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The local converse theorem for \(\operatorname {SO}(2n+1)\) and applications. (English) Zbl 1049.11055
In the theory of admissible representations of \(p\)-adic groups, one of the basic problems is to characterize an irreducible admissible representation up to isomorphism. For irreducible generic representations of general linear groups, a characterization was given by G. Henniart [Invent. Math. 113, 339–350 (1993; Zbl 0810.11069)] in terms of twisted local gamma factors, and this is known as the Local Converse Theorem.
The main purpose of this paper is to prove the Local Converse Theorem for irreducible admissible generic representations of split odd special orthogonal groups. Moreover, two very important applications of this to the theory of admissible representations are given: firstly, the Langlands functoriality, from admissible representations of odd special orthogonal groups to those of general linear groups, is established for irreducible generic supercuspidal representations; secondly, the local Langlands reciprocity law, from irreducible generic supercuspidal representations of odd special orthogonal groups to certain symplectic representations of the Weil group, preserving \(L\)-, \(\varepsilon\)- and \(\gamma\)-factors, is proved.
Let us give some more details. Let \(k\) be a \(p\)-adic field (of characteristic zero), let \(\psi\) be a given non-trivial additive character of \(k\), and let \(\sigma\), \(\sigma'\) be irreducible admissible generic representations of \(\text{SO}_{2n+1}(k)\). For \(\rho\) an irreducible admissible generic representation of \(\text{GL}_l(k)\), the twisted local gamma factor \(\gamma(\sigma\times\rho,s,\psi)\) has been studied by D. Soudry [Rankin-Selberg convolutions for SO\(_{2\ell+1}\times \text{GL}_ n\): Local theory, Mem. Am. Math. Soc. 500 (1993; Zbl 0805.22007)] or, in a more general situation, by F. Shahidi [Ann. Math. (2) 132, 273–330 (1990; Zbl 0780.22005)].
The Local Converse Theorem (Theorem 1.2) says that, if \(\gamma(\sigma\times\rho,s,\psi)=\gamma(\sigma'\times\rho,s,\psi)\), for all irreducible supercuspidal representations \(\rho\) of \(\text{GL}_l(k)\), \(l\leq 2n-1\), then \(\sigma\) and \(\sigma'\) are isomorphic.
The first step in the proof is to reduce to the case when \(\sigma\) and \(\sigma'\) are supercuspidal. This is achieved using Jacquet’s subquotient theorem, the multiplicativity of twisted local gamma factors, and a careful analysis of the nature and locations of their poles (§5).
The case of supercuspidal representations is rather more involved, combining local and global methods. The first ingredient is the explicit local Howe duality for irreducible generic supercuspidal representations of \(\text{SO}_{2n+1}(k)\) and \(\widetilde{\text{Sp}}_{2n}(k)\), the metaplectic (double) cover of the symplectic group (Theorem 2.2).
The second ingredient is the (weak) Langlands functorial lift from \(\text{SO}_{2n+1}(\mathbb A)\) to \(\text{GL}_{2n}(\mathbb A)\), where \(\mathbb A\) is the ring of adeles of a number field, due to J. W. Cogdell, H. H. Kim, I. I. Piatetski-Shapiro and F. Shahidi [Publ. Math., Inst. Hautes Étud. Sci. 93, 5–30 (2001; Zbl 1028.11029)]. In particular, using an identity of twisted local gamma factors for the local components of the weak Langlands functorial lift, it is shown (Theorem 3.3) that, for each irreducible generic supercuspidal representation \(\sigma\) of \(\text{SO}_{2n+1}(k)\), there is a unique irreducible generic representation of \(\tau\) of \(\text{GL}_{2n}(k)\) with the same twisted local gamma factors. Moreover, \(\tau\) takes the from \(\eta_1\times\cdots\times\eta_t\), where the \(\eta_j\) are non-isomorphic irreducible unitary supercuspidal self-dual representations of \(\text{GL}_{2n_j}(k)\) (\(n=\sum_{j=1}^t n_j\)) and each local \(L\)-function \(L(\eta_j,\Lambda^2,s)\) has a pole at \(s=0\).
