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**Démonstration de la conjecture \(\tau\). (Proof of the \(\tau\)-conjecture.).**
*(French)*
Zbl 1025.11012

Let \(G\) be a (simply connected) semisimple group over a number field \(k\), denote by \(\mathbb A\) the adeles of \(k\) and let \(v\) be a place of \(k\). Then \(G(k_v)\) acts on the space of automorphic forms \(\mathcal A_G=L^2(G(k)\backslash G(\mathbb A))\) and denote by \(\mathcal A_G^0\) the subspace of forms with zero integral. If \(G(k_v)\) is not compact then, by strong approximation, the trivial representation does not appear in \(\mathcal A_G^0\).

In this paper, the author proves the “\(\tau\)-conjecture” of A. Lubotzky and R. J. Zimmer [Isr. J. Math. 66, 289-299 (1989; Zbl 0706.22010)]: the support of \(\mathcal A_G^0\) in the unitary dual of \(G(k_v)\) is separated from the trivial representation.

The idea of the proof is as follows. We may assume \(G(k_v)\) has rank \(1\), for otherwise Kazhdan’s property \((T)\) implies the result. Now, if \(H\) is a semisimple subgroup of \(G\) such that \(H(k_v)\) is also of rank \(1\), then the \(\tau\)-conjecture for \(G\) is implied by that for \(H\) [M. Burger and P. Sarnak, Invent. Math. 106, 1-11 (1991; Zbl 0774.11021)] for \(v\) a real place [the author and F. Ullmo, Equidistribution des points de Hecke, to appear in “Contributions to Automorphic Forms, Geometry and Arithmetic”. To commemorate the sixtieth birthday of J. Shalika, H. Hida, D. Ramakrishnan, F. Shahidi (eds.) (Johns Hopkins University Press, Baltimore)] for \(v\) \(p\)-adic).

The first step in the paper is then to determine the semisimple groups which have no proper semisimple subgroup: \(SL(2)\); \(SL(1,D)\), for \(D\) a division algebra of prime degree over an extension of \(k\); and \(SU(D,*)\), where \(D\) is a division algebra of prime degree over a quadratic extension \(E\) of a finite extension \(K\) of \(k\) and \((*)\) is an \(E/K\)-involution of the second kind.

The case of \(SL(2)\) may be treated directly, and the restriction on the rank of \(G\) reduces the case \(SL(1,D)\) to a quaternionic algebra which is similar, using Jacquet-Langlands functoriality. The bulk of the paper is then concerned with the final case \(SU(D,*)\) where the author proceeds via a base change to \(D^\times\). This is achieved by direct comparison in a trace formula.

In this paper, the author proves the “\(\tau\)-conjecture” of A. Lubotzky and R. J. Zimmer [Isr. J. Math. 66, 289-299 (1989; Zbl 0706.22010)]: the support of \(\mathcal A_G^0\) in the unitary dual of \(G(k_v)\) is separated from the trivial representation.

The idea of the proof is as follows. We may assume \(G(k_v)\) has rank \(1\), for otherwise Kazhdan’s property \((T)\) implies the result. Now, if \(H\) is a semisimple subgroup of \(G\) such that \(H(k_v)\) is also of rank \(1\), then the \(\tau\)-conjecture for \(G\) is implied by that for \(H\) [M. Burger and P. Sarnak, Invent. Math. 106, 1-11 (1991; Zbl 0774.11021)] for \(v\) a real place [the author and F. Ullmo, Equidistribution des points de Hecke, to appear in “Contributions to Automorphic Forms, Geometry and Arithmetic”. To commemorate the sixtieth birthday of J. Shalika, H. Hida, D. Ramakrishnan, F. Shahidi (eds.) (Johns Hopkins University Press, Baltimore)] for \(v\) \(p\)-adic).

The first step in the paper is then to determine the semisimple groups which have no proper semisimple subgroup: \(SL(2)\); \(SL(1,D)\), for \(D\) a division algebra of prime degree over an extension of \(k\); and \(SU(D,*)\), where \(D\) is a division algebra of prime degree over a quadratic extension \(E\) of a finite extension \(K\) of \(k\) and \((*)\) is an \(E/K\)-involution of the second kind.

The case of \(SL(2)\) may be treated directly, and the restriction on the rank of \(G\) reduces the case \(SL(1,D)\) to a quaternionic algebra which is similar, using Jacquet-Langlands functoriality. The bulk of the paper is then concerned with the final case \(SU(D,*)\) where the author proceeds via a base change to \(D^\times\). This is achieved by direct comparison in a trace formula.

Reviewer: Shaun Stevens (Norwich)