Multiplicity-one theorem

Let k be a field, let G/k be a reductive group, let Z_A denote the center of G_A, let ρ denote a one-dimensional unitary representation of Z_A which is trivial on Z_A\cap G_k. Consider the Hilbert space C_ρ of cusp forms on G_A associated with the character ρ; the group G_A 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 G_A occurs with multiplicity at most 1 in C_ρ.

(1) Proven for G=GL_n on page 187 of The Multiplicity One Theorem for GL_n by J. A. Shalika, The Annals of Mathematics, Second Series, Vol. 100, No. 2 (Sep., 1974), pp. 171-193.