These two ingredients can be put together (Proposition 3.4) to show that, if \(\pi\) is the local \(\psi\)-Howe lift of \(\sigma\) and \(\tau\) as above is the local lift of \(\sigma\) to \(\text{GL}_{2n}(k)\), then the gamma function \(\gamma(\pi\times\tau,s,\psi)\) has a pole of order \(t\) at \(s=1\). This implies that \(\pi\) is the (unique) local backward lift of \(\tau\) to an irreducible generic supercuspidal representation of \(\widetilde{\text{Sp}}_{2n}(k)\) [see D. Ginzburg, S. Rallis and D. Soudry, J. Inst. Math. Jussieu 1, No. 1, 77–123 (2002; Zbl 1052.11032)].
Putting all this together with the Local Converse Theorem for \(\text{GL}_{2n}(k)\), we see (Theorem 4.1) that, if \(\sigma\), \(\sigma'\) are irreducible generic supercuspidal representations of \(\text{SO}_{2n+1}(k)\) with the same twisted local gamma factors, then their local \(\psi\)-Howe lifts are isomorphic, so that \(\sigma\) and \(\sigma'\) are isomorphic also.
Two immediate consequences of the Local Converse Theorem are that the weak Langlands functorial lift is injective (Theorem 5.2) and that two irreducible generic cuspidal automorphic representations of \(\text{SO}_{2n+1}(\mathbb A)\) are equivalent if and only if their local components are equivalent at almost all places (Rigidity Theorem 5.3).
The applications of the Local Converse Theorem to the Langlands conjectures are given in §6. First, local Langlands functoriality is established (Theorem 6.1): let \(\mathcal S\mathcal O^{\text{igsc}}_{2n+1}(k)\) be the set of equivalence classes of irreducible generic supercuspidal representations of \(\text{SO}_{2n+1}(k)\), and let \(\mathcal G\mathcal L^{\text{ifl}}_{2n}(k)\) be the set of equivalence classes of irreducible admissible representations of \(\text{GL}_{2n}(k)\) of the form \(\eta_1\times\cdots\times\eta_t\) as above; then there is a unique bijective map from \(\mathcal S\mathcal O^{\text{igsc}}_{2n+1}(k)\) to \(\mathcal G\mathcal L^{\text{ifl}}_{2n}(k)\) which preserves twisted local \(L\)-, \(\epsilon\)-, and \(\gamma\)-factors. Given the results above, the main thing to be checked here is surjectivity, which is again proved via Howe duality and the local backward lifting to \(\widetilde{\text{Sp}}_{2n}(k)\).
Finally, the local Langlands reciprocity law is proved (Theorem 6.4): there is a unique bijection from the set \(\mathcal G^0_{2n}(k)\) of conjugacy classes of \(2n\)-dimensional, admissible, completely reducible, multiplicity-free, symplectic complex representations of the Weil group of \(k\) onto \(\mathcal S\mathcal O^{\text{igsc}}_{2n+1}(k)\), which preserves twisted local \(L\)-, \(\epsilon\)- and \(\gamma\)-factors. This is proved using the functoriality above, by showing that the image of \(\mathcal G^0_{2n}(k)\) under the local Langlands correspondence for \(\text{GL}_{2n}\) is precisely the image \(\mathcal G\mathcal L^{\text{ifl}}_{2n}(k)\) of the functorial lift. In terms of the complete local Langlands reciprocity conjecture, an immediate consequence of the Local Converse Theorem is that each (conjectural) local \(L\)-packet of irreducible admissible representations of \(\text{SO}_{2n+1}(k)\) contains at most one generic member.
We remark that the proof of Theorem 6.3, due to Henniart, can (currently) be found in: G. Henniart, “Correspondance de Langlands et fonctions \(L\) des carrés extérieur et symétrique”, Prépublication M/03/20 de l’Institut des Hautes Études Scientifiques, at
http://www.ihes.fr/PREPRINTS/M03/M03-20.pdf.

MSC:
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields
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