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

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [Ku2] introduced a ring homomorphism $ \xi^P_\lambda: Rep^$\mathbb{C}${\lambda-poly}(L) \to H^*(G/P, $\mathbb{C}$)$,where$ Rep^$\mathbb{C}${\lambda-poly}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(\lambda )$ of G with highest weight $\lambda$). In this paper we study this homomorphism for G=Sp(2n) and its maximal parabolic subgroups $P_{n-k}$ for any $1\le k\le n$ (with the choice of $V(\lambda )$ to be the defining representation $V({\omega }_1)$ in ${\mathbb{C}}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $ \xi_{n,k}: Rep^$\mathbb{C}$_{\omega_1-poly}(Sp(2k)) \to H^*(IG(n-k, 2n), $\mathbb{C}$)$. Our main result asserts that ${\xi }_{n,k}$ is injective when n tends to $\mathrm{\infty }$ keeping k fixed. Similar results are obtained for the odd orthogonal groups.