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)&times; to C&times;. Let L20(G(K)/G(A), $ω$) denote the space of cusp forms with central character &omega; on G(A). This space decomposes into a direct sum of Hilbert spaces
 * $$L^2_0(G(K)\backslash G(\mathbf{A}),\omega)=\widehat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi$$

where the sum is over irreducible subrepresentations and m$\pi$ 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 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.

Strong multiplicity one theorem
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.