In the mathematical theory of automorphic representations, a multiplicity-one theorem is a result about the representation theory of an adelic reductive algebraic group. The multiplicity in question is the number of times a given abstract group representation is realised in a certain space, of square-integrable functions, given in a concrete way.
A multiplicity one theorem may also refer to a result about the restriction of a representation of a group G to a subgroup H. In that context, the pair (G, H) is called a strong Gelfand pair.
Let G be a reductive algebraic group over a number field K and let A denote the adeles of K. Let Z denote the centre of G and let be a continuous unitary character from Z(K)\Z(A)× to C×. Let L20(G(K)/G(A), ) denote the space of cusp forms with central character ω on G(A). This space decomposes into a direct sum of Hilbert spaces
| 2 | |
| L | |
| 0(G(K)\backslash |
| G(A),\omega)=\widehat{oplus} | |
| (\pi,V\pi) |
m\piV\pi
The group of adelic points of G, G(A), is said to satisfy the multiplicity-one property if any smooth irreducible admissible representation of G(A) occurs with multiplicity at most one in the space of cusp forms of central character , i.e. m is 0 or 1 for all such .
The fact that the general linear group, GL(n), has the multiplicity-one property was proved by for n = 2 and independently by and for n > 2 using the uniqueness of the Whittaker model. Multiplicity-one also holds for SL(2), but not for SL(n) for n > 2 .
The strong multiplicity one theorem of and states that two cuspidal automorphic representations of the general linear group are isomorphic if their local components are isomorphic for all but a finite number of places.