Representation ring of Levi subgroups versus cohomology ring of flag varieties II

Abstract

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ ξ^P_λ: Rep^\mathbb{C}_{λ-poly}(L) \to H^*(G/P, \mathbb{C})$, where $ Rep^\mathbb{C}_{λ-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(λ)$ of G with highest weight $λ$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\leq k\leq n$ (with the choice of $V(λ) $ to be the defining representation $V(ω_1) $ in $\mathbb{C}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $ ξ_{n,k}: Rep^\mathbb{C}_{ω_1-poly}(Sp(2k)) \to H^*(IG(n-k, 2n), \mathbb{C})$. Our main result asserts that $ ξ_{n,k}$ is injective when n tends to $\infty$ keeping k fixed. Similar results are obtained for the odd orthogonal groups.

19 pages. arXiv admin note: text overlap with arXiv:1508.06826