Perverse sheaves on affine Grassmannians and Langlands duality. (English) Zbl 0987.14015

Mirković, Ivan; Vilonen, Kari

Math. Res. Lett. 7, No. 1, 13-24 (2000).

The paper outlines the construction of the Langlands duality, i.e. the equivalent of the category of \(k\)-representations of the Langlands dual group \(^LG\) and the category of perverse sheaves with coefficients in \(\mathbb{K}\) on the affine Grassmannian associated in \(G\) for a complex algebraic reductive group \(G\). The result for \(\mathbb{K}\) the field of complex numbers is due to V. Ginzburg (alg-geom/9511007). The authors generalize it to any commutative ring \(\mathbb{K}\), and claim new proofs for some crucial steps of the complex case, but the text only outlines the constructions and announces the results.

Reviewer: Alexei Rudakov (Trondheim)

MSC:

14F43	Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14M15	Grassmannians, Schubert varieties, flag manifolds
20G05	Representation theory for linear algebraic groups
18F20	Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects)

Keywords:

Satake isomorphism; Langlands duality; perverse sheaves; affine Grassmannian