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The unitary dual of \(p\)-adic \(G_2\). (English) Zbl 0896.22006
The author determines the set \(\widehat G_2\) of equivalence classes of irreducible unitarizable representations of a simply connected split simple group \(G\) of type \(G_2\) over a nonarchimedean field \(F\) of characteristic zero. The corresponding problem in the complex and real cases were respectively solved by M. Duflo [Bull. Soc. Math. Fr. 107, 55-96 (1979; Zbl 0407.22014)] and D. Vogan [Invent. Math. 116, No. 1-3, 677-791 (1994; Zbl 0808.22003)]. It is natural to think that these results will also play a role in the understanding of both local and global Howe correspondences (see G. Savin [Invent. Math. 118, No. 1, 141-160 (1994; Zbl 0858.22015), J. Reine Angew. Math. 434, 115-126 (1993; Zbl 0762.22013)], B. H. Gross and G. Savin [Can. Math. Bull. 40, No. 3, 376-384 (1997; Zbl 0881.22010)], K. Magaard and G. Savin [Compos. Math. 107, No. 1, 89-123 (1997; Zbl 0878.22011)], the latter authors and J.-S. Huang [J. Reine Angew. Math. 500, 65-81 (1998)]).
Let us describe the main results of the paper under review. Let \(F\) be a nonarchimedean field of characteristic zero, with ring of integers \({\mathfrak O}_F\), let \({\mathfrak P}_F\) and \({\mathbb F}_q\) respectively denote the maximal ideal of \({\mathfrak O}_F\) and the residual field of \(F\). Let \(T\) be a split torus and \(B=TU\) a Borel subgroup of \(G\). Denote by \(\{\alpha,\beta,\alpha+\beta,2\alpha+\beta,3\alpha+\beta,3\alpha+2\beta\}\) a corresponding set of positive roots, and by \(X(T)\) the group of rational characters of \(T\). We have \(X(T)={\mathbb Z}(2\alpha+\beta)\oplus{\mathbb Z}(\alpha+\beta)\), and the positive chamber in \(X(T)\otimes_{{\mathbb Z}}{\mathbb R}\) is \({\mathcal C}^+=\{s(2\alpha+\beta)+s'(\alpha+\beta):s>s'>0\}\).
Write \(I(\cdot\otimes\cdot)\) for the normalized induction from \(B\) to \(G\). Let \(\chi_1\), \(\chi_2\) be unitary characters of \(F^\times\), and \(s_1>s_2>0\). Then the representation \(I(|\text{det}(\;)|_F^{s_1}\chi_1\otimes|\text{det}(\;)|_F^{s_2}\chi_2)\) has a unique irreducible quotient denoted by \(J(s_1,s_2,\chi_1,\chi_2)\). Further the intertwining operator \[ A(s_1,s_2,\chi_1,\chi_2,w_0)\colon I(|\text{det}(\;)|_F^{s_1}\chi_1\otimes| \text{det}(\;)|_F^{s_1}\chi_2)\rightarrow I(|\text{det}(\;)|_F^{-s_1}\chi_1^{-1}\otimes|\text{det}(\;)|_F^{-s_2}\chi_2^{-1}), \] defined by \[ A(s_1,s_2,\chi_1,\chi_2,w_0)f(g):=\int_{U\cap w_0\overline U w_0^{-1}}f(w_0^{-1}ug)du, \] \[ f\in I(|\text{det}(\;)|_F^{s_1}\chi_1\otimes|\text{det}(\;)|_F^{s_2}\chi_2), \] where \(w_0\) denotes the element of maximal length in the Weyl group and \(\overline U\) denotes the unipotent radical opposed to \(U\), has image isomorphic to \(J(s_1,s_2,\chi_1,\chi_2)\).
It is a result of C. D. Keys [Pac. J. Math. 101, 351-388 (1982; Zbl 0488.22035)] that a unitary principal series \(I(\chi_1\otimes\chi_2)\) is reducible if and only if \(\chi_1\) and \(\chi_2\) are different characters of order \(2\). It is of multiplicity \(1\) and length \(2\). The author shows that a nonunitary series \(I(\chi_1\otimes\chi_2)\) is irreducible if and only if \(\chi_1\neq|\text{det}(\;)|_F^{\pm 1}\), \(\chi_2\neq|\text{det}(\;)|_F^{\pm 1}\), \(\chi_1\chi_2\neq|\text{det}(\;)|_F^{\pm 1}\), \(\chi_1\chi_2^{-1}\neq|\text{det}(\;)|_F^{\pm 1}\), \(\chi_1^2\chi_2\neq|\text{det}(\;)|_F^{\pm 1}\), \(\chi_1\chi_2^2\neq|\text{det}(\;)|_F^{\pm 1}\).
He obtains the following classification of all the square integrable representations which are supported on a minimal parabolic subgroup:
(1) If \(\chi\) is of order \(2\) or \(3\), then the induced representation \(I(|\text{det}(\;)|_F\chi\otimes\chi)\) contains a unique irreducible subrepresentation \(\pi(\chi)\), which is square integrable, and the only equivalences among these representations are \(\pi(\chi)\simeq\pi(\chi^{-1})\);
(2) the induced representation \(I(1\otimes|\text{det}(\;)|_F)\) contains exactly two irreducible subrepresentations \(\pi(1)\) and \(\pi(1)'\), they are nonequivalent and square integrable;
(3) if \(\pi\) is a square integrable representation of \(G\) supported on a minimal parabolic subgroup, then \(\pi\) is either the Steinberg representation, or it is equivalent to exactly one of the representations considered in (1) and (2).
