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On the nonnegativity of \(L(\frac12,\pi)\) for \(\text{SO}_{2n+1}\). (English) Zbl 1067.11026

Let \(\mathbb A\) be the adele ring of a number field \(k\). According to the authors, the goal of their paper is to show that \(L({1\over 2},\pi)\geq 0\) for any symplectic cuspidal representation \(\pi\) of the group \(\text{GL}_n(\mathbb A)\). As a consequence of that result, the authors prove that \(L^S({1\over 2},\sigma)\geq 0\) for a cuspidal generic representation \(\sigma\) of the group \(\text{SO}(2n+1,\mathbb A)\), where \(L^S(s,\sigma)\) is the partial \(L\)-function corresponding to \(S\), a finite set of primes of \(k\); the authors remark that this theorem “applies equally well to the completed \(L\)-function as defined by F. Shahidi in [Am. J. Math. 103, 297–355 (1981; Zbl 0467.12013)]”.
The authors’ third main theorem concerns the \(\varepsilon\)-root numbers of the \(L\)-functions \(L(s,\pi,\wedge^2)\) and \(L(s,\pi,\text{sym}^2)\) associated to a self-dual cuspidal representation \(\pi\) of the group \(\text{GL}_n(\mathbb A)\); it asserts that \[ \varepsilon(\textstyle{{1\over 2}},\pi,\wedge^2)= \varepsilon(\textstyle{{1\over 2}},\pi, \text{sym}^2)= 1. \] Since this paper had been written, the first author generalized the main theorem of the reviewed paper to tensor product \(L\)-functions of symplectic type, [E. M. Lapid, Int. Math. Res. Not. 2003, No. 2, 65–75 (2003; Zbl 1046.11032)], and proved that some other root numbers of orthogonal type are also equal to 1 [E. M. Lapid, Compos. Math. 140, No. 2, 274–286 (2004; Zbl 1052.11034)].
Reviewer: B. Z. Moroz (Bonn)

MSC:

11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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