A Gauss-Bonnet theorem for constructible sheaves on reductive groups

Abstract

In this paper, we prove an analog of the Gauss-Bonnet formula for constructible sheaves on reductive groups. This formula holds for all constructible sheaves equivariant under the adjoint action and expresses the Euler characteristic of a sheaf in terms of its characteristic cycle. As a corollary from this formula we get that if a perverse sheaf on a reductive group is equivariant under the adjoint action, then its Euler characteristic is nonnegative. In the sequel by a constructible complex we will always mean a bounded complex of sheaves of C-vector spaces whose cohomology sheaves are constructible with respect to some finite algebraic stratification. We now formulate the main results. Let G be a complex reductive group, and let F be a constructible complex on G. Denote by CC(F) the characteristic cycle of F . It is a linear combination of Lagrangian subvarieties CC(F) = ∑ cαTXαG (see [9]). Here and in the sequel T ∗ XG denotes the closure of the conormal bundle to the smooth locus of a subvariety X ⊂ G. With X one can associate a nonnegative number gdeg(X) called the Gaussian degree of X. It is equal to the number of zeros of a generic left-invariant differential 1-form on G restricted to X. The precise definitions of the Gaussian degree and of the Gauss map are given in section 2. Theorem 1.1. If F is equivariant under the adjoint action of G, then its Euler characteristic can be computed in terms of the characteristic cycle by the following formula χ(G,F) = ∑ cαgdeg(Xα). For a perverse sheaf the multiplicities cα of its characteristic cycle are nonnegative [7]. The Gaussian degrees of Xα are also nonnegative by their definition, see below. Thus Theorem 1.1 immediately implies the following important corollary. Corollary 1.2. If F is a perverse sheaf equivariant under the adjoint action of G, then its Euler characteristic is nonnegative.