Geometric Langlands duality and representations of algebraic groups over commutative rings

2004/01/18 by Ivan Mirković, Mirkovic, I., Kari Vilonen +1 · 11 citations
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.math/0401222

openalex publication_date 2004/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we give a geometric version of the Satake isomorphism. Given a connected complex reductive algebraic group, we show that the category of representations of its Langlands dual is naturally equivalent to a certain category of perverse sheaves on the complex affine Grassmannian. We can work with perverse sheaves with coefficients in an arbitrary commutative ring and then we recover the representation theory of the split form of the dual group over the commutative ring.

Cited by

• Tilting modules for reductive algebraic groups: characters and support varieties
• Intersection cohomology without spaces
• Eigenforms and graphs of Hecke operators with wild ramification
• Isotopy invariance and stratified 𝔼₂-structure of the Ran Grassmannian
• Hecke modifications of conformal blocks outside the critical level
• Pointwise purity, derived Satake, and Symplectic duality
• Twisted Whittaker model and factorizable sheaves
• Geometric Endoscopy and Mirror Symmetry
• Modules over the small quantum group and semi-infinite flag manifold
• Relation between two geometrically defined bases in representations of GL_n
• A weight multiplicity formula for Demazure modules

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