Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Abstract

Let LG n denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2 n. For each strict partition λ = ( λ 1 , … , λ k ) with λ 1 ⩽ n there is a Schubert variety X ( λ ) . Let T denote a maximal torus of the symplectic group acting on LG n . Consider the T-equivariant cohomology of LG n and the T-equivariant fundamental class σ ( λ ) of X ( λ ) . The main result of the present paper is an explicit formula for the restriction of the class σ ( λ ) to any torus fixed point. The formula is written in terms of factorial analogue of the Schur Q-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for LG n . As another consequence of the main result, we obtained a presentation of the ring H T ∗ ( LG n ) .