Let be a local field with ring of integers , and let be a geometrically connected split reductive group over . The spherical Hecke algebra is the ring of -valued compactly supported functions on the double coset space under convolution. Although convolution algebras are generally non-commutative, an argument known as Gelfand's trick? implies that is commutative.
The Satake isomorphism is a ring isomorphism
where
denotes the cardinality of the residue field of ,
denotes the Langlands dual group of , a split reductive group over , and
denotes the representation ring of .
The geometric Satake equivalence categorifies the Satake isomorphism in the setting of local geometric Langlands. To be precise, let now denote a reductive group over with Langlands dual group also over . Let denote the loop group of and let denote the arc group of . Then one has an equivalence of symmetric monoidal abelian categories
where:
is the affine Grassmannian of ,
denotes the category of -equivariant D-modules, with the convolution monoidal structure, and
denotes the category of algebraic representations of .
The construction of the correct commutativity constraint on the left hand side of this equivalence is subtleβwe warn that this is an additional data and not merely a property of the monoidal structure.
Wikipedia, Satake isomorphism
Benedict Gross, On the Satake isomorphism (pdf)
Dennis Gaitsgory, Graduate Seminar. Geometric Representation theory. Fall 2009βSpring 2010, Feb. 23: Affine Grassmannian-II: the Satake equivalence (Ryan Reich) (pdf)
example 1.7 in these rough notes, apparently taken in some talk by Jacob Lurie: Talk by Jacob Lurie
Last revised on July 22, 2024 at 00:15:24. See the history of this page for a list of all contributions to it.