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Ramanujan duals. II. (English) Zbl 0774.11021
In Part I of this paper [Bull. Am. Math. Soc., New. Ser. 26, No. 2, 253- 257 (1992; Zbl 0762.22009) (joint with J. S. Li)], the authors introduced the concept of the automorphic dual $$\widehat G_{\operatorname{Aut}}$$ of a semisimple group $$G$$ defined over $$\mathbb{Q}$$. Roughly speaking, $$\widehat G_{\operatorname{Aut}}$$ consists of the unitary representations of $$G(\mathbb{R})$$ which occur in $$L^ 2(\Gamma(N)\setminus G(\mathbb{R}))$$ for some $$N\geq 1$$. Whereas in Part I it was proved that $$\text{Ind}^{G(\mathbb{R})}_{H(\mathbb{R})} \widehat H_{\operatorname{Aut}}\subset \widehat G_{\operatorname{Aut}}$$ for any semisimple subgroup $$H$$ of $$G$$, i.e., any $$\pi$$ (weakly) contained in $$\text{Ind}^{G(\mathbb{A})}_{H(\mathbb{A})}\tau$$ for some $$\tau$$ in $$\widehat H_{\operatorname{Aut}}$$ also lies in $$\widehat G_{\operatorname{Aut}}$$, in Part II the main result is that $$\text{Res}^{G(\mathbb{R})}_{H(\mathbb{R})} \widehat G_{\operatorname{Aut}}\subset \widehat H_{\operatorname{Aut}}$$.
To illustrate possible applications, let $$\widehat G_{\text{Raman}}$$ denote the intersection of $$\widehat G_{\operatorname{Aut}}$$ with the class 1 unitary representations of $$G(\mathbb{R})$$. Now suppose $$G=\text{Res}^ k_{\mathbb{Q}} SO(q)$$, where $$q$$ is a quadratic form over a totally real field $$k$$, such that $$q$$ has signature $$(n,1)$$ over $$\mathbb{R}$$ and all other conjugates are definite. Then the main result alluded to above is used to prove that $$\widehat G_{\text{Raman}}\subset i\mathbb{R}^ +\cup [0,\rho- 1/2]\cup\{\rho\}$$ (here the class 1 unitary representations of $$G(\mathbb{R})$$ have been identified with the set $$i\mathbb{R}^ +\cup [0,\rho])$$.
In fact, assuming the Ramanujan conjecture “at infinity” for $$GL(2,F)$$ (where $$F$$ is any number field), this last containment can be improved to one with $$\rho-1$$ in place $$\rho-1/2$$ equivalently, in terms of the first eigenvalue $$\lambda_ 1$$ of the Laplacian acting on $$L^ 2(\Gamma\setminus \mathbb{H}^ n)$$, where $$\mathbb{H}^ n$$ is the hyperbolic $$n$$- space, and $$\Gamma$$ is a congruence subgroup, $$\lambda_ 1\geq (2n-3)/4$$ (resp. $$\lambda_ 1\geq n-2$$, assuming Ramanujan …). In case $$k=\mathbb{Q}$$ and $$n\geq 4$$, $$G(\mathbb{Q})$$ is isotropic, and similar results were proved independently and earlier (using Kloosterman sums) by J. Elstrodt, F. Grunewald and J. Mennicke [Invent. Math. 101, 641-685 (1990; Zbl 0737.11013)] and J. S. Li, I. I. Piatetski- Shapiro and P. Sarnak [Proc. Indian Acad. Sci., Math. Sci. 97, 231-237 (1987; Zbl 0659.10028)].
In the present paper, the main result above is applied taking $$H\approx SO(3,1)$$ (where the required upper bounds on $$\widehat H_{\text{Raman}}$$ are already known).

##### MSC:
 11F55 Other groups and their modular and automorphic forms (several variables) 11F72 Spectral theory; trace formulas (e.g., that of Selberg) 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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##### References:
 [1] [A] Arthur, J.: On Some Problems Suggested by the Trace Formula. (Lect. Notes Math., vol. 1041, pp. 1-50) Berlin Heidelberg New York: Springer 1983 [2] [BS] Borel, A., Serre, J.P.: Cohomologie d’immeubles et de Groupes S-arithmétiques. Topology15, 211-232 (1976) · Zbl 0338.20055 [3] [BLSI] Burger, M., Li, J.S., Sarnak, P.: Ramanujan Duals and automorphic Spectrum. (Preprint) · Zbl 0762.22009 [4] [Ca] Cassels, J.W.S.: Rational Quadratic Forms. Reading, MA: Academic Press 1978 · Zbl 0395.10029 [5] [Dd] Dixmier, J.: C*-Algebras. Amsterdam: North Holland 1982 [6] [E-G-M] Elstrodt, J., Grunewald, F., Mennicke, J.: Kloosterman Sums for Clifford Algebras and a Lower Bound for the Positive Eigenvalues of the Laplacian for Congruence Subgroups Acting on Hyperbolic Spaces. Invent. Math.101, 641-685 (1990) · Zbl 0737.11013 [7] [G-J] Gelbart, S.S., Jacquet, H.: A Relation Between Automorphic Representations of GL(2) and GL(3). Ann. Sci. Éc. Norm. Supér., IV. Sér. II., 471-542 (1978) · Zbl 0406.10022 [8] [G-V] Gangolli, R., Varadarajan, V.S.: Harmonic Analysis of Spherical Functions on Real Reductive Groups. (Ergeb. Math. Grenzgeb., vol. 101) Berlin Heidelberg New York: Springer 1988 · Zbl 0675.43004 [9] [J-L] Jacquet, H., Langlands, R.P.: Automorphic Forms on GL(2). (Lect. Notes Math., vol. 114) Berlin Heidelberg New York: Springer 1970 [10] [K] Kneser, M.: Strong Approximation, in Algebraic Groups and Discontinuous Subgroups. Proc. Symp. Pure Math. IX., 187-196 (1966) [11] [Ko] Kostant, B.: On the Existence and Irreducibility of a Certain Series of Representations. Bull. Am. Math. Soc.75, 627-642 (1969) · Zbl 0229.22026 [12] [L-P-S] Li, J.S., Piatetskii-Shapiro, I.I., Sarnak, P.: Poincaré Series for SO(n, 1). Proc. Indian Acad. Sci., Math. Sci.97, 231-237 (1987) · Zbl 0659.10028 [13] [P] Platonov, V.P.: Arithmetic Theory of Algebraic Groups. Russ. Math. Surv.37, 1-62 (1982) · Zbl 0513.20028 [14] [R] Ratner, M.: On Measure Rigidity of Unipotent Subgroups of Semisimple Groups. (Preprint) · Zbl 0745.28010 [15] [S] Sarnak, P.: The Arithmetic and Geometry of Some Hyperbolic 3-manifolds. Acta Math.151, 253-295 (1983) · Zbl 0527.10022 [16] [San] Sansuc, J.J.: Groupe de Brauer et Arithmétique des Groupes Algébriques Linéaires sur un Corps de Nombres. J. Reine Angew. Math.327, 12-80 (1981) · Zbl 0468.14007 [17] [V] Vigneras, M.F.: Quelques Remarques sur la Conjecture ?1-1/4. Séminaire de Théorie des Nombres, Paris 1981-82. (Prog. Math.) Boston Basel Stuttgart: Birkhäuser
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