THE GENUINE OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP

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1 THE GENUINE OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP ALESSANDRA PANTANO, ANNEGRET PAUL, AND SUSANA A. SALAMANCA-RIBA Abstract. We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real an at least as regular as that of the oscillator representation. In a previous paper we exhibite a certain family of representations satisfying these conitions, obtaine by cohomological inuction from the tensor prouct of a one-imensional representation an an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary ual of the metaplectic group. 1. Introuction In [4], we formulate a conjecture that provies a classification of all the genuine unitary omega-regular representations of the metaplectic group. (Roughly, a representation is calle omega-regular if its infinitesimal character is real an at least as regular as that one of the oscillator representation. See Definition.1.) In this paper, we prove that conjecture. In particular, we show that all such representations are obtaine by cohomological parabolic inuction from the tensor prouct of a one-imensional representation an an oscillator representation. The reaer may think of the notion of omega-regular representations as a generalization, in the context of genuine representations of the metaplectic group, of the iea of strongly regular representations. (Recall that an infinitesimal character is calle strongly regular if it is at least as regular as the infinitesimal character of the trivial representation.) Then our classification appears, for the case of the metaplectic group, as a generalization of the main result of [10], which asserts that, for real reuctive Lie groups, every irreucible unitary representation with strongly regular infinitesimal character is cohomologically inuce from a unitary character. For ouble covers of other linear groups one can efine a similar notion of omega-regular for which every genuine unitary omega-regular representation shoul be obtaine by cohomological inuction from one of a small set of basic unitary representations. We make this more explicit after Theorem 1.4 below. All representations consiere in this paper have real infinitesimal character. Let M p(n) be the metaplectic group of rank n, i. e., the two-fol connecte cover of the symplectic group Sp(n, R). We recall the construction of the A q (Ω) representations of Mp(n). Choose a theta stable parabolic subalgebra q = l + u of sp(n, C), an let L be the Levi subgroup of Mp(n) corresponing to l. We may 1991 Mathematics Subject Classification. Primary E46. Key wors an phrases. Metaplectic group; oscillator representation; bottom layer map; cohomological inuction; Parthasarathy s Dirac Operator Inequality; pseuospherical principal series. This material is base on work supporte by NSF grants DMS , DMS , DMS an DMS

2 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA ientify L with a quotient of [ u ] L = Ũ (p i, q i ) Mp(m). i=1 Here, for i = 1... u, Ũ (p i, q i ) is a connecte two-fol cover of U (p i, q i ). However, we will abuse notation an say L L. (See Section for etails.) u If C λ is a genuine (unitary) character of Ũ(p i, q i ), an ω is an irreucible summan of one of the two oscillator representations of Mp(m), we set consiere as a representation of L. Let (see Section for notation). i=1 Ω = C λ ω, A q (Ω) := R q (Ω) Remark 1.1. Our (new) efinition of A q (Ω) is slightly ifferent from the one given in [4], because we o not require Ω to be in the goo range for q. Definition 1.. A representation Y of L is in the goo range for q if its infinitesimal character γ Y satisfies γ Y + ρ(u), α > 0, α (u). Here ρ(u) enotes one half the sum of the roots in (u). For brevity of notation, an omega-regular representation will be calle ω-regular. In [4], we prove the following result. Proposition 1.3 (Proposition 3 of [4]). If the representation Ω of L is in the goo range for q, then the representation A q (Ω) of Mp(n) is nonzero, irreucible, genuine, ω-regular an unitary. Our main result is the following converse of this statement. Theorem 1.4. Let X be an irreucible genuine ω-regular an unitary representation of Mp(n). Then X = A q (Ω), for some theta stable parabolic subalgebra q of g an some representation Ω = C λ ω, in the goo range for q, of the Levi subgroup corresponing to q. Theorem 1.4 suggests that the four oscillator representations form a basic set of builing blocks for all genuine omega-regular unitary representations of the metaplectic group, via cohomological parabolic inuction. Using the results of [15] an [], one can verify that similar statements hol true for the simply connecte split group of type G an the nonlinear ouble cover of GL(n, R). Here, omegaregularity might be efine in terms of 1 ρ, an the basic representations are the pseuospherical representations which correspon to the trivial representation of the linear group uner the Shimura corresponence of [1]. This suggests the following generalization of omega-regular for a ouble cover G of a split linear group. Fix a choice of positive roots with respect to the split Cartan, an consier metaplectic roots as in [1], Definition 4.4. Define γ ω to be one half the sum of the non-metaplectic positive roots, plus one fourth the sum of the metaplectic positive roots, an call a representation of G omega-regular if its infinitesimal character

3 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 3 is at least as regular as γ ω (as in Definition.1). This efinition agrees with that for M p(n) an the examples consiere above, an reuces to the strongly regular efinition if G is linear. If G is a nonlinear ouble cover of a split real group as in [1], then the set of basic representations for G shoul inclue the (conjecturally unitary) pseuospherical representations at infinitesimal character γ ω, i. e., the Shimura lifts of the trivial representation of the corresponing linear group (e. g., the even oscillator representations of M p(n)). This collection of pseuospherical representations at infinitesimal character γ ω may or may not exhaust the set of basic representations for G. We conjecture that a similar notion of omega-regularity, an a similar small collection Π ω (G) (finite if G is semisimple) of basic genuine, unitary, an omegaregular representations can be efine for any ouble cover G of a reuctive linear real Lie group. Conjecture 1.5. For each ouble cover H of a real reuctive linear Lie group there is a set Π ω (H) of basic representations as above, with the following property. Suppose that G is a ouble cover of a linear reuctive real Lie group. If X is the Harish-Chanra moule of a genuine irreucible omega-regular unitary representation of G then there is a Levi subgroup L of G an a representation Y Π ω (L) such that X is obtaine from Y by cohomological parabolic inuction. Remark 1.6. For linear ouble covers, such as the trivial ouble cover, or the square root of the eterminant cover of U(p, q), the basic representations are the (genuine) unitary one-imensional representations. For these groups, the notion of omega-regularity shoul coincie with strong regularity, an the conjecture recovers Salamanca-Riba s result [10]. In [4], we prove Theorem 1.4 for a metaplectic group of rank. The proof was base on a case by case calculation. For each (genuine) Ũ()-type µ that is the lowest K-type of an A q (Ω) representation, we showe that there exists a unique unitary an ω-regular representation of Mp(4) with lowest K-type µ. For each genuine Ũ()-type µ that is not the lowest K-type of an A q (Ω) representation, we showe that every ω-regular representation of Mp(4) with lowest Ũ()-type µ must be nonunitary. The main tool in the proof of both claims was Parthasarathy s Dirac Operator Inequality (cf. [6]). This scheme worke for all genuine Ũ()-types, except for the (unique) fine Ũ()-type which occurs in the genuine non-pseuospherical principal series. In this case, we explicitly compute the intertwining operator that gives the invariant Hermitian form on the representation space, an showe that its signature is inefinite. The case-by-case calculation we use to prove Theorem 1.4 for the case n = is not suitable for a generalization to arbitrary n. For the general case, we apply a reuction argument similar to the one use in [9], but we also nee some of the non-unitarity results an non-unitarity certificates obtaine in [5]. (The question of the unitarity of the ω-regular principal series of Mp(n) was the motivation for [5].) We sketch the proof of Theorem 1.4 (for arbitrary n). Let X be a genuine amissible irreucible unitary representation of M p(n). The first step is to realize X as the lowest K-type constituent of a moule of the form R q (X 1 X 0 ),

