Howe Duality Correspondence of (O(p, q), osp(2, 2))

Transcription

1 Howe Duality Correspondence of (O(p, q), osp(, )) Dan Lu November 19, 009 Abstract The local theta correspondence for reductive dual pairs of subgroups of the symplectic group formulated by R. Howe has repeatedly provided insight into representation theory of reductive groups and the theory of automorphic forms. The most heavily studied example of such a pair is (O(p, q), SL(, R)). In this talk we will discuss a variant of this pair, involving the indefinite orthogonal groups O(p, q) and the Lie superalgebra osp(, ). We will classify the irreducible representations of osp(, ) subject to some technical restrictions and construct the correspondence for the pair (O(p, q), osp(, )) in the oscillator-spin representation. The Casimir operator plays an important role in setting up the correspondence. 1 Introduction In the paper [H1], Roger Howe formulated the notion of a reductive dual pair of subgroups of the symplectic group. A reductive dual pair is a pair of subgroups (G, G ) of the symplectic group such that G is the centralizer of G and vice versa. Let ω be the osillator representation of Sp(n, R) realized on a Hilbert space H. Howe has shown that for a real reductive dual pair (G, G ) Sp(n, R), there is a bijection between some irreducible admissible representations of G(a double cover of G) and some representations of G (a double cover of G ) which can be realized as quotients by ω (Sp)-invariant closed subspaces of H where H is the subspace of smooth vectors in H. We call this bijection the oscillator duality correspondence or Howe correspondence. After that a lot of work has been done for this particular representation. Many explicit constructions of the oscillator duality correspondence have been established for various dual pairs. In that paper, Howe also formulated the notion of a reductive dual pair in the orthosymplectic Lie superalgebra and the oscillator representation was extended to a representation for the orthosymplectic Lie superalgebra, which we call oscillator-spin representation for the orthosymplectic Lie superalgebra denoted by ω. The object of this talk is to discuss the Howe correspondence in the oscillator-spin representation of (O(p, q), osp(, )) osp((p + q), (p + q)). In this representation, the Casimir operators of O(p, q) and osp(, ) are exactly the same. We show that the Howe correspondence exists for the representations with nonzero eigenvalues for the Casimir operators. When q = 1, we can even give the Howe correspondence of the representations with zero eigenvalue for the Casimir operators. Thus the Howe correspondence is completely proved in this case.

2 Structure of osp(, ) and its irreducible representations In order to get the correspondence of (O(p, q), osp(, )) in the oscillator-spin reprensentation, we first study the structure of this Lie superalgebra and classify its irreducible representations subject to some technical restrictions. The irreducible osp(, )-representations which can appear in the correspondence belong to this class. We begin with a short review of representations of sl(, R). Let {h, e +, e } be the standard basis of sl(, R) and Ω sl = h + (e + e + e e + ) be its Casimir operator. We will describe all irreducible h-semisimple, h-admissible and Ω sl -quasisimple sl -modules in this section. This is a classical result which has been known for a long time. Theorem.1. ([HT]) All h-admissible, h-semisimple quasisimple irreducible sl -modules are classified into the following 4 classes: 1) V λ, λ / Z ) V λ, λ / Z + 3) F λ, λ Z + 4) U(ν +, ν ) ν +, ν / Z Theorem.. ([Kn]) All irreducible representations of SL(, R) must be infinitesimally equivalent to one of the following: 1) Discrete series V λ, λ Z + ; V λ, λ Z. ) Finite dimensional representations F m, m Z +. 3) Principal series S s,+ = U( s, s ), s / Z, Ss, = U( s+1 ), s / Z + 1, s 1 Since the fundamental group of SL(, R) is Z, for each n N, there exists an n- fold covering group SL n (, R) of SL(, R). In this talk, we are particularly interested in the -fold covering of SL(, R) denoted by SL(, R). This is the group which relates to the representation of osp(, ) in the duality correspondence (O(p, q), osp(, )). We say a representation of SL(, R) is genuine if it does not factor to SL(, R). Corollary.3. The irreducible genuine SL(, R) representations are infinitesimally equivalent to the followings: 1) Lowest weight modules, V λ, λ Z + 1. ) Highest weight modules, V λ, λ Z ) Principal series U( ν + 1 4, ν ) or U(ν + 3 4, ν ), ν / Z + 1. Now we look at osp(, ) in detail. First we give the matrix form of the elements osp(, ). Form [K], we see a matrix in osp(, ) is of the form osp(, ) 0 = { k 0 x x 1 0 k y y 1 y 1 x 1 a b y x c a k k a b 0 0 c a (.1) = sl so }

