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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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