nLab geometric Satake equivalence

Contents

Contents

Idea

Let KK be a local field with ring of integers π’ͺ\mathcal{O}, and let GG be a geometrically connected split reductive group over π’ͺ\mathcal{O}. The spherical Hecke algebra is the ring β„‹\mathcal{H} of Z\mathbf{Z}-valued compactly supported functions on the double coset space G(π’ͺ)\G(K)/G(π’ͺ)G(\mathcal{O}) \backslash G(K) / G(\mathcal{O}) under convolution. Although convolution algebras are generally non-commutative, an argument known as Gelfand's trick? implies that β„‹\mathcal{H} is commutative.

The Satake isomorphism is a ring isomorphism

β„‹βŠ—Z[q Β±1/2]≃R( LG)βŠ—Z[q Β±1/2]\mathcal{H} \otimes \mathbf{Z}[q^{\pm 1/2}] \simeq R({}^L G) \otimes \mathbf{Z}[q^{\pm 1/2}]

where

The geometric Satake equivalence categorifies the Satake isomorphism in the setting of local geometric Langlands. To be precise, let now GG denote a reductive group over C\mathbf{C} with Langlands dual group LG{}^L G also over C\mathbf{C}. Let β„’G\mathcal{L}G denote the loop group of GG and let β„’ +G\mathcal{L}^+G denote the arc group of GG. Then one has an equivalence of symmetric monoidal abelian categories

DMod(Gr G) β„’ +G≃Rep( LG)\mathrm{DMod}(\mathrm{Gr}_G)^{\mathcal{L}^+G} \simeq \mathrm{Rep}({}^L G)

where:

  • Gr G=β„’G/β„’ +G\mathrm{Gr}_G = \mathcal{L}G/\mathcal{L}^+G is the affine Grassmannian of GG,

  • DMod(Gr G) β„’ +G\mathrm{DMod}(\mathrm{Gr}_G)^{\mathcal{L}^+G} denotes the category of β„’ +G\mathcal{L}^+G-equivariant D-modules, with the convolution monoidal structure, and

  • Rep( LG)\mathrm{Rep}({}^L G) denotes the category of algebraic representations of LG{}^L G.

The construction of the correct commutativity constraint on the left hand side of this equivalence is subtle–we warn that this is an additional data and not merely a property of the monoidal structure.

References

Last revised on July 22, 2024 at 00:15:24. See the history of this page for a list of all contributions to it.