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.

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^×. Fix a Haar measure on G(A) and let L²₀(G(K)/G(A), ω) denote the Hilbert space of measurable complex-valued functions, f, on G(A) satisfying

f(γg) = f(g) for all γ ∈ G(K)
f(gz) = f(g)ω(z) for all z ∈ Z(A)
$\int _{Z(\mathbf {A} )G(K)\backslash G(\mathbf {A} )}|f(g)|^{2}\,dg<\infty$
$\int _{U(K)\backslash U(\mathbf {A} )}f(ug)\,du=0$ for all unipotent radicals, U, of all proper parabolic subgroups of G(A).

This is called 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)\backslash G(\mathbf {A} ),\omega )={\hat {\bigoplus }}_{(\pi ,V_{\pi })}m_{\pi }V_{\pi }

where the sum is over irreducible subrepresentations and m_{π 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. m_π 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).

References

Blasius, Don (1994), "On multiplicities for SL(n)", Israel Journal of Mathematics, 88: 237–251
Cogdell, James (2004), Lecturs on L-functions, converse theorems, and functoriality for GL(n), pp. 1–105, ISBN 978-0-821-83516-6
Jacquet, Hervé; Langlands, Robert (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114, Springer-Verlag
Piatetski-Shapiro, I.I. (1979), Multiplicity one theorems, Proceedings of the Symposium in Pure Mathematics, vol. 33-I, AMS, pp. 209–212
Shalika, Joseph (1974), "The multiplicity one theorem for GL(n)", Annals of Mathematics, 100: 171–193