Cominuscule points and Schubert varieties

William Graham, Victor Kreiman

Published 2017-01-21Version 1

Let X be a generalized flag variety for a semisimple algebraic group G, T a maximal torus in G, and Y a Schubert variety in X. In this paper, we define what it means for a T-fixed point in Y to be cominuscule. We derive formulas expressing the Hilbert series and multiplicity of Y at a cominuscule point in terms of the restrictions of classes in T-equivariant K-theory and cohomology to that point. Thus, we can calculate Hilbert series and multiplicities in cases where these were previously unknown. If X is of cominuscule type, then every T-fixed point in Y is cominuscule, and our formulas specialize to previously known formulas. The formulas for Schubert varieties are special cases of more general formulas valid at what we call generalized cominuscule points of schemes with torus actions.