Bisymplectic Grassmannians of planes

The bisymplectic Grassmannian ({{,mathrm{I_2Gr},}}(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 projective variety which admits an action of a torus with a finite number of fixed points. In this work, we study its equivariant cohomology with complex coefficients when (k=2); the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyperplane class by any Schubert class. Moreover, we study in detail the case of ({{,mathrm{I_2Gr},}}(2, {mathbb {C}}^6)), which is a quasi-homogeneous variety, we analyse its deformations, and we give a presentation of its cohomology.