Unitary genuine principal series of the metaplectic group

Alessandra Pantano, Annegret Paul, Susana A. Salamanca-Riba
2010 Representation Theory: An Electronic Journal of the AMS  
This paper presents some recent progress on the classification of the unitary genuine irreducible representations of the metaplectic group Mp(2n). Our focus will be on Langlands quotients of genuine minimal principal series; the main result is an embedding of the set of unitary parameters of such representations into the union of spherical unitary parameters for certain split orthogonal groups. The latter are known from work of D. Barbasch; hence we obtain the non-unitarity of a large
more » ... f a large (conjecturally complete) set of parameters for Langlands quotients of genuine principal series of Mp(2n). If the spherical Langlands quotient J(δ 0 , ν p ) of SO(p + 1, p) 0 and/or the spherical Langlands quotient J(δ 0 , ν q ) of SO(q + 1, q) 0 are not unitary, then the genuine Langlands quotient J(δ p,q , ν) of Mp(2n) is also not unitary. Because the spherical unitary dual of real split orthogonal groups is known (by work of Barbasch), we obtain explicit non-unitarity certificates for genuine principal series for Mp(2n). Here is an equivalent formulation of Theorem 1.1. UNITARY GENUINE PRINCIPAL SERIES OF THE METAPLECTIC GROUP 203 Theorem 1.2. Let G = Mp(2n) and let ν = (ν 1 , . . . , ν n ) be a real character of A. Write ν = (ν p |ν q ), as in (1.4). There is a well-defined injection Conjecture 1.3. The map (1.5) is a bijection. As an example, we include the picture of the spherical unitary duals of SO(2, 1) 0 and SO(3, 2) 0 , and of the δ 2,1 -complementary series of Mp(6). See Figure 1 . We briefly discuss the relation between Conjecture 1.3 and the "omega-regular conjecture" introduced in [14]. In [14] , we defined the notion of an omega-regular representation for the metaplectic group. Roughly, a representation of Mp(2n) is omega-regular if its infinitesimal character is at least as regular as that of the oscillator representation. Generalizing the idea of an admissible A q (λ)-module of [17], we displayed a family of unitary omega-regular representations called A q (Ω)modules, and conjectured that these modules exhaust the genuine omega-regular unitary dual of Mp(2n). If π is an irreducible principal series representation of the metaplectic group, then π is an A q (Ω)-module if and only if it is an even oscillator representation. Then, for principal series representations, the "omega-regular conjecture" states that a genuine omega-regular irreducible principal series representation of Mp(2n) is unitary if and only if it is an even oscillator representation. This result follows from Theorem 1.2. Hence, we obtain the following corollary. Corollary 1.4. The "omega-regular conjecture" is true for all principal series representations of Mp(2n). UNITARY GENUINE PRINCIPAL SERIES OF THE METAPLECTIC GROUP 205 genuine δ, W δ coincides with the Weyl group of the system of good roots, hence with the Weyl group of G δ . Therefore, this matching of intertwining operators ties the unitarity of a genuine Langlands quotient J(δ, ν) of Mp(2n) to the unitarity of a spherical representation of G δ . Proving Conjecture 1.3 amounts to showing that for each pair of parameters ν p ∈ CS(SO(p + 1, p) 0 , δ 0 ) and ν q ∈ CS(SO(q + 1, q) 0 , δ 0 ), the Langlands quotient J(δ p,q , (ν p |ν q )) of Mp(2(p + q)) is unitary. We do this completely for p + q ≤ 3. For the general case, we produce two large families of spherical unitary parameters for SO(p + 1, p) 0 × SO(q + 1, q) 0 which give rise to δ p,q -complementary series of Mp(2(p+q)). One can construct many more examples; we plan to pursue this in a future paper. Some comments are in order. Barbasch's method of computing some intertwining operators purely in terms of Weyl group representations is central to this paper. The same technique has proven successful in the past: it was used in [5] to prove that the spherical unitary dual of a real split group embeds into the spherical unitary dual of the corresponding p-adic split group, and in [3] to prove that the pseudospherical unitary dual of non-trivial coverings of split simple groups embeds into the spherical unitary dual of certain linear groups. Generalizing these ideas to genuine non-pseudospherical representations is non-trivial, because the intertwining operators are harder to compute, due to the presence of bad roots. We give an outline of the paper. Since understanding the techniques used in the determination of the spherical unitary dual of split real classical groups is crucial for our argument, we summarize the main results of [5] in Section 2. Section 3 is devoted to the structure of the group Mp(2n). The genuine complementary series of Mp(2n) are introduced in Section 4. In Section 5, we discuss the structure of the space Hom M (µ, δ) and the corresponding W δ -representation. Next, in Section 6, we review the theory of intertwining operators for minimal principal series of Mp(2n) and we prove some preliminary results. In Section 7, we explain the role of petite K-types for producing non-unitarity certificates, and we state the main theorem. The actual proof of the main theorem is contained in Section 10. In Section 8, we present some evidence for Conjecture 1.3. The irreducible representations of Weyl groups of type C are described in Section 9. Finally, in the appendix, we give an explicit description of the spherical unitary dual of split groups of type B as in [4] . Sections 5, 6 and 7 are based on unpublished work of D. Barbasch and the first author on non-unitarity certificates for non-spherical principal series (for more general cases than Mp(2n)), cf. [8] . We thank D. Barbasch for generously sharing his ideas. In addition, we would like to thank D. Vogan, P. Trapa and D. Ciubotaru for offering valuable suggestions along the way, and the authors of [3] for providing such an interesting inspiration. Spherical unitary dual of real classical split groups The purpose of this section is to summarize some of the results we need about the spherical unitary dual of real split groups. For a more detailed description, the interested reader can consult [6] . Let G be the set of real points of a connected linear reductive group defined over R. Let K be a maximal compact subgroup of G, let MA be the Levi factor of a minimal parabolic subgroup of G (with compact part M ) and let W be the Weyl 206 A. PANTANO, A. PAUL, AND S. A. SALAMANCA-RIBA 20. B. Speh, D.A. Vogan, Jr., Reducibility of generalized principal series representations, Acta
doi:10.1090/s1088-4165-10-00367-5 fatcat:wfxer2cbonctdfy34qvto4yzei