Endomorphisms of the Cohomology Algebra of the Even Orthogonal Grassmannian

Abstract

Let $M_{n,k}$ denote the even orthogonal Grassmanian, $SO(2n) / (U(k) \times SO(2n-2k) )$. We study endomorphisms of the rational cohomology algebra of $M_{n,k}$. We prove that an endomorphism of the rational cohomology algebra of $M_{n,k}$, which maps all the Chern classes of the canonical $k$-plane bundle over $M_{n,k}$ to zero, or maps all the Pontrjagin classes of the canonical, real, oriented $(2n-2k)$-plane bundle over $M_{n,k}$ to zero, is the zero endomorphism. Additionally, we prove that if an endomorphism of the rational cohomology algebra of $M_{n,k}$ vanishes on $H^{2}(M_{n,k}; \mathbb{Q})$, and admits a splitting, then the splitting equals zero.