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 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.

39 pages, v2: fixed some mistakes, revision of Section 8