SMOOTH REPRESENTATIONS OF p-ADIC REDUCTIVE GROUPS
نویسنده
چکیده
Smooth representations of p-adic groups arise in number theory mainly through the study of automorphic representations, and thus in the end, for example, from modular forms. We saw in the first lecture by Matt Emerton that a modular form, thought of as function on the set of lattices with level N structure, we obtain a function in C(GL2(Z)\GL2(R) × GL2(Z/N),C) satisfying certain differential equations (coming from holomorphicity and the weight of the modular form) and growth conditions (related to the cusps). As was suggested, this space has an adelic interpretation: by the strong approximation theorem, GL2(AQ) = GL2(Q)GL2(Ẑ)GL2(R), and a simple argument shows that C(GL2(Z)\GL2(R)×GL2(Z/N),C) ∼= C(GL2(Q)\GL2(AQ)/U(N),C),
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