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Representation ring of Levi subgroups versus cohomology ring of flag varieties II
Journal of Algebra ( IF 0.8 ) Pub Date : 2020-08-01 , DOI: 10.1016/j.jalgebra.2020.02.029
Shrawan Kumar , Sean Rogers

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 ξ λ P : Rep λ − poly C ( L ) → H ⁎ ( G / P , C ) , where Rep λ − poly C ( 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 ( 2 n ) and its maximal parabolic subgroups P n − k for any 1 ≤ k ≤ n − 1 (with the choice of V ( λ ) to be the defining representation V ( ω 1 ) in C 2 n ). Thus, we obtain a C -algebra homomorphism ξ n , k : Rep ω 1 − poly C ( Sp ( 2 k ) ) → H ⁎ ( I G ( n − k , 2 n ) , C ) . Our main result asserts that ξ n , k is injective when n tends to ∞ keeping k fixed. Similar results are obtained for the odd orthogonal groups.

中文翻译:

Levi 亚群的表示环与标志变体 II 的上同调环

摘要 对于任何还原群 G 和抛物线子群 P 及其 Levi 子群 L,[5] 中的第一作者引入了环同态 ξ λ P : Rep λ − poly C ( L ) → H ⁎ ( G / P , C ) ,其中 Rep λ − poly C ( L ) 是 L 的复表示环的某个子环(取决于对 G 具有最高权重 λ 的不可约表示 V ( λ ) 的选择)。在本文中,我们研究了 G = Sp ( 2 n ) 及其最大抛物线子群 P n − k 的同态,对于任何 1 ≤ k ≤ n − 1(选择 V ( λ ) 作为定义表示 V ( ω 1 ) 在 C 2 n )。因此,我们得到一个 C -代数同态 ξ n , k : Rep ω 1 − poly C ( Sp ( 2 k ) ) → H ⁎ ( IG ( n − k , 2 n ) , C ) 。我们的主要结果断言 ξ n , k 是单射的,当 n 趋于 ∞ 保持 k 固定时。对于奇数正交组获得了类似的结果。
更新日期:2020-08-01
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