4 4 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA where q = l + u is a theta stable subalgebra of sp(n, C) with Levi subgroup L Ũ(r, s) Mp(), X 1 is a genuine strongly regular irreucible representation of Ũ(r, s), an X 0 is the irreucible Langlans quotient of a genuine ω-regular principal series representation of M p(). If X 1 X 0 is unitary, then the results of [9] imply that X 1 A q (λ ) for some theta stable parabolic subalgebra q an some parameter λ, an the results of [5] tell us that X 0 must be the even half of an oscillator representation of Mp(). In this case, by a version of inuction by stages, X must be of the esire form. If X 1 X 0 is not unitary, then there must be an (L K)-type µ L which etects non-unitarity (in the sense that the invariant Hermitian form on µ L is not positive efinite). The results of [9], which rely heavily on Parthasarathy s Dirac Operator Inequality, together with the calculations in [5] (see Lemma 5.) give us specific information about what µ L coul look like. Recall that, associate to the cohomological inuction functor R q, there is the bottom layer map, which takes (L K)-types (in (X 1 X 0 )) to K-types (in R q (X 1 X 0 )). Let µ be the image of µ L uner this map. If µ were nonzero, then by a theorem of Vogan, µ woul occur in the lowest K-type constituent X of R q (X 1 X 0 ) an woul carry the same signature as µ L ; hence X woul be not unitary. Because we are assuming that X is unitary, we euce that µ L must be mappe to 0. It turns out that, in this case, there exists a ifferent theta stable parabolic subalgebra q of sp(n, C) with Levi subgroup L Ũ(r, s ) Mp( + ), such that X is the lowest K-type constituent of R q (X 1 X 0). Here X 1 is an A q3 (λ 3 ) moule of Ũ(r, s ), an X 0 is the o half of an oscillator representation of Mp( + ). As in the previous case, using inuction by stages, we get the esire result. In [4], we also classifie the non-genuine unitary ω-regular representations of Mp(4), i. e., the ω-regular part of the unitary ual of Sp(4, R). We plan to aress the generalization of this result to metaplectic groups of arbitrary rank in a future paper. The paper is organize as follows. In Section, we set up the notation an recall some properties of the cohomological inuction construction. We outline the proof of our main theorem in Section 3. The argument is essentially reuce to two main propositions, which we prove in Sections 6 an 7, an a series of technical lemmas that are presente in Section 4 (the casual reaer may want to skip this section). Aitional results neee for the proof are inclue in Sections 5 an 8. Once again, we thank Davi Vogan an Jeffrey Aams for posing the problem, an for their patience an encouragement along the way. We also thank the referee for several helpful suggestions.. Definitions an Preliminary Results We begin with some notation. For the metaplectic group of rank r, we enote by ω an irreucible summan of an oscillator representation an we write ω r for the corresponing infinitesimal character. Recall that there are four such summans, namely, the o an even halves of the holomorphic an antiholomorphic oscillator representations, respectively. For any compact connecte group, we often ientify irreucible representations of the group with their highest weight. For G = Mp(n), set g 0 = sp(n, R) an g = sp(n, C), an let t 0 an t be the real an complexifie Lie algebra of a compact Cartan subgroup T of G. Let, enote a fixe non-egenerate G-invariant θ-invariant symmetric bilinear

5 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 5 form on g 0, negative efinite on k 0 an positive efinite on p 0 ; use the same notation for its complexification, an its various restrictions an ualizations. Definition.1. Let γ it 0. Choose a positive system + (γ) (g, t) such that α, γ 0 for all α + (γ), an let ω n be the representative of the infinitesimal character of the oscillator representation of G which is ominant with respect to + (γ). We call γ ω-regular if the following regularity conition is satisfie: α, γ ω n 0, α + (γ). We say that a representation of G is ω-regular if its infinitesimal character is. Given a θ-stable parabolic subalgebra q = l+u of g, write L for the corresponing subgroup of Mp(n). Then L is a ouble cover of a Levi subgroup L of Sp(n, R) of the form (.1) L u = U(p i, q i ) Sp(m, R). i=1 For each i = 1...u, let Ũ(p i, q i ) be the inverse image of U(p i, q i ) (in (.1)) uner the covering map p : Mp(n) Sp(n, R), an similarly for Sp(m, R). Then Ũ(p i, q i ) is the (connecte) square root of the eterminant cover of U(p i, q i ), an Sp(m, R) = Mp(m). The groups Ũ(p i, q i ) an Sp(m, R) intersect in the kernel of the covering map p, an there is a surjective map [ u ] (.) L = Ũ (p i, q i ) Mp(m) L i=1 given by multiplication insie M p(n). Since the factors in (.) commute, we have that genuine irreucible representations of L are in corresponence with tensor proucts of genuine irreucible representations of the factors of L. In orer to keep our notation simpler, we will ientify L with L, an just write [ u ] (.3) L = Ũ (p i, q i ) Mp(m). ( Now let R (g,k) q i=1 ) i be the functors of cohomological parabolic inuction carrying (l, L K)-moules to (g, K)-moules (cf. [V], Def ). We will occasionally apply these functors to ifferent settings. When the group an the Lie algebra are clear from the context, we will omit the superscript (g, K) an use the more stanar notation R i q. In our situation, we only use the egree i = im(u k) (usually enote by S), hence we may omit the superscript i as well. Definition.. An A q (Ω) representation is a genuine representation of G of the following form. Let q = l + u be a theta stable parabolic subalgebra of g, with corresponing Levi subgroup [ L (as in Equation 1). Let C λ be a genuine oneimensional representation of Ũ(p i, q i ) an let ω be an irreucible u ] summan i=1 of an oscillator representation of M p(m). We efine A q (Ω) := R q (Ω).

