Bisymplectic Grassmannians of planes

The bisymplectic Grassmannian I$_2$Gr$(k, V)$ parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano variety which admits an action of a torus with a finite number of fixed points. In this work we study its equivariant cohomology when $k = 2$; the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyper-plane class by any Schubert class. Moreover, we study in detail the case of I$_2$Gr$(2, \mathbb{C}^6)$, which is a quasi-homogeneous variety, we analyze its deformations and we give a presentation of its cohomology.