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

TL;DR

This paper explicitly studies the homomorphism connecting representation rings of Levi subgroups to cohomology rings of flag varieties for classical and exceptional groups, analyzing its properties and limits.

Contribution

It provides explicit descriptions of the homomorphism for $SO(2n)$ and exceptional groups, and investigates its behavior as the rank tends to infinity.

Findings

Explicit homomorphism for $SO(2n)$ and maximal parabolics

Injectivity of the homomorphism in the limit as $n$ approaches infinity

Explicit calculations for exceptional groups (except $E_8$)

Abstract

For any reductive group $G$ and a parabolic subgroup $P$ with its Levi subgroup $L$, 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 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 $G=SO(2n)$ and its maximal parabolic 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. We further 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. We also 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 $E_8$.