6 6 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA Remark.3. In Definition., the rank m of the metaplectic factor of L is allowe to be equal to 0 or n; in these cases, the representation A q (Ω) of Mp(n) is an A q (λ) representation or an oscillator representation, respectively. We will prove Theorem 1.4 in Section 3. Now we recall some properties of the functors of cohomological inuction. Let X be an irreucible amissible (g, K)-moule. Let ( (.4) λa, q a, L a, X La) be the θ-stable parameters associate to X via Vogan s classification of amissible ( ) representations, so that X is the unique lowest K-type constituent of R Sa q a X L a, with S a = im (u a k) an X La a minimal principal series of L a (cf. [13]). Recall that the parameter λ a it 0 etermines the theta stable parabolic subalgebra q a = l a + u a. In particular, the set of roots in u a is given by (u a ) = {α (g, t) λ a, α > 0. Choose an fix the positive system of compact roots (.5) + c = {ɛ i ɛ j 1 i < j n so that ominant weights, an therefore highest weights of K-types, are given by strings of weakly ecreasing half-integers. The parameter λ a is weakly ominant for + c as well. If (.6) λ a = g 1,..., g {{ 1,..., g t,..., g {{ t 0,..., 0 {{ g t,..., g t,..., g {{ 1,..., g 1 {{ r 1 r t s t s 1 with g 1 > > g t > 0, then the centralizer of λ a on G is of the form [ t ] L a = Centr G (λ a ) Ũ (r i, s i ) Mp(), an each factor is quasisplit. Note that the parameter (.7) ξ = 1, 1,..., 1 {{ 0, 0,..., 0 {{ 1, 1,..., 1 {{ r s is a singularization of λ a (cf. i=1 [11]). Here r = t i=1 r i an s = t i=1 s i. Set L = Cent G (ξ) Ũ (r, s) Mp() an q = l + u. Then L L a, l a l, u a = u + (l u a ) an q = l + u q a. Proposition.4 below lists a few results regaring the functors of cohomological inuction an their restriction to K (see [13] an [3] for proofs). Most of these results are gathere together in [11], but are state there for the functors L S q instea of R S q. Note that in our context, the two constructions are isomorphic by a result ue to Enright an Wallach (cf. Theorem 5.3 of [14]). Proposition.4. Suppose that X is an irreucible amissible (g, K)-moule of Mp(n). Define L an q as above, an let S = im (u k). Then

7 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 7 (i) There is a unique irreucible (l, L K)-moule X L associate to X so that X can be realize as the unique lowest K-type constituent of ( X L ). R (K,k) q k R S q (ii) If µ L is (the highest weight of) an irreucible representation V µ L of L K an µ = µ L + ρ (u p) is + (k, t)-ominant, then every irreucible constituent of R (K,k) (Vµ ) ( ) S L q k has highest weight µ. If µ is not ominant for ( ) S (Vµ ) + (k, t), then L = 0. (iii) There is a natural injective map, the bottom layer map, of K-representations ( ) S (X L B X L : ) R S ( q X L ). R (K,k) q k Moreover, there is a one-to-one corresponence (with multiplicities) between the lowest K-types of X an the lowest (L K)-types of X L. (iv) The moule X is enowe with a nonzero invariant Hermition form, G if an only if the moule X L is enowe with a nonzero invariant Hermitian form, L. (v) The bottom layer map is unitary. That is, for X Hermitian, on each K-type in the bottom layer of X (cf. Remark.6), the signature of, G matches the signature of, L on the corresponing (L K)-type of X L. (vi) If γ XL it 0 is a representative of the infinitesimal character of X L, then γ X = γ XL + ρ(u) is a representative of the infinitesimal character of X. (vii) If λ L a is the Vogan classification parameter associate to X L, then λ a = λ L a + ρ(u). Proof. Because ξ in (.7) is a singularization of λ a, these facts follow from Lemma.7 an Theorem.13 in [11]. More precisely, part (i) an (vii) follow from Theorem.13 (b); (ii) from Lemma.7; (iii) an (v) follow from Theorem.13 (); (iv) is prove in more generality in Proposition 5. an Corollary 5.3 of [9], but it is also Theorem.13 (c) of [11]. Part (vi) is Proposition in [13]. Remark.5. If X has real infinitesimal character, then X L an X are Hermitian by Lemma 6.5 of [9]; the argument given there is easily seen to exten to the case of the metaplectic group. Consequently, we have that in our setting, the forms, L an, G of Proposition.4 (iv) always exist. Remark.6. The image of the bottom layer map B X L (as in part (iii) of Proposition.4) is calle the bottom layer of X. We say that an (L K)-type µ L survives in the bottom layer if µ = µ L + ρ (u p) is + (k, t)-ominant (as in part (ii) of Proposition.4). Note that, in this case, µ is the highest weight of a K-type in the bottom layer of X.

8 8 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA Lemma.7. Retain the notation of Proposition.4. Set L = L 1 L 0 with L 0 = Mp() an L 1 = Ũ (r, s), an write XL X 1 X 0, with X i an irreucible (l i, L i K)-moule (i = 0, 1). If X is ω-regular, then X 1 is strongly regular for L 1 an X 0 is ω-regular for L 0. Proof. Let γ XL = ( ) γ X1, γ X0 be (a representative of) the infinitesimal character of X L, an write γ X = γ XL + ρ(u) as in part (vi) of Proposition.4. Choose a positive system + (g, t) of roots so that γ X is ominant with respect to +, an choose the representative of the infinitesimal character ω n which is ominant with respect to +. Because X is ω-regular, we have γ X, α ω n, α, α +. If we normalize our form so that α, α = for all short roots α, then this is equivalent to saying that γ X, α 1, α +. Now let + (l) = + (l, t) = [ + (l 1, t)] [ + (l 0, t)]. Because ρ(u) is orthogonal to the roots of l, we have γ XL, α 1, α + (l). Note that the roots of l 1 are orthogonal to γ X0, hence γ X 0, α 1, α + (l 0, t) an X 0 is ω-regular for L 0. Similarly, the roots of l 0 are orthogonal to γ X1, hence γ X 1, α 1, α + (l) (l 1, t). This implies that X 1 is strongly regular for L Proof Of Theorem 1.4 The proof of Theorem 1.4 relies on a series of auxiliary lemmas an propositions. We will state these results as neee along the way an postpone the longer proofs to later sections. Fix X as in Theorem 1.4, i. e., let X be a genuine irreucible ω-regular unitary representation of M p(n). By virtue of Proposition.4 an Lemma.7, we can assume that our genuine irreucible ω-regular unitary representation X is (the unique lowest K-type constituent of) a representation of the form R S q (X 1 X 0 ) with X 1 an irreucible genuine strongly regular (l 1, L 1 K)-moule for L 1 = Ũ (r, s), an X 0 an irreucible genuine ω-regular (l 0, L 0 K)-moule for L 0 = Mp(). Note that, because L 0 is a factor of L a (an X La is a minimal principal series representation of L a ), X 0 must be a minimal principal series representation of L 0. The following proposition asserts that, in this setting, X 1 is a goo A q (λ ) moule.

