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 [9] introduced a ring homomorphism ξλP:Repλ−polyC(L)→H⁎(G/P,C), where Repλ−polyC(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 Pn−k for any 2≤k≤n−1 (with the choice of V(λ) to be the defining representation V(ω1) in C2n). Thus, we obtain a C-algebra homomorphism ξn,kD:Repω1−polyC(SO(2k))→H⁎(OG(n−k,2n),C). We determine this homomorphism explicitly in the paper. We further analyze the behavior of ξn,kD when n tends to ∞ keeping k fixed and show that ξn,k becomes injective in the limit. We also determine explicitly (via some computer calculation) the homomorphism ξλP for all the exceptional groups G (with a specific ‘minimal’ choice of λ) and all their maximal parabolic subgroups except E8.