Regular points in affine Springer fibers

Abstract

studied by Kazhdan and Lusztig in [KL]. For x = gG(O) ∈ X the G(O)-orbit (for the adjoint action) of Ad(g−1)(u) in g(O) depends only on x, and its image under g(O) g(C) is a well-defined G(C)-orbit in g(C). We say that x ∈Xu is regular if the associated orbit is regular in g(C). (Recall that an element of g(C) is regular if the nilpotent part of its Jordan decomposition is a principal nilpotent element in the centralizer of the semisimple part of its Jordan decomposition.) We write X reg for the (Zariski open) subset of regular elements in X. From now on we assume that u is regular semisimple with centralizer T, a maximal torus in G over F. Assume further that u is integral, by which we mean that X is nonempty. Kazhdan and Lusztig [KL] show that X is then a locally finite union of projective algebraic varieties, and in [KL, Sec. 4, Cor. 1] they show that the open subset X reg of X u is nonempty (and hence dense in at least one irreducible component of X). The action of T(F ) on X clearly preserves the subsets X andX reg. Bezrukavnikov [B] proved thatX u reg forms a single orbit under T(F ). (Actually Kazhdan–Lusztig and Bezrukavnikov consider only topologically nilpotent elements u, but the general case can be reduced to their special case by using the topological Jordan decomposition of u.) The goal of this paper is to characterize regular elements in X (for integral regular semisimple u as just described). When T is elliptic (in other words, F anisotropic modulo the center of G), the characterization gives no new information. At the other extreme, in the split case, the characterization gives a clear picture of what it means for a point in X to be regular. We will now state our characterization in the split case, leaving the more technical general statement to the next section (see Theorem 1). Fix a split maximal torus