Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Abstract

Let $L{G}_n$ denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension $2n.$ For each strict partition $\lambda =({\lambda }_1,...,{\lambda }_k)$ with ${\lambda }_1\le n$ there is a Schubert variety $X(\lambda ).$ Let $T$ denote a maximal torus of the symplectic group acting on $L{G}_n.$ Consider the $T$-equivariant cohomology of $L{G}_n$ and the $T$-equivariant fundamental class $\sigma (\lambda )$ of $X(\lambda ).$ The main result of the present paper is an explicit formula for the restriction of the class $\sigma (\lambda )$ 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 $L{G}_n.$ As another consequence of the main result, we obtained a presentation of the ring $H_T^{\ast }(L{G}_n).$