2022/07/10 by Kumar, Shrawan, Xie, Jiale
#14L235 #14M15 #14N15 #17B10 #Algebraic Topology (math.AT) #FOS: Mathematics #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2207.04537
For any reductive group G and a parabolic subgroup P with its Levi subgroup L, the first author [Ku2] introduced a ring homomorphism ξPλ: Rep^ℂλ-poly(L) → H^*(G/P, ℂ), where Rep^ℂλ-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 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: Rep^ℂω1-poly(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 ξPλ for all the exceptional groups G (with a specific `minimal' choice of λ) and all their maximal parabolic subgroups except E8.