Multiplicity-one theorem: Difference between revisions
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Created page, stated theorem, gave reference to Shalika's paper |
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let ''G/k'' be a reductive group, |
let ''G/k'' be a reductive group, |
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let ''Z<sub>A</sub>'' denote the center of ''G<sub>A</sub>'', |
let ''Z<sub>A</sub>'' denote the center of ''G<sub>A</sub>'', |
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let ''ρ'' denote a one-dimensional unitary representation of ''Z_A'' which is trivial on ''Z_A |
let ''ρ'' denote a one-dimensional unitary representation of ''Z_A'' which is trivial on ''Z_A ∩ G_k''. Consider the Hilbert space ''C<sub>ρ</sub>'' of cusp forms on ''G<sub>A</sub>'' associated with the character ''ρ''; the group ''G<sub>A</sub>'' acts on that space on the right by unitary transformations. This representation decomposes into a direct sum of irreducible representations each having finite multiplicity. |
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<h2>Multiplicity-one theorem</h2> |
<h2>Multiplicity-one theorem</h2> |
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Revision as of 04:25, 28 January 2010
Let k be a field, let G/k be a reductive group, let ZA denote the center of GA, let ρ denote a one-dimensional unitary representation of Z_A which is trivial on Z_A ∩ G_k. Consider the Hilbert space Cρ of cusp forms on GA associated with the character ρ; the group GA acts on that space on the right by unitary transformations. This representation decomposes into a direct sum of irreducible representations each having finite multiplicity.
Multiplicity-one theorem
Each irreducible unitary representation of GA occurs with multiplicity at most 1 in Cρ.
(1) Proven for G=GLn on page 187 of The Multiplicity One Theorem for GLn by J. A. Shalika, The Annals of Mathematics, Second Series, Vol. 100, No. 2 (Sep., 1974), pp. 171-193.