9 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 9 Proposition 3.1. Let X be an irreucible unitary (g, K)-moule of M p(n). Assume that X is( genuine an ω-regular; realize X as the unique lowest K-type constituent of R ) S q X L, as in Proposition.4, an write L = L 1 L 0 = Ũ (r, s) Mp() XL X 1 X 0, as in Lemma.7. Suppose that r + s 0. Then there exist a theta stable parabolic subalgebra q an a representation C λ of the Levi factor corresponing to q, with λ in the goo range for q, such that X 1 A q (λ ). Using Proposition 3.1, we will prove the following result. Proposition 3.. In the setting of Proposition 3.1, assume > 0 an allow r + s to be possibly equal to zero. Then one of the following (mutually exclusive) options occurs: (i) X 0 is an even oscillator representation with lowest K-type µ 0 = ± (1/, 1/,..., 1/). (ii) There exist a subgroup L = L 1 L 0 = Ũ(r, s ) Mp(( + 1)) G also containing L a, a theta stable subalgebra q = l + u q an representations X i of L i (i = 0, 1), such that (a) Ũ (r, s ) Ũ (r, s), with either r = r + 1 or s = s + 1, (b) X 1 is a goo A q3 (λ 3 ) (or r + s = 0), (c) X 0 is an o oscillator representation with lowest K-type µ 0 = (3/, 1/,... 1/) or ( 1/, 1/,... 1/. 3/), an () X can be realize as the unique lowest K-type constituent of R S q (X 1 X 0). Putting all these results together, we obtain the following corollary. Corollary 3.3. Every genuine irreucible ω-regular unitary representation X of M p(n) satisfies one of the following two properties: (i) X is (the unique lowest K-type constituent of) a representation of the form ( X L ) R S q with L = Ũ (r, s) Mp() = L 1 L 0 an X L = X 1 X 0. Here (a) X 1 is a goo A q (λ ) moule for Ũ (r, s), unless r + s = 0, an (b) X 0 is an even oscillator representation of Mp() with lowest K-type µ 0 = ± (1/, 1/,... 1/), unless = 0. (ii) X is (the unique lowest K-type constituent of) a representation of the form ( R S q X L ) with L = Ũ (r, s ) Mp(( + 1)) = L 1 L 0 an X L = X 1 X 0. Here (a) X 1 is a goo A q3 (λ 3 ) moule for Ũ (r, s ), unless r + s = 0, an (b) X 0 is an o oscillator representation of Mp(( + 1)) with lowest K-type µ 0 = (3/, 1/,... 1/) or ( 1/, 1/,... 1/. 3/).

10 10 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA In orer to conclue the proof of Theorem 1.4, we must show that (in both cases) our genuine irreucible ω-regular unitary representation X of M p(n) can also be realize as an A q (Ω) representation, for some theta stable parabolic subalgebra q an some representation Ω = C λ ω in the goo range for q. For this, we nee two more results. Proposition 3.4. Let q = l+u g be a θ-stable parabolic subalgebra. Assume that L = N G (q) is a irect prouct of two reuctive subgroups L = L 1 L 0. (Here L can be of the form Ũ (r, s) Mp(), as in Proposition.4, or Ũ (r, s ) Mp((+1)), as in Proposition 3..) Suppose further that we have a representation X R q (A q (λ ) ω) where ω is an irreucible summan of an oscillator representation, q = l + u l 1, an A q (λ ) is goo for q. Then there is a theta stable parabolic subalgebra q ω = l ω + u ω of g such that X R qω (C λ ω) = A qω (Ω) with Ω = C λ ω. Proof. Assume that X R q (A q (λ ) ω), where q = l + u l 1. Set q b = q l 0 = (l + u ) l 0. Then q b is a theta stable parabolic subalgebra of l, an by Lemma 3.5 (which we will state below), we have that Therefore, Now set A q (λ ) ω R (l1,l1 K) q (C λ ) R(l0,L0 K) ( ) X R q R (l,l K) q b (C λ ω). (3.1) q ω = q b + u = (l l 0 ) {{ + (u + u) {{ l ω l 0 (ω) R (l,l K) q b (C λ ω). u ω. Note that q ω is a parabolic subalgebra of g. Since X 1 = Aq (λ ), by Proposition 3.5 in [10], we know that l l a l 1 an u u a l 1. We have (3.) l ω = l l 0 l, u u + u = u ω an q ω q. Hence, by inuction in stages (cf. [13], Proposition 6.3.6), we fin that ( ) R q R q (l,l K) b (C λ ω) R qω (C λ ω) = A qω (Ω), an Proposition 3.4 is prove. Lemma 3.5. For i = 1,, let G i be a reuctive Lie group with maximal compact subgroup K i, complexifie Lie algebra g i, an a theta stable parabolic subalgebra q i = l i + u i. Moreover, let X i be an (l i, L i K i )-moule, an let S i = im(u i k i ). Then R S1 q 1 (X 1 ) R S q (X ) R S1+S q 1 q (X 1 X ). Proof. This follows by tracing through the efinitions of the various cohomological inuction functors, using stanar techniques an ientities in homological algebra, incluing an appropriate Künneth formula.

11 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 11 Finally, we nee to show that our representation Ω is inee in the goo range for q ω. Proposition 3.6. In the setting of Proposition 3.4, let X be the irreucible (lowest K-type constituent of the) representation R qω (C λ ω). Assume, moreover, that λ is in the goo range for q. Then Ω = C λ ω is in the goo range for q ω. This conclues the proof of Theorem 1.4. The proofs of Propositions 3.1, 3. an 3.6 will be given in Sections 6, 7 an 8, respectively. Before we turn to those proofs, we nee to prove several technical lemmas concerning lowest K-types an to recall some results about genuine minimal principal series representations of M p(). We aress these issues in the next two sections. 4. Technical lemmas The purpose of this section is to escribe lowest K-types of irreucible, genuine an ω-regular representations of M p(n), an to present some results that will be neee for the proofs of Propositions 3.1, 3. an 3.6. We begin, in great generality, with irreucible amissible (g,k)-moules M p(n). Lemma 4.1. Let X be an irreucible amissible (g,k)-moule of Mp(n), an let µ be a lowest Ũ(n)-type of X. Let λ a an q a = l a + u a be the Vogan classification parameter an the theta stable parabolic subalgebra associate to X, respectively, as in (.4). Then µ is of the form (4.1) µ = λ a + ρ(u a p) ρ(u a k) + δ La, where δ La is a fine (L a K)-type. Proof. This is a known result ue to Vogan. It follows from Proposition.4 (ii)-(iii) (when q = q a ) an from the fact that (the highest weight of) the representation µ La in that Proposition is of the form λ a ρ (u a ) + δ La, with δ La a fine (L a K)-type. See Proposition 3..7 of [7] for a etaile proof in the case of G = U(p, q). Next, we restrict our attention to irreucible amissible (g,k)-moules of M p(n) that are genuine an ω-regular. Lemma 4.. Let X be an irreucible amissible (g, K)-moule of Mp(n). Assume that X is genuine an ω-regular. Realize X as the ( unique lowest K-type constituent of a cohomologically inuce representation R ) S q X L, with L = L 1 L 0 = Ũ (r, s) Mp() an XL X 1 X 0, as in Proposition.4. Let λ a = g 1,..., g {{ 1,..., g t,..., g {{ t 0,..., 0 {{ g t,..., g t,..., g {{ 1,..., g 1 {{ r 1 r t s t s 1 be the Vogan classification parameter of X, as in Equation.6. (i) If X 1 is an A q (λ ) moule of Ũ(r, s), then the representation X has a unique lowest Ũ(n)-type µ. (ii) For i = 1,..., t, if r i s i, then g i Z + 1. In this case, the infinitesimal character of X contains an entry ±g i, an the infinitesimal character of X 0 oes not.