3 Again we let e + = osp(, ) 1 = { e = 0 0 x x y y 1 y 1 x y x be a basis for sl. Additionally, the element is a basis for so. d = Define d = h = δ = h = } (.) (.3) δ = (.4) After simple calculation, we have the following commutation relations. Since for x, y osp 1 (, ), [x, y] = xy + ( 1) 1 1 yx = xy + yx = {x, y}, we use {, } to indicate the operation for x, y osp 1 (, ). For the Lie subalgebra sl so : [h, e + ] = e + [h, e ] = e [e +, e ] = h [h, x] = 0 Here x is anything in sl so. Between the Lie algebra and the super part. [e +, d] = δ [e +, δ ] = 0 [e +, d ] = δ [e +, δ] = 0 [e, d] = 0 [e, δ ] = d [e, d ] = 0 [e, δ] = d [h, d] = d [h, δ ] = δ [h, d ] = d [h, δ] = δ [h, d] = d [h, δ ] = δ [h, d ] = d [h, δ] = δ (.5) Anticommutation relations for the super part. d = {d, δ } = (δ ) = 0 (d ) = {d, δ} = δ = 0 {d, d } = e {d, δ} = h h {d, δ } = h + h {δ, δ } = e + (.6) From the above commutation relations, we get:

4 Theorem.4. osp(, ) is a classical Lie superalgebra. The pairs {d, δ } and {d, δ} each span a two dimensional irreducible module for sl with eigenvalue +1 and 1 for h respectively. As for sl, it is useful to know the center of U(osp(, )). Theorem.5. Define Ω osp = Ω sl h ([d, δ] + [d, δ ]) in U(osp(, )). Then Ω osp is in the center of U(osp(, )), in other words, Ω osp commutes with every element in U(osp(, )). Moreover, Ω osp generates the center of U(osp(, )) as an (associative) algebra, i.e. Z(U(osp(, )) = C[Ω osp ]. Having dealt with the structure of osp(, ) and given the commutation relations for the basis vectors of osp(, ), we are able to classify its h-admissible, h, h -semisimple Ω osp - quasisimple irreducible representations, Since the Lie part is a direct sum of sl and so, we classify the irreducible osp(, )- modules according to their structures considered as sl so -modules. From Theorem.1, we know the h-semisimple, h-admissible quasisimple irreducible representations of sl are classified into four different classes. Since so is an abelian Lie algebra, so its irreducible representation can be identified with C. Also since so commutes with sl, each h, h - semisimple irreducible sl so -module has only one eigenvalue for h. We use V λ,µ, V λ,µ, F λ,µ, U(ν +, ν, µ) to denote the irreducible sl so -modules of the same sl -types as in Theorem.1 with h -eigenvalue µ. We will analyze h-admissible h, h -semisimple quasisimple osp(, )-modules which contain sl so -modules from these four different classes. For osp(, ), we consider q = sl so E where E is the span of d and δ. It is easy to check that q is a Lie subsuperalgebra, and actually we can get the following theorem which is helpful for us to classify irreducible h-admissible, h, h -semisimple, Ω osp -quasisimple osp(, )-modules. Theorem.6. Suppose V is an h, h -semisimple, h-admissible, Ω osp -quasisimple, irreducible osp(, )-module. a) Set W 1 = ker d ker δ. Then W 1 is invariant under sl so, and is a non-zero, irreducible, h, h -semisimple, quasisimple sl so -module. b) Set W = Indq osp(,) W 1. Then W is an indecomposable osp(, ) module, and V is a quotient of W. Theorem.7. Suppose V is an h, h -semisimple, h-admissible, Ω osp -quasisimple, irreducible osp(, )-module and W 1 = ker d ker δ. Let W = Ind osp(,) q W 1. Then W is irreducible if and only if Ω osp W 0. Based these two Theorems above, we are able to classify the irreducible osp(, )- modules. We omit elaborate computations here. 1) Irreducible lowest weight modules W (λ, µ)(λ / Z ). i) µ ±λ

