zbMATH — the first resource for mathematics

Automorphic forms constructed from Whittaker vectors. (English) Zbl 0692.10029
Let G be a semisimple Lie group of split rank 1 and \(\Gamma\) a discrete subgroup of finite covolume. Forming a generalized Poincaré series of weak Whittaker functions attached to the nonunitary principal series a meromorphic family M(s) of functions on \(\Gamma\) \(\setminus G\) is obtained that satisfy all axioms of automorphic forms except the condition of moderate growth. It is shown that the principal part at a pole \(s_ 0\), Re \(s_ 0\geq 0\) is square integrable and that for a fixed K-type all automorphic forms up to a finite dimensional subspace are obtained. For the trivial K-type all automorphic forms are derived this way. The Fourier coefficients of the M-series are computed and a functional equation is derived.
Trying to generalize this to higher rank leads to the observation that in certain cases (proved here for \(G=SO(n,1)^ d)\) the moderate growth condition for automorphic forms is redundant. This may be called a Koecher principle. The conjecture is stated that a Koecher principle holds for all G of rank\(\geq 2\) and \(\Gamma\) irreducible.
Reviewer: A.Deitmar

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E40 Discrete subgroups of Lie groups
22E46 Semisimple Lie groups and their representations
Full Text: DOI
[1] Arthur, J, Eisenstein series and the trace formula, (), 153-274
[2] Borel, A, Introduction to automorphic forms, (), 199-210
[3] Borel, A; Garland, H, Laplacian and the discrete spectrum of an arithmetic group, Amer. J. math., 105, 309-335, (1983) · Zbl 0572.22007
[4] Borel, A; Wallach, N.R, Continuous cohomology discrete subgroups and representations of reductive groups, () · Zbl 0980.22015
[5] Good, A, Local analysis of Selberg’s trace formula, () · Zbl 0525.10013
[6] Goodman, R, Differential operators of infinite order on a Lie group, II, Indiana univ. math. J., 21, 283-409, (1971) · Zbl 0246.22011
[7] Goodman, R; Wallach, N.R, Whittaker vectors and conical vectors, J. funct. anal., 39, 199-279, (1980) · Zbl 0475.22010
[8] Harish-Chandra, Automorphic forms on semi-simple Lie groups, () · Zbl 0199.46403
[9] Jacquet, H, Fonctions de Whittaker associées aux groupes de Chevalley, Bull. soc. math. France, 95, 243-309, (1967) · Zbl 0155.05901
[10] Knapp, A.W; Stein, E.M, Intertwining operators for semisimple groups, Ann. of math., 93, 489-578, (1971) · Zbl 0257.22015
[11] Konstant, B, On Whittaker vectors and representation theory, Invent. math., 48, 101-184, (1978)
[12] Langlands, R.P, On the functional equations satisfied by Eisenstein series, () · Zbl 0204.09603
[13] Lehner, J, Discontinuous groups and automorphic functions, () · Zbl 0178.42902
[14] Lynch, E, Generalized Whittaker vectors and representation theory, ()
[15] Neunhöffer, H, Uber die analytische fortsetzung von Poincaré-reihen, (), 33-90
[16] Niebur, D, A class of non-analytic automorphic functions, Nagoya math. J., 52, 133-145, (1973) · Zbl 0288.10010
[17] Osborne, S; Warner, G, The theory of Eisenstein systems, (1981), Academic Press New York · Zbl 0489.43009
[18] Petersson, H, Uber die entwicklungscoeffizientem der automorphen formen, Acta math., 58, 169-215, (1932) · JFM 58.1110.01
[19] Rankin, R.A, On modular forms and functions, (1977), Cambridge Univ. Press London/New York · Zbl 0359.10021
[20] Schiffman, G, Integrales d’entrelacement et fonctions de Whittaker, Bull. soc. math. France, 99, 3-72, (1971)
[21] Shalika, J, The multiplicity one theorem for GL(n), Ann. of math., 100, 171-193, (1974) · Zbl 0316.12010
[22] Slater, L.J, Confuent hypergeometric functions, (1960), Cambridge Univ. Press London/New York · Zbl 0086.27502
[23] Smart, R, On modular forms of dimension −2, Trans. amer. math. soc., 116, 86-107, (1965) · Zbl 0144.08302
[24] \scJ. F. Treves, “Basic Linear P. D. E.,” Academic Press, Orlando, FL.
[25] Vogan, D, The algebraic structure of the representations of semisimple Lie groups I, Ann. of math., 109, 1-60, (1979) · Zbl 0424.22010
[26] Wallach, N.R, Lie algebra cohomology and holomorphic continuation of generalized Jacquet integrals, (), to appear · Zbl 0714.17016
[27] Wallach, N.R, Asymptotic expansions of generalized matrix entries of representations of real reductive groups, Lie group representations, I, () · Zbl 0553.22005
[28] Wallach, N.R, On the constant term of a square integrable automorphic form, operator algebras and group representations, () · Zbl 0554.22004
[29] Wallach, N.R, Real reductive groups, I, (1988), Academic Press San Diego · Zbl 0666.22002
[30] Whittaker, E.T; Watson, G.N, A course of modern analysis, (1963), Cambridge Univ. Press London · Zbl 0108.26903
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.