12 1 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA (iii) Suppose that X has a unique lowest Ũ(n)-type µ. (a) µ is of the form (4.) µ = λ a + ρ(u a p) ρ(u a k) + µ 0, where µ 0 is the lowest Ũ()-type of X 0. (b) Moreover, if r i = s i, then g i Z. Proof. Recall that X 0 is a genuine minimal principal series of Mp(), therefore it has a unique lowest K-type (cf. [5]). If X 1 is an A q (λ ) moule of Ũ(r, s), then X 1 has a unique lowest K-type, as well. The uniqueness of the lowest K-type of X then follows from part (iii) of Proposition.4. For part (ii), recall from Section 3.1 of [8] the relationship between the parameter λ a, the infinitesimal character, the lowest K-types, an the Langlans parameters of a representation. There, the theory is lai out in etail for representations of the symplectic group. The corresponing statements for genuine representations of M p(n) can be obtaine by making slight moifications. In the case of ω-regular (hence non-singular) representations, all limits of iscrete series representations are iscrete series, an all the lowest K-types of a stanar moule appear in the same irreucible representation. Because X is a genuine representation with associate parameter λ a as in (.6), we can realize X as the Langlans subquotient of an inuce representation from a cuspial parabolic subgroup with Levi component MA isomorphic to (a quotient of) Mp(a) GL(, R) b GL(1, R), where a = t r i s i an b = t min {r i, s i. If r i = s i + 1 for some i, then the entry g i i=1 i=1 coincies with an entry of the Harish-Chanra parameter of a genuine iscrete series representation of Mp(a). Similarly for g i, if r i = s i 1. Therefore, g i Z + 1 whenever r i s i. In this case, we also fin that g i is an entry of the infinitesimal character of X. Then the fact that X 0 oes not contain an entry ±g i follows from the ω-regular conition. For part (iii), assume that our representation has a unique lowest Ũ(n)-type µ. Write µ = λ a + ρ(u a p) ρ(u a k) + δ La (as in Lemma 4.1). By Proposition.4, this equals Now look at the restriction to L 0 K: Hence µ 1 + µ 0 + ρ(u p). λ a l0 k = 0 µ 1 l0 k = 0 ρ(u p) l0 k = [ρ(u a p) ρ(u a k)] l0 k. δ La l0 k= µ 0. Next, we show that δ La l1 k= 0. Because X has a unique lowest Ũ(n)-type, the representation X 1 must have the same property. In orer to have a representation of L 1 with a unique lowest Ũ(r) Ũ(s)-type, every factor Ũ(r i, s i ) of L a with r i = s i shoul carry the trivial fine Ũ(r i) Ũ(s i)-type, because non-trivial fine

13 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 13 Ũ(r i ) Ũ(s i)-types on such factors come in pairs. This shows that the restriction of δ La to L 1 K is zero, hence δ La = µ 0 an µ = λ a + ρ(u a p) ρ(u a k) + µ 0, completing the proof of part (a). For part (b), write r i = s i + ε i for all i = 1,..., t. Note that ε i = 0 or ±1, because each factor of [ t ] L a = Ũ (r i, s i ) Mp() i=1 is quasisplit. Then (4.3) ρ (u a p) ρ (u a k) =..., f i,..., f {{ i,... c,. {{.., c..., h i,..., h i,..., {{ r i s i with (4.4) f i = (r j s j ) + ε i + 1 = j<i j<i h i = (r j s j ) + ε i 1 = j<i j<i c = r s. ε j + ε i + 1 ε j + ε i 1 If r i = s i for some i = 1,..., t, then ε i = 0. The corresponing entry f i of ρ (u a p) ρ (u a k) is then in Z+ 1. Because genuine Ũ(n)-types have half-integral entries, this implies that g i Z. The proof of Lemma 4. is now complete. Remain in the setting of Lemma 4.. Let µ be a lowest K-type of X. Recall that µ satisfies Equation 4.1: µ = λ a + ρ(u a p) ρ(u a k) + δ La. Write (4.5) µ = a 1,..., a {{ 1,..., a t,..., a {{ t c 1,..., c {{ b t,..., b t,..., b {{ 1,..., b 1, {{ r 1 r t s t s 1 accoring to how the coorinates break into the factors of the subgroup L a. Then the fine (L K)-type δ La is of the form (4.6) δ La = y 1,..., y {{ 1,..., y t,..., y {{ t z 1,..., z {{ y t,..., y t,..., y {{ 1,..., y 1, {{ r 1 r t s t s 1 with y i = 0 or ± 1, an z i = ± 1 (cf. weakly ecreasing. Proposition 6 of [8]). Note that the {z i are Remark 4.3. If r i = 0, then a i oes not occur as a coorinate of µ, but it is still convenient to efine (4.7) a i = g i + f i + y i,

14 14 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA with f i as in Equation 4.4. (Because s i > 0 in this case, the quantities y i an ( )g i can be etermine by the coorinates of δ La an λ a, respectively.) Similarly, if s i = 0 (an r i > 0), we efine b i by (4.8) b i = g i + h i + y i. We obtain sequences {a i an {b i of half-integers satisfying (4.9) a 1 a a t an b t b t 1 b 1. Lemma 4.4. Retain all the previous notation. Then (i) a t b t. (ii) Assume that the K-type µ satisfies the aitional conition (4.10) a t = c 1. (a) ε t 0 an g t = 1. (b) If ε t = 0, then δ La (cf. (4.1)) must be of the form (4.11) δ La =..., 1,..., 1 1 {{,..., 1, 1 {{,..., 1 {{ 1,..., 1,... {{ p q s t for some p > 0 an q = p. (c) If ε t = 1, then δ La (cf. (4.1)) must be of the form δ La =..., 0,. {{.., 0 1,..., 1, 1 {{,..., 1 {{ 0,..., 0,... {{ s t+1 s t p q for some p > 0 an q = p. () c > b t. Proof. For part (i), observe that a t = g t + (r s) (r t s t ) + ε t y t = g t + (r s) + 1 (1 ε t) + y t, c 1 = (r s) + z 1, c = (r s) + z, an b t = g t + (r s) (r t s t ) + ε t 1 + y t = g t + (r s) + 1 ( 1 ε t) + y t. Hence, we obtain s t (4.1) (4.13) a t c 1 = g t + 1 (1 ε t) + y t z 1, c b t = g t + 1 (1 + ε t) + z y t, an a t b t = g t + 1, because g t 1. This proves (i). Now assume that µ satisfies Equation The coorinate g t of λ a is either an integer or a half-integer. We consier the two cases separately.

