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

Abstract

For any reductive group $G$ and a parabolic subgroup $P$ with its Levi subgroup $L$, the first author [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 $G=SO(2n)$ and its maximal parabolic subgroups $P_{n-k}$ for any $2\le k\le n-1$ (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 $\hat{I}{¾}_{n,k}^D:Re{p}_{\ddot{I}{‰}_1-poly}^{\mathbb{C}}(SO(2k))\to H^{\ast }(OG(n-k,2n),\mathbb{C})$. We determine this homomorphism explicitly in the paper. We further analyze the behavior of $\hat{I}{¾}_{n,k}^D$ when $n$ tends to $\infty$ keeping $k$ fixed and show that $\hat{I}{¾}_{n,k}$ becomes injective in the limit. We also determine explicitly (via some computer calculation) the homomorphism $\hat{I}{¾}_{\hat{I}}^P»$ for all the exceptional groups $G$ (with a specific `minimal' choice of $\hat{I}»$) and all their maximal parabolic subgroups except $E_8$.

28 pages