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

For any reductive group $G$ and a parabolic subgroup $P$ with its Levi subgroup $L$, the first author [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 $G=SO(2n)$ and its maximal parabolic subgroups $P_{n-k}$ for any $2\leq k\leq n-1$ (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}^D: Rep^\mathbb{C}_{ω_1-poly}(SO(2k)) \to H^*(OG(n-k, 2n), \mathbb{C})$. We determine this homomorphism explicitly in the paper. We further analyze the behavior of $ ξ_{n,k}^D$ when $n$ tends to $\infty$ keeping $k$ fixed and show that $ ξ_{n,k}$ becomes injective in the limit. We also determine explicitly (via some computer calculation) the homomorphism $ ξ^P_λ$ for all the exceptional groups $G$ (with a specific `minimal' choice of $λ$) and all their maximal parabolic subgroups except $E_8$.