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^×. 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

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

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).

Strong multiplicity one theorem

The strong multiplicity one theorem 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.

References

• Blasius, Don (1994), "On multiplicities for SL(n)", Israel Journal of Mathematics, 88: 237–251
• Cogdell, James W. (2004), "Lectures on L-functions, converse theorems, and functoriality for GL_n", in Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (eds.), Lectures on automorphic L-functions, Fields Inst. Monogr., vol. 20, Providence, R.I.: American Mathematical Society, pp. 1–96, ISBN 978-0-8218-3516-6, MR2071506 {{citation}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)
• Jacquet, Hervé; Langlands, Robert (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, vol. 114, Springer-Verlag
• Jacquet, H.; Shalika, J. A. (1981), "On Euler products and the classification of automorphic representations. I", American Journal of Mathematics, 103 (3): 499–558, doi:10.2307/2374103, ISSN 0002-9327, MR618323 Jacquet, H.; Shalika, J. A. (1981), "On Euler products and the classification of automorphic representations. II", American Journal of Mathematics, 103 (4): 777–815, ISSN 0002-9327, MR618323
• Piatetski-Shapiro, I. I. (1979), "Multiplicity one theorems", in Borel, Armand; Casselman., W. (eds.), Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos. Pure Math., XXXIII, Providence, R.I.: American Mathematical Society, pp. 209–212, ISBN 978-0-8218-1435-2, MR546599
• Shalika, J. A. (1974), "The multiplicity one theorem for GL_n", Annals of Mathematics. Second Series, 100: 171–193, ISSN 0003-486X, MR0348047