Existence of affine pavings for varieties of partial flags associated to nilpotent elements

TL;DR

This paper generalizes a known theorem about affine pavings of Springer fibers in classical groups to the broader setting of partial flag varieties, revealing similar affine paving structures.

Contribution

It extends the affine paving result from Springer fibers to varieties of partial flags associated with nilpotent elements.

Findings

Affine pavings exist for these generalized varieties.

The structure of these varieties parallels that of classical Springer fibers.

The results apply to a broad class of partial flag varieties.

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 Lie(G). Given a nilpotent element e in Lie(G), one calls Springer fiber the subvariety formed by the Borel subalgebras which contain e. Springer fibers have in general a complicated structure (not irreducible, singular). Nevertheless, a theorem by C. De Concini, G. Lusztig, and C. 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 to the context of partial flag varieties, that is, subvarieties of the quotient G/P by a parabolic subgroup (instead of a Borel subgroup). The main result of the paper is a generalization of De Concini, Lusztig, and Procesi's theorem to this context.