5 W is composed of four sl so -modules depicted below. V λ+1,µ+1 V λ,µ V λ+,µ V λ+1,µ 1 ii) µ = λ (λ 0) iii) µ = λ (λ 0) W (λ, λ) = V λ,λ V λ+1,λ 1 W (λ, λ) = V λ, λ V λ+1, λ+1 )Irreducible highest weight modules W (λ, µ) (λ / Z + ). i) µ ±λ W is composed of four sl so -modules depicted below. V λ 1,µ+1 V λ,µ Vλ,µ V λ 1,µ 1 ii) µ = λ (λ 0). iii) µ = λ (λ 0) W (λ, λ) = V λ,λ V λ 1,λ+1 W (λ, λ) = V λ, λ V λ 1, λ 1 3) Irreducible finite dimensional modules F osp (λ, µ) (λ Z ). i) Trivial module F osp (0, 0). ii) µ ±λ, a) λ 1, F osp (λ, µ) is composed of four finite dimensional sl so -module. F λ+1,µ+1 F λ,µ F λ+,µ F λ+1,µ 1 b) λ = 1, F osp ( 1, µ) sl so = F 1,µ F 0,µ 1 F 0,µ+1

6 iii) µ = λ (λ 0) iv) µ = λ (λ 0) F osp (λ, λ) = F λ, λ F λ+1, λ+1 F osp(λ,λ) = F λ,λ F λ+1,λ 1 4) Irreducible modules W (ν +, ν, µ), ν +, ν / Z, µ C i) µ ±(ν + + ν ) W is composed of four sl so -modules depicted below. U(ν +, ν + 1, µ + 1) U(ν +, ν, µ) U(ν + + 1, ν + 1, µ) U(ν +, ν + 1, µ 1) ii) µ = ν + + ν W sl so = U(ν +, ν, µ) U(ν +, ν + 1, µ 1) iii) µ = (ν + + ν ) W sl so = U(ν +, ν, µ) U(ν +, ν + 1, µ + 1) Theorem.8. All h, h -semisimple, h-admissible, Ω osp -quasisimple irreducible representations of osp(, ) are classified into four groups listed in the lemmas above: 1) W (λ, µ), λ C, λ / Z, µ C; ) W (λ, µ), λ / Z +, µ C ; 3) F osp (λ, µ), λ Z ; 4) W (ν +, ν, µ), ν +, ν / Z, µ C. Proof. In Theorem.6, we proved that each h-admissible, h, h -semisimple, Ω osp -quasisimple irreducible osp(, )-module V is a quotient of an induced module from an irreducible sl so -module which is annihilated by d and δ. And we have found all irreducible quotients in the osp(, )-modules induced from four different classes of sl so -modules. Thus V must be isomorphic to one of the osp(, )-modules listed in the theorem. Let V be an h-admissible, h, h -semisimple, Ω osp -quasisimple, irreducible osp(, )-module with given basis vectors. The set of formal vectors on V is all possibly infinite linear combinations of these basis vectors in V. We write it as V. In the following sections, we are interested in understanding the action of osp(, ) on V. We may think of V as sort of completion of V, and this becomes a useful analytic tool, which will be used in setting up the duality correspondence. Theorem.9. Let W (λ 1, µ 1 ) and W (λ, µ ) be the lowest and highest weight irreducible osp(, )-modules λ i ±µ i (i = 1, ). In this case λ 1 / Z and λ / Z +. Let K(a, b) be the formal vectors in the h, h -eigenspace with h, h -eigenvalues a, b in (W (λ 1, µ 1 ) W (λ, µ )). Then

