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Some applications of Gel’fand pairs to number theory. (English) Zbl 0733.11018

The author discusses a generalization of the classical theory of Gel’fand pairs (G,H) where G is a compact group and H a closed subgroup, to the case where G is locally compact totally disconnected, H is closed and the space \(H\setminus G\) carries a G-invariant measure. This generalization is due to I. M. Gel’fand and D. A. Kazhdan [Lie Groups Represent., Proc. Summer Sch. Bolyai János Math. Soc. 1971, 95-118 (1975; Zbl 0348.22011)] and J. N. Bernstein [Lect. Notes Math. 1041, 50-102 (1984; Zbl 0541.22009)] and was developed to study the representation theory of p-adic groups. The author also discusses some number theoretic results on the central critical values of automorphic L- functions fitting into this framework.

MSC:

11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
22D12 Other representations of locally compact groups
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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