For any root \(\gamma\) we shall denote by \(s_\gamma\) the corresponding reflection in the Weyl group. Let \(L_\alpha\) and \(L_\beta\) be the standard Levi subgroups (\(\simeq\text{GL}_2(F)\)) of the maximal standard parabolic subgroups \(P_\alpha=L_\alpha U_\alpha\) and \(P_\beta=L_\beta U_\beta\) which correspond to the roots \(\alpha\) and \(\beta\), respectively. For an admissible representation \((\sigma,W)\) of \(\text{GL}_2(F)\) and \(\gamma\in\{\alpha,\beta\}\) we set \[ I_\gamma(s,\sigma):=\text{Ind}_{P_\gamma}^G(|\text{det}(\;)|_F^s\otimes\sigma). \] When \((\sigma,W)\) is a tempered irreducible representation and \(s>0\), the above representation has a unique irreducible quotient, say \(J_\gamma(s,\sigma)\). Further, we have an intertwining operator \(A(s,\sigma,w_\gamma)\colon I_\gamma(s,\sigma)\rightarrow I_\gamma(-s,\tilde\sigma)\), where \(w_\alpha:=s_{3\alpha+2\beta}\) and \(w_\beta:=s_{2\alpha+\beta}\), defined by \[ A(s,\sigma,w_\gamma)f(g):=\int_{U_\gamma}f(w_\gamma^{-1}ug)d u, \] with image isomorphic to \(J_\gamma(s,\sigma)\).
For a character \(\chi\) of \(F^\times\), we denote by \(\delta(\chi)\) the unique irreducible subrepresentation of the induced representation \((|\text{det}(\;)|^{1\over 2}\chi)\times(|\text{det}(\;)|^{-{1\over 2}}\chi)\) of \(\text{GL}_2(F)\), in the notation of A. V. Zelevinsky [Ann. Sci. Ec. Norm. Supér., IV. Sér. 13, 165-210 (1980; Zbl 0441.22014)]. It is square integrable if and only if \(\chi\) is unitary. For unitary characters \(\chi_1\), \(\chi_2\), we shall denote by \(\sigma(\chi_1,\chi_2)\) the tempered irreducible representation \(\chi_1\times\chi_2\). From the properties of the duality operation defined by the reviewer [Trans. Am. Math. Soc. 347, No. 6, 2179-2189 (1995; Zbl 0827.22005), 348, No. 11, 4687-4690 (1996; Zbl 0861.22012)], it follows that \(I_\gamma(s,\delta(\chi))\) reduces if and only if \(I_\gamma(s,\chi\circ\det)\) reduces.
The author proves that, for \(\chi\) a unitary character and \(s\in{\mathbb R}\), the following hold:
(1) \(I_\alpha(s,\delta(\chi))\) and \(I_\alpha(s,\chi\circ\det)\) reduce if and only if \(s=\pm{1\over 2}\), \(\chi^2=1\), or \(s=\pm {3\over 2}\), \(\chi=1\), or \(s=\pm{1\over 2}\), \(\chi^3=1\);
(2) \(I_\beta(s,\delta(\chi))\) and \(I_\beta(s,\chi\circ\det)\) reduce if and only if \(s=\pm{1\over 2}\), \(\chi^2=1\) or \(s=\pm {5\over 2}\), \(\chi=1\);
(3) if \(\chi\) is a unitary character and \(s>0\) a real number, then:
(a) \(J_\gamma(s,\delta(\chi))\), for \(\gamma\in\{\alpha,\beta\}\), is unitarizable if and only if \(\chi^2=1\) and \(s\leq{1\over 2}\);
(b) if \((\sigma,W)\) is a tempered (non square integrable) irreducible representation of \(\text{GL}_2(F)\), then:
(i) \(J_\alpha(s,\sigma)\) is unitarizable if and only if one of the following conditions is satisfied:
(i1) \(\sigma\simeq\sigma(\chi,\chi^{-1})\) and \(s\leq{1\over 2}\);
(i2) \(\sigma\simeq\sigma(1,\chi)\), \(\chi\) has order \(2\), and \(s\leq {1\over 3}\);
(ii) \(J_\beta(s,\sigma)\) is unitarizable if and only if one of the following conditions is satisfied:
(ii1) \(\sigma\simeq\sigma(\chi,\chi^{-1})\), \(\chi^3\neq 1\), and \(s\leq{1\over 2}\);
(ii2) \(\sigma\simeq\sigma(\chi,\chi^{-1})\), \(\chi^3= 1\), and \(s\leq{1\over 2}\) or \(s=1\);
(ii3) \(\sigma\simeq\sigma(1,\chi)\), \(\chi\) has order \(2\), and \(s\leq 1\).
He also classifies the unitarizable nontempered Langlands quotients that are supported on maximal parabolic subgroups, and, in particular, proves that, if \(\chi\) is of order \(3\), then \(J_\beta(1,\sigma(\chi,\chi^{-1}))\) is unitarizable.
Reviewer: A.M.Aubert (Paris)

MSC:
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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