Gelfand-Kazhdan criterion
نویسنده
چکیده
Proofs that endomorphism (convolution) algebras are commutative most often depend upon identifying an anti-involution to interchange the order of factors, but which nevertheless acts as the identity on the algebra or suitable subalgebras. Silberger gave such an argument for the spherical Hecke algebras of p-adic reductive groups, for example. Apparently the first occurrence of the Gelfand-Kazhdan criterion idea is in I.M. Gelfand, ‘Spherical functions on symmetric spaces’, Dokl. Akad. Nauk SSSR 70 (1950), pp. 5-8. An extension of that idea appeared in I.M. Gelfand and D. Kazhdan, ‘Representations of the group GL(n, k) where k is a local field’, in Lie Groups and their Representations, Halsted, New York, 1975, pp. 95-118. A relatively recent survey of some of this is given by B. Gross, ‘Some applications of Gelfand pairs to number theory’, Bull. A.M.S. 24, no. 2 (1991), pp. 277-301.
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