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

Abstract

For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author in [5] 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=Sp(2n) and its maximal parabolic subgroups Pn−k for any 1≤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,k:Repω1−polyC(Sp(2k))→H⁎(IG(n−k,2n),C). Our main result asserts that ξn,k is injective when n tends to ∞ keeping k fixed. Similar results are obtained for the odd orthogonal groups.