Knowlpedia

Construction

Convolution of sheaves on the affine Grassmannian

The sheaf obtained by external product on the convolution Grassmannian followed by proper pushforward along multiplication.

Core idea

Let Gr⁡G×~Gr⁡G\operatorname{Gr}_G\widetilde\times\operatorname{Gr}_G be the convolution Grassmannian and let

m:Gr⁡G×~Gr⁡G⟶Gr⁡Gm:\operatorname{Gr}_G\widetilde\times\operatorname{Gr}_G \longrightarrow\operatorname{Gr}_G

compose the two successive modifications. For L+GL^+G-equivariant constructible complexes F,G\mathcal F,\mathcal G, their convolution is

F⋆G=m!(F⊠~G).\mathcal F\star\mathcal G =m_!\bigl(\mathcal F\widetilde\boxtimes\mathcal G\bigr).

The map mm is ind-proper on the relevant finite-dimensional supports, so m!=m∗m_!=m_*. With the Satake normalization, convolution preserves and defines the tensor product used in .

References
  1. Ivan Mirković and Kari Vilonen, “Geometric Langlands duality and representations of algebraic groups over commutative rings,” Annals of Mathematics 166 (2007), 95–143. arXiv.