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For any reductive group $G$ and a parabolic subgroup $P$ with its Levisubgroup $L$, the first author [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 thecomplexified representation ring of $L$ (depending upon the choice of anirreducible representation $V(\lambda)$ of $G$ with highest weight $\lambda$).In this paper we study this homomorphism for $G=SO(2n)$ and its maximalparabolic subgroups $P_{n-k}$ for any $2\leq k\leq n-1$ (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 $\xi_{n,k}^D: Rep^\mathbb{C}_{\omega_1-poly}(SO(2k)) \to H^*(OG(n-k, 2n),\mathbb{C})$. We determine this homomorphism explicitly in the paper. Wefurther analyze the behavior of $ \xi_{n,k}^D$ when $n$ tends to $\infty$keeping $k$ fixed and show that $ \xi_{n,k}$ becomes injective in the limit. Wealso determine explicitly (via some computer calculation) the homomorphism $\xi^P_\lambda$ for all the exceptional groups $G$ (with a specific `minimal'choice of $\lambda$) and all their maximal parabolic subgroups except $E_8$.

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