15 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 15 First assume that g t Z. By Lemma 4., ε t = 0, so a t Z y t an c 1 Z + z 1. In orer for µ to be genuine, the coorinates of the fine (L a K)-type δ La satisfy { { y t Z y t = 0 z 1 Z + 1 z 1 = ± 1. must This says that the representation δ La of (L a K) is trivial on the Ũ(s t, s t )-factor of L a, an non-trivial on the Mp()-factor: δ La =..., 0,. {{.., 0 1,..., 1, 1 {{,..., 1 {{ 0,,..., 0,... {{. s t s t p q Equation 4.1 then gives a t c 1 = { g t if p > 0, g t + 1 if p = 0. Because g t > 0, this contraicts the fact that µ satisfies conition (4.10). Hence g t cannot be an integer. Next, assume that g t Z+ 1. We nee to show that g t = 1. Recall that ε t is either 0 or ±1. We consier the two cases separately. If ε t = 0, the fine representation δ La of (L a K) must be non-trivial on both the Ũ(s t, s t )-factor an the Mp()-factor of L a. Then either (4.14) δ La =..., 1,..., 1 1 {{,..., 1, 1 {{,..., 1 {{ 1,..., 1,... {{ s t p q s t or (4.15) δ La =..., 1,..., 1 1 {{,..., 1, 1 {{,..., 1 1 {{,..., 1,... {{ p q s t for some p, q 0 such that p + q =. Equation 4.15 contraicts conition (4.10), because the ifference a t c 1 is always positive; hence Equation 4.14 must hol. We fin a t c 1 = 0 g t = 1 an p > 0. Note that, in this case, c b t > 0 by Equation This proves (b) (an also () for the case ε t = 0). If ε t 0, the fine representation δ La of (L a K) must be trivial on the Ũ(s t+ε t, s t )- factor of L a an non-trivial on the Mp()-factor of L a. Hence δ La =..., 0,. {{.., 0 1,..., 1, 1 {{,..., 1 {{ 0,..., 0,... {{. s t s t p q s t

16 16 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA Just as above, we get that a t c 1 = 0 g t = 1, ε t = 1 an p > 0. Moreover, c b t > 0 always in this case. This conclues the proofs of (a), (c) an (). The proof of Lemma 4.4 is now complete. Finally, we look at the case in which the irreucible, amissible, genuine an ω-regular (g, K)-moule X of M p(n) is also unitary. ( Realize X as the unique lowest K-type constituent of a representation R ) S q X L, with L = L 1 L 0 = Ũ (r, s) Mp() an XL X 1 X 0, as in Proposition.4. Proposition 3.1 then implies that X 1 a goo A q (λ ) moule. The next Lemma escribes the coorinates of the lowest K-type of X, uner some technical assumptions that are neee for the proof of Claim (A) in Proposition 3.. Lemma 4.5. Let X be an irreucible, amissible, genuine an ω-regular (g, K)- moule of Mp(n). Assume that X 1 (as above) is a goo A q (λ ) moule. Let µ be the unique lowest K-type of X an let λ a be its Vogan classification parameter. Write the coorinates of µ an λ a as in Equations 4.5 an.6, respectively. Set If g t = 1, ε t = 1 an s t > 0, then x = max {i < t r i > 0 y = max {i < t s i > 0. (i) b t b y. (ii) If g t 1 Z or if g t 1 Z + 1 an g t 1 7, then a x a t. Proof. Recall that, by Lemma 4., X has a unique lowest Ũ(n)-type µ (of the form (4.)). Equations 4.4 give (4.16) b t b t 1 = g t + h t ( g t 1 + h t 1 ) = g t + g t 1 + ε t 1 + ε t = g t 1 + ε t 1 (because g t = 1/ an ε t = 1). Note that g t 1 is a half-integer greater than g t = 1/, an ε t 1 is 0 or ±1. Then b t b t 1 1. (This is obvious if g t 1 Z + 1 ; when g t 1 is an integer, it follows from the fact that ε t 1 = 0 because b t b t 1 must be an integer.) Since the entries {b i t i=1 are weakly increasing, we also fin that b t b y 1. In orer to show that the ifference b t b y is, in fact, at least, we consier the lowest (K L 1 )-type µ 1 of X 1. This is a (Ũ(r) Ũ(s)) -type of the form (4.17) µ 1 = (µ ρ(u p)) eu(r) e U(s). Note that ρ(u p) is constant on Ũ(r) an Ũ(s) so, to unerstan the ifference b t b y, it is sufficient to look at the (ifference among) coorinates of µ 1. Write u (4.18) L = Ũ(p i, q i ). i=1

17 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 17 Section 8 of [9] gives a number of properties of lowest K-types of (goo) A q (λ) moules. Assuming we have chosen q so that L is maximal (there is a unique such choice), µ 1 must be of the form (4.19) µ 1 = n 1,..., n {{ 1,..., n u,..., n {{ u 0,..., 0 {{ m u,..., m u,..., m {{ 1,..., m 1 {{ p 1 p u q u q 1 with n j > n k an m k > m j for all 1 j < k u. Moreover, Proposition 8.6 of [9] implies that (4.0) n j n k q i + q i+1 an m k m j p i + p i+1 whenever j i an k i + 1. Let so that m z exists. Then we get z = max {i < u q i > 0, b t b y = m u m z p u + p u 1 (by (4.0) for i = u 1). By Proposition.4 (vii), an Proposition 3.5 of [10], L a L 1 u(r t, s t ) u(p u, q u ), so that L, therefore r t p u an s t q u. In fact, s t = q u, because ρ(u p) is constant on Ũ(r) an Ũ(s), an b t b t 1 > 0. Hence we can write b t b y p u + p u 1 r t = s t + ε t. This proves (i). For the proof of (ii), we istinguish two cases. If g t 1 is an integer, then Lemma 4. implies that ε t 1 = 0, hence s t 1 = r t 1 > 0 an x = y = t 1. Furthermore, because L a L 1 L, we have u(r t 1, s t 1 ) u(p u 1, q u 1 ). This implies Then (p u, q u ) = (r t, s t ) an q u 1 s t 1 1. a t 1 a t = n u 1 n u q u + q u =, proving (ii). If g t 1 7, then using Equations 4.4, we get a t 1 a t = g t 1 g t + f t 1 f t = g t 1 g t ε t 1 + ε t = g t 1 ε t 1 1. This conclues the proof of the lemma.

