A geometric Schur functor. (English) Zbl 1331.20061
Summary: We give geometric descriptions of the category \(C_k(n,d)\) of rational polynomial representations of \(\mathrm{GL}_n\) over a field \(k\) of degree \(d\) for \(d\leq n\), the Schur functor and Schur-Weyl duality. The descriptions and proofs use a modular version of Springer theory and relationships between the equivariant geometry of the affine Grassmannian and the nilpotent cone for the general linear groups. Motivated by this description, we propose generalizations for an arbitrary connected complex reductive group of the category \(C_k(n,d)\) and the Schur functor.
MSC:
| 20G43 | Schur and \(q\)-Schur algebras |
| 20G05 | Representation theory for linear algebraic groups |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 17B08 | Coadjoint orbits; nilpotent varieties |
| 20C30 | Representations of finite symmetric groups |
Keywords:
modular Springer theory; Schur algebras; Schur functors; Schur-Weyl duality; perverse sheaves; nilpotent cones; affine Grassmannians; categories of polynomial representations; general linear groups; representations of symmetric groupsReferences:
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