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 $\hat{I}{¾}_{\hat{I}}^P»:Re{p}_{\hat{I}»-poly}^{\mathbb{C}}(L)\to H^{\ast }(G/P,\mathbb{C})$, where $Re{p}_{\hat{I}»-poly}^{\mathbb{C}}(L)$ is a certain subring of the complexified representation ring of L (depending upon the choice of an irreducible representation $V(\hat{I}»)$ of G with highest weight $\hat{I}»$). 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 $V(\hat{I}»)$ to be the defining representation $V(\ddot{I}{‰}_1)$ in ${\mathbb{C}}^{2n}$). Thus, we obtain a $\mathbb{C}$-algebra homomorphism $\hat{I}{¾}_{n,k}:Re{p}_{\ddot{I}{‰}_1-poly}^{\mathbb{C}}(Sp(2k))\to H^{\ast }(IG(n-k,2n),\mathbb{C})$. Our main result asserts that $\hat{I}{¾}_{n,k}$ 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