Representation ring of Levi subgroups versus cohomology ring of flag varieties. II. (English) Zbl 1451.14145
Summary: For any reductive group \(G\) and a parabolic subgroup \(P\) with its Levi subgroup \(L\), the first author introduced in the first part [Math. Ann. 366, No. 1–2, 395–415 (2016; Zbl 1359.14044)] a ring homomorphism \(\xi_\lambda^P : \operatorname{Rep}_{\lambda - \operatorname{poly}}^{\mathbb{C}}(L) \to H^\ast(G / P, \mathbb{C})\), where \(\operatorname{Rep}_{\lambda - \operatorname{poly}}^{\mathbb{C}}(L)\) 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 \(G = \operatorname{Sp}(2 n)\) and its maximal parabolic subgroups \(P_{n - k}\) for any \(1 \leq k \leq n - 1\) (with the choice of \(V(\lambda)\) to be the defining representation \(V( \omega_1)\) in \(\mathbb{C}^{2 n}\)). Thus, we obtain a \(\mathbb{C} \)-algebra homomorphism \(\xi_{n, k} : \operatorname{Rep}_{\omega_1 - \operatorname{poly}}^{\mathbb{C}}(\operatorname{Sp}(2 k)) \to H^\ast(I G(n - k, 2 n), \mathbb{C})\). Our main result asserts that \(\xi_{n, k}\) is injective when \(n\) tends to \(\infty\) keeping \(k\) fixed. Similar results are obtained for the odd orthogonal groups.
MSC:
14M15 | Grassmannians, Schubert varieties, flag manifolds |
14L35 | Classical groups (algebro-geometric aspects) |
20G05 | Representation theory for linear algebraic groups |
20G07 | Structure theory for linear algebraic groups |
14L30 | Group actions on varieties or schemes (quotients) |
14M17 | Homogeneous spaces and generalizations |
17B67 | Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras |
22E10 | General properties and structure of complex Lie groups |
Citations:
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