Multiplicity-one theorem

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.

Definition

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 ω be a continuous unitary character from Z(K)\Z(A)× to C×. Let L20(G(K)/G(A), ω) denote the space of cusp forms with central character ω on G(A). This space decomposes into a direct sum of Hilbert spaces

L

0

2

(

G

(

K

)

∖

G

(

A

)

,

ω

)

=

⨁

^

(

π

,

V

π

)

m

π

V

π

{\displaystyle L_{0}^{2}(G(K)\backslash G(\mathbf {A} ),\omega )={\widehat {\bigoplus }}_{(\pi ,V_{\pi })}m_{\pi }V_{\pi }}

where the sum is over irreducible subrepresentations and mπ are non-negative integers.

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 ω, i.e. mπ is 0 or 1 for all such π.

Results

The fact that the general linear group, GL(n), has the multiplicity-one property was proved by Jacquet & Langlands (1970) for n = 2 and independently by Piatetski-Shapiro (1979) and Shalika (1974) 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 (Blasius 1994).