Equivariant quantum cohomology of the odd symplectic Grassmannian

Abstract

The odd symplectic Grassmannian $$\mathrm {IG}:=\mathrm {IG}(k, 2n+1)$$ parametrizes k dimensional subspaces of $${\mathbb {C}}^{2n+1}$$ which are isotropic with respect to a general (necessarily degenerate) symplectic form. The odd symplectic group acts on $$\mathrm {IG}$$ with two orbits, and $$\mathrm {IG}$$ is itself a smooth Schubert variety in the submaximal isotropic Grassmannian $$\mathrm {IG}(k, 2n+2)$$ . We use the technique of curve neighborhoods to prove a Chevalley formula in the equivariant quantum cohomology of $$\mathrm {IG}$$ , i.e. a formula to multiply a Schubert class by the Schubert divisor class. This generalizes a formula of Pech in the case $$k=2$$ , and it gives an algorithm to calculate any multiplication in the equivariant quantum cohomology ring.