## Student Number Theory Seminar 2019-20

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2/24/20: John O'Brien -- *The Satake equivalence I: The classical formulation*

**Abstract:** When studying the representation theory of reductive groups, one runs into a mysterious phenomenon: a certain duality between certain groups. In 1963, IchirÅ Satake gave one of the first attempts of explaining this duality--the Satake isomorphism between the spherical Hecke algebra of one group over a local field with the complexified representation ring of the Langlands dual group. In this duo of talks, we will discuss two formulations of the Satake equivalence: the classical formulation in terms of algebras, and the more recent geometric formulation in terms of categories related to the affine Grassmannian, an infinite-dimensional space. For this first talk, we will give an overview of the classical formulation.