18 18 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA 5. Genuine Principal Series of M p() Genuine minimal principal series of Mp() were stuie in etails in [5]. In this section, we summarize the non-unitarity results that are neee for this paper. We refer the reaer to [5] for more etails. Lemma 5.1 (cf. [5]). (i) Every genuine minimal principal series representation of Mp() has a unique lowest Ũ()-type of the form µ δp,q = 1,..., 1, 1 {{,..., 1 {{ p q with p + q =. Here δ p,q is the character of the finite subgroup M of M p() use to construct the inuce representation. (ii) For every pair of non-negative integers p an q with p+q =, the Langlans quotients of genuine minimal principal series of Mp() with lowest Ũ()- type µ δp,q are parametrize by -tuples of real numbers ν = (ν 1,..., ν ) := (ν p ν q ), where ν 1 ν ν p 0 an ν p+1 ν p+ ν p+q 0. The infinitesimal character of the corresponing representation is equal to ν. (iii) Irreucible genuine pseuospherical representations of M p(), i. e., those with lowest Ũ()-type µ δ,0 or µ δ0,, are uniquely etermine by their infinitesimal character. Lemma 5.. Let J(δ p,q, ν) be the Langlans quotient of a genuine minimal principal series representation of Mp(), with lowest Ũ()-type µ δ p,q an parameter ν = (ν p ν q ) as in Lemma 5.1. (i) If ν p > 1 or if ν i ν i+1 > 1 for some 1 i p 1, then J (δ p,q, ν) is not unitary, an the Ũ()-type δ 1 = 1,... 1, 1 {{,..., 1, 3 {{ p 1 q etects non-unitarity. (ii) If ν p+q > 1 or if ν i ν i+1 > 1 for some p+1 i p+q 1, then J (δ p,q, ν) is not unitary, an the Ũ()-type δ = 3, 1,... 1, 1 {{,..., 1 {{ p q 1 etects non-unitarity. (iii) If J (δ p,q, ν) is ω-regular, pq 0, ν p ω p an ν q ω q, then both δ 1 an δ etect non-unitarity. As usual, ω r enotes the infinitesimal character of the oscillator representation of M p(r).

19 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 19 Proof. The main ingreients for the proof of Lemma 5. are certain non-unitarity results containe in [1] an [5]. Note that the assumptions in part (i) of Lemma 5. are precisely conitions (1) an (3) of Proposition 5 of [5]; similarly, the assumptions in part (ii) coincie with conitions () an (4) of that proposition. Therefore, in both cases, the non-unitarity of the Langlans quotient J(δ p,q, ν) of Mp() follows irectly from Proposition 5 of [5]. We are left with the problem of ientifying a Ũ()-type on which the intertwining operator changes sign. Recall from 7 of [5] that each Ũ()-type µ in J(δ p,q, ν) carries a representation ψ µ of the stabilizer W δp,q = W (C p ) W (C q ) of δ p,q. This group is isomorphic to the Weyl group of G δp,q = SO(p + 1, p) SO(q + 1, q); hence, associate with ψ µ, there is an intertwining operator for spherical Langlans quotients of G δp,q. If µ is petite, the Mp()-intertwining operator on µ with parameters {δ = δ p,q, ν = (ν p ν q ) coincies with the G δp,q -intertwining operator on ψ µ with parameters {δ = triv, ν = (ν p ν q ). If ν p > 1 or if ν i ν i+1 > 1 for some 1 i p 1, the spherical Langlans quotient of SO(p + 1, p) with parameter ν p is not unitary; the reflection representation σ R = (p 1) (1) of W (C p ) etects non-unitarity (see Lemma 14.6 of [1]). For all choices of ν q, the spherical Langlans quotient of SO(p + 1, p) SO(q + 1, 1) with parameter (ν p ν q ) is also not unitary, an the representation σ R triv of W δp,q etects non-unitarity. The computations in 10.1 of [5] show that the Ũ()-type δ 1 = 1,... 1, 1 {{,..., 1, 3 {{ p 1 q carries the representation σ R triv of W δp,q. Because δ 1 is petite, the Mp()- intertwining operator on δ 1 with parameters {δ p,q, (ν p ν q ) matches the G δp,q - spherical operator with parameter (ν p ν q ), hence it is not positive semi-efinite. This shows that J(δ p,q, (ν p ν q )) is not unitary, an conclues the proof of part (i) of the lemma. The proof of part (ii) is analogous: the Ũ()-type δ carries the representation triv σ R of W δp,q, hence the Mp()-intertwining operator on δ with parameters {δ p,q, (ν p ν q ) is not positive semi-efinite an the Langlans quotient J(δ p,q, (ν p ν q )) of Mp() is not unitary. For part (iii) of the lemma, note that if ν = (ν p ν q ) is ω-regular an ν p ω p, then ν p > 1 or ν i ν i+1 > 1 for some 1 i p 1. Similarly, if ν = (ν p ν q ) is ω-regular an ν q ω q, then ν p+q > 1 or ν i ν i+1 > 1 for some p + 1 i p + q 1.

20 0 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA Therefore, the assumptions of both part (i) an part (ii) of the lemma must hol, an both δ 1 an δ etect non-unitarity. 6. Proof of Proposition 3.1. In this section, we give the proof of Proposition 3.1. For convenience, we begin by restating the result. Proposition. Let X be an irreucible unitary (g, K)-moule of M p(n). Assume that X( is genuine an ω-regular. Realize X as the unique lowest K-type constituent of R ) S q X L, with L = L 1 L 0 = Ũ (r, s) Mp() an XL X 1 X 0, as in Proposition.4. Suppose that r + s 0. Then there exist a theta stable parabolic subalgebra q an a representation C λ of the Levi factor corresponing to q, with λ in the goo range for q, such that X 1 A q (λ ). Proof. Note that, by Proposition.4, the (l, L K)-moule X L is irreucible, an so are the Harish-Chanra moules X i of L i (i = 0, 1). Let µ be a lowest K-type of X, an let µ L be the (L K)-type of X L corresponing to µ via the bottom layer map (cf. Proposition.4 (ii)-(iii)); then (6.1) µ L = µ 1 + µ 0 with µ i = µ Li a lowest (L i K)-type of X i. By Lemma.7, the representation X 1 of L 1 = Ũ (r, s) is strongly regular (an irreucible). Assume, by way of contraiction, that X 1 is not a goo A q (λ) moule. We will show that X must be non-unitary, reaching a contraiction. By Theorem 1. in [9] for the case G = SU(p, q), if X 1 is not a goo A q (λ) moule, then X 1 not unitary, an there exists an (L 1 K)-type η 1 such that for some β (l 1 p), an the form η 1 = µ 1 + β, L1 V (µ1) V (η 1) is inefinite. (In [9], the infinitesimal character of X 1 is assume to be integral; however, the proof of Theorem 1. only uses the fact that it is real an strongly regular.) If η = µ + β = (µ 1 + β) + µ 0 + ρ (u p) is also ominant, then, by the bottom layer argument, η occurs in X an the Hermitian form on V (µ) V (η) is inefinite (cf. Proposition.4 (iii) an (v)). This implies that X is not unitary, reaching a contraiction. So we may assume that η = µ + β is not ominant. Moreover, we can assume that µ is weakly ominant with respect to our fixe choice of + c in (.5). Because β is a short non-compact root, if η 1 = µ 1 + β is ominant but η = µ + β is not ominant, then one of the following two options must occur: (6.) = 0 an a t = b t, or (6.3) a t = c 1 or c = b t.

