Some applications of Gelfand pairs to number theory

The classical theory of Gelfand pairs has found a wide range of applications, ranging from harmonic analysis on Riemannian symmetric spaces to coding theory. Here we discuss a generalization of this theory, due to Gelfand-Kazhdan, and Bernstein, which was developed to study the representation theory of p-adic groups. We also present some recent number-theoretic results, on local e-factors and on the central critical values of automorphic L-functions, which fit nicely into this framework.

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