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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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