this context.Communicated by Corrado de Concini