7 a) For l Z such that the parameters a ±b in K(a, b), there is an one dimensional kernel for {d, d, e } in each of the following spaces, K(λ 1 + λ 1 + l, µ 1 + µ 1), K(λ 1 + λ, µ 1 + µ ), K(λ 1 + λ 1 + l, µ 1 + µ + 1), and there is a two dimensional kernel for {d, d, e } in K(λ 1 + λ + l, µ 1 + µ ). b) For l Z + such that the parameters a ±b in K(a, b), there is an one dimensional kernel for {δ, δ, e + } in each of the following spaces, K(λ 1 + λ l, µ 1 + µ 1), K(λ 1 + λ, µ 1 + µ ), K(λ 1 + λ l, µ 1 + µ + 1), and there is a two dimensional kernel for {δ, δ, e + } in K(λ 1 + λ + + l, µ 1 + µ ). Corollary.10. Let W (λ 1, µ 1 ) and W (λ, µ ) be the lowest weight and highest weight irreducible osp(, )-module, µ i ±λ i (i = 1, ), λ 1 Z, λ Z +. Consider the formal vectors in (W (λ 1, µ 1 ) W (λ, µ )), a) if λ 1 + λ Z, each {e, d, d }-null vector in K(a, b) with a ±b in Theorem.9 generates an irreducible lowest weight osp(, )-module W (a, b). b) if λ 1 + λ Z, each {e, d, d }-null vector in K(a, b) with a ±b in Theorem.9 generates an irreducible lowest weight osp(, )-module W (a, b) if and only if a 0. 3 Some degenerate principal series representations of O(p, q) In this section, we discuss a class of representations which are induced from a maximal parabolic subgroup of O(p, q), usually called degenerate principal series. In [HT], one can find the structures of some of these degenerate principal series, the ones which are actually obtained from inducing a character on the Levi component of the maximal parabolic subgroup. These representations occur in the duality correspondence of (O(p, q), SL ). However, what we want to find now is the correspondence of (O(p, q), osp(, )). More representations will appear in this correspondence. For this purpose, we will extend the class of representations in this section by inducing from a particular class of finite dimensional representations of the maximal parabolic subgroup. Except in a few exceptional cases, we are able to determine the structures of these new representations by using Zuckerman s translation functor. The exceptional cases are those on which the action of the Casimir operator of O(p, q) vanishes. Let O(p, q) be the orthogonal group acting on R p+q with the basis {ɛ 1,..., ɛ p+q } and g be its Lie algebra. We choose an abelian subalgebra a of g, a = R(e 1,p+q + e p+q,1 ) where e ij denotes the (p + q) (p + q) matrix with 1 in the intersection of the ith row and jth column and zero otherwise. The maximal compact subgroup K of O(p, q) is isomorphic to O(p) O(q) and k is the Lie algebra of K. Let g = k a n be an Iwasawa decomposition of g with a a and we assume the parabolic subalgebra contains n. Let P = MAN be the maximal parabolic subgroup that stabilizes the isotropic line R(ɛ 1 + ɛ p+q ) with the Langlands decomposition A = {expta, t R}, M = O(p 1, q 1) Z where O(p 1, q 1) acts on the subspace spanned by {ɛ,..., ɛ p+q 1 }. (3.1)

