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

$L^2_0(G(K)\backslash G(\mathbf{A}),\omega)=\hat{\bigoplus}_{(\pi,V_\pi)}m_\pi V_\pi$

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 of Piatetski-Shapiro (1979) and Jacquet & Shalika (1981) 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 (1): 237–251, doi:10.1007/BF02937513, ISSN 0021-2172, MR1303497 
• 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, Lectures on automorphic L-functions, Fields Inst. Monogr., 20, Providence, R.I.: American Mathematical Society, pp. 1–96, ISBN 978-0-8218-3516-6, MR2071506, http://www.math.osu.edu/~cogdell/ 
• Jacquet, Hervé; Langlands, Robert (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, 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, JSTOR 2374050, MR618323 
• Piatetski-Shapiro, I. I. (1979), "Multiplicity one theorems", in Borel, Armand; Casselman., W., 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, http://www.ams.org/publications/online-books/pspum331-index 
• Shalika, J. A. (1974), "The multiplicity one theorem for GL_n", Annals of Mathematics. Second Series 100: 171–193, ISSN 0003-486X, JSTOR 1971071, MR0348047