21 OMEGA-REGULAR UNITARY DUAL OF THE METAPLECTIC GROUP 1 Remark 6.1. Lemma 4.4 shows that (6.) is not possible, an that the two ientities in (6.3) can not hol simultaneously. Without loss of generality, we may then assume that > 0 an a t = c 1. We first look at the case in Lemma 4.4 when ε t = 1 an g t = 1. By Lemma 4. (ii), the infinitesimal character γ X of X contains an entry 1, but γx0 oes not. Recall that X 0 is a genuine principal series representation of L 0 = Mp(). We claim that in this case, X 0 is not unitary, an that a Ũ()-type etecting non-unitarity survives uner the bottom layer map, leaing to a contraiction. We make use of the non-unitarity results for genuine minimal principal series of M p() containe in Section 5. Write X 0 = J (δp,q, ν) as in Lemma 5.; note that p > 0, by Lemma 4.4. Then, by ω-regularity, ν p is strictly greater than 1. By part (i) of Lemma 5., we know that the representation X 0 is not unitary, an the signature of the Hermitian form changes on δ 1. If µ 1 is as in Equation 6.1, the weight η = µ 1 + δ 1 + ρ (u p) is K-ominant. By Proposition.4 (iii) an (v), our original representation X is not unitary, which contraicts our assumption. Now assume that ε t = 0. Recall that the fine K-type δ La is of the form (4.11); in particular, its restriction to the subgroup Ũ(s t, s t ) of L a is given by δ La Ũ(st,s = t) 0,..., 0, 1,..., 1 {{ 0,..., 0 {{ 1,..., 1, 0,..., 0. {{ Hence, the representation µ 1 Ũ(st,s t), insie Mp(n), is of the form 0,..., 0, a,..., a {{ 0,. {{.., 0 a 1,..., a 1, 0,..., 0 {{ s t s t s t for some a 1 Z. By Lemma 8.8 an Lemma 6.3 (b) in [9], if β = (1, ; 0,... 0, 1) is the root ɛ 1 ɛ st in U(s t, s t ), then µ 1 U(st,s t) + β etects the signature change an survives in the bottom layer. Note that the root in Mp(n, R) corresponing to β is 0,..., 0, 1, 0,..., 0 {{ 0,. {{.., 0 1, 0,..., 0, 0,..., 0. {{ s t s t Therefore, X is not unitary, once again a contraiction. The proof of Proposition 3.1 is now complete. 7. Proof of Proposition 3. We now restate an prove Proposition 3.. Proposition. Let X be an irreucible unitary (g, K)-moule of M p(n). Assume that X( is genuine an ω-regular; realize X as the unique lowest K-type constituent of R ) S q X L, with L = L 1 L 0 = Ũ (r, s) Mp() an XL X 1 X 0, s t

22 A. PANTANO, A. PAUL, AND S. SALAMANCA-RIBA as in Proposition.4. Suppose that > 0. Then one of the following (mutually exclusive) options occurs: (i) X 0 is an even oscillator representation with lowest Ũ()-type µ 0 = ± (1/, 1/,..., 1/). (ii) There exist a subgroup L = L 1 L 0 = Ũ(r, s ) Mp(( + 1)) G also containing L a, a theta stable subalgebra q = l + u q an representations X i of L i (i = 0, 1), such that (a) Ũ (r, s ) Ũ (r, s), with either r = r + 1 or s = s + 1, (b) X 1 is a goo A q3 (λ 3 ) (or r + s = 0), (c) X 0 is an o oscillator representation with lowest K-type µ 0 = (3/, 1/,... 1/) or ( 1/, 1/,... 1/. 3/), an () X can be realize as the unique lowest K-type constituent of R S q (X 1 X 0). Proof. Let λ a be the Vogan classification parameter of X. Recall that, by Proposition 3.1, X 1 is a goo A q (λ ) moule; then, by Lemma 4., X has a unique lowest Ũ(n)-type µ. To prove Proposition 3., we first show that the representation X 0 of Mp() is pseuospherical, i. e., that X 0 J(δ p,q, ν) with pq = 0. Assume, by way of contraiction, that X 0 J(δ p,q, ν) with p > 0 an q > 0, an set ν = (ν p ν q ) (as in Lemma 5.1). Write the lowest K-type µ of X as in Equation 4.5 an the Vogan s classification parameter λ a as in (.6). Recall that the half integers a i an b i are efine even if r i = 0 or s i = 0, an that they satisfy (4.9). By an argument similar to the proof of Lemma 4.4, (at least) one of the following conitions must hol: a t > c 1 or c > b t. Notice that, since p > 0 an q > 0, the Ũ()-types δ 1 = 1,... 1, 1 {{,..., 1, 3 {{ an δ = 3, 1,... 1, 1 {{,..., 1 {{ p 1 q p q 1 (of Lemma 5.) are obtaine from the lowest Ũ()-type µ 0 of X 0 by aing or subtracting a short root. One can check that if a t > c 1, then δ survives in the bottom layer; if c > b t, then δ 1 survives in the bottom layer. We istinguish two cases. First, suppose that ν p ω p an ν q ω q ; then (by Case 3 of Lemma 5.) X 0 is not unitary an both δ 1 an δ etect non-unitarity. Because (at least) one of the two Ũ()-types survives in the bottom layer, we conclue that X is not unitary an we reach a contraiction. Next, assume that ν p = ω p or ν q = ω q. (Note that the two options can not hol at the same time, because X 0 is ω-regular.) If ν p = ω p, then the entries of ν q are all greater than or equal to p + 1 ; in particular, ν p+q > 1. Hence, we are in Case (ii) of Lemma 5.: X 0 is not unitary an δ etects non-unitarity. Similarly, if ν q = ω q, then X 0 is not unitary an δ 1 etects non-unitarity. In either case, the