Equivariant K-Theory and Tangent Spaces to Schubert Varieties

Abstract

Tangent spaces to Schubert varieties of type A were characterized by Lakshmibai and Seshadri [LS84]. This result was extended to the other classical types by Lakshmibai [Lak95, Lak00b], and [Lak00a]. We give a uniform characterization of tangent spaces to Schubert varieties in cominuscule $\mathit{G}/\mathit{P}$. Our results extend beyond cominuscule $\mathit{G}/\mathit{P}$; they describe the tangent space to any Schubert variety in $\mathit{G}/\mathit{B}$ at a point $\mathit{x}\mathit{B}$, where x is a cominuscule Weyl group element in the sense of Peterson. Our results also give partial information about the tangent space to any Schubert variety at any point. Our method is to describe the tangent spaces of Kazhdan–Lusztig varieties, and then to recover results for Schubert varieties. Our proof uses a relationship between weights of the tangent space of a variety with torus action and factors of the class of the variety in torus equivariant K-theory. The proof relies on a formula for Schubert classes in equivariant K-theory due to Graham [Gra02] and Willems [Wil06] and on a theorem on subword complexes due to Knutson and Miller [KM04, KM05].