8 Let σ be a representation of M, and ν be a character of A. We call the induced representation π(p, σ ν) = Ind G P (σ ν 1) a generalized principal series representation or a degenerate principal series. The representation space for Ind G P (σ ν 1) is with norm G acts by f = {f C (G, V ) f(mang) = e (ν+ρ A) log a σ(m)f(g)} (3.) K f(k) dk, ρ A = 1 (dimµ)µ, where Γ + = {roots of (g, a) positive for N}. µ Γ + π(p, σ ν)(g)f(x) = f(xg) Here we use the normalized induction. These degenerate principal series are unitary when Re ν = 0. In [HT], Howe and Tan have analyzed the degenerate principal series Ind G P (ɛ ν 1) where ɛ = 0, 1 indicates the trivial or sign representation on Z factor of M. In the oscillator-spin representation, we need to tensor the representation space of the oscillator representation with the exterior algebra, thus it is necessary for us to determine the structure of Ind G P (ɛ ν 1) Λk (R p+q ) as O(p, q)-representation. The main tool we use is the Zuckerman s translation functor. Let + (h C, g C ) be a fixed choice of positive system [Kn] and M be an H-C module with infinitesimal character γ such that Re γ is dominant under + (h C, g C ). Suppose V σ is a finite dimensional irreducible representation of g with highest weight σ with respect to this same positive system of roots. The (g, K)-module M V σ has a summand with infinitesimal character γ +σ, which is again a H-C module. Let P γ γ+σ be the projection on this summand and let φ γ γ+σ be the functor M P γ σ+γ (M V σ). Let V σ be the contragradient module to V σ. Then M V σ has a direct summand with infinitesimal character γ σ. Let P γ γ σ be the projection on this summand and let ψ γ γ σ be the functor M P γ γ σ (M V σ). Theorem 3.1. For 1 k p+q 1, ν C, Re ν 0, let λ k be the highest weight of O(p, q) acting on Λ k (R p+q ) and χ u(ν) be the infinitesimal character of Ind G P (ɛ ν 1). Then when ν p+q k 1, φ u(ν) u(ν)+λ k is exact and i) If Reν p+q k 1, φ u(ν) u(ν)+λ k Ind G P (ɛ ν 1) = IndG P (ɛ σ k 1 (ν + 1) 1). ii) If Reν < p+q k 1, φ u(ν) u(ν)+λ k Ind G P (ɛ ν 1) = IndG P (ɛ σ k ν 1). In particular, when ν Z for p + q even or ν Z + 1 for p + q odd, IndG P (ɛ σ k ν 1) is irreducible. Here σ k is the representation of O(p 1, q 1) on Λ k (R p+q ). 4 Howe Correspondence of (O(p, q), osp(, )) We realize the oscillator representation for this pair (O(p, q), osp(, )) in the Schrödinger model L (R p+q ) Λ (R p+q ) such that for any g O(p, q), ω p,q (g)( n f i v i )(x) = n f i (g 1 x) gv i

9 for n f i v i L (R p+q ) Λ (R p+q ). In this realization the smooth vectors are S(R p+q ) Λ (R p+q ). If we differentiate the representation on S(R n ) Λ (R n ), we find that a basis for the action of Lie algebra o p,q consists of: x j y j x j x k + xj xk xk xj x k x j j, k = 1,, p y k + yj yk yk yj y k y j j, k = 1,, p (4.1) + y k + xj yk + yk xj y k x j 1 j p, 1 k q The operators in osp(, ), which commute the action of O(p, q) in this realization, have the following forms: e = i p,q = i ( x x p y1 yq ) e + = i r p,q = i (x x p y1 x q) sl p q h = x i + y i + p + q x i y i h = p xi xi + d = i( d = ( p p j=1 q j=1 x i xi + x i xi yj yj p + q q j=1 q j=1 y j yj ) δ = i( y j yj ) δ = so p x i xi + p x i xi q y j yj ) j=1 q y j yi osp(, ) 1 j=1 (4.) The operators satisfy the (anti)commutation relations in (.5) and they form a basis of osp(, ). Theorem 4.1. In the oscillator-spin representation ω p,q of (O(p, q), osp(, )), ω p,q (Ω p,q ) = ω p,q (Ω osp(,) ). From Theorem 4.1, we see that the eigenvalues of the Casimir operator of O(p, q) and osp(, ) are the same in the oscillator-spin representation, this will help us find out the correspondence of their representations. Theorem 4. (Howe correspondence for (O(p), osp(, ))). Let ω be the oscillator representation of osp(p, p) realized in the space L (R p ) Λ (R p ). Then L (R p ) Λ (R p ) is isomorphic to λ p 0 W (p, p ) λ p m,a W (m 1 + p, a p ) λ p det W (p, p ) (5.3) m 1,1 a p 1 Here W (λ, µ) is the lowest weight irreducible representation of osp(, ) and λ p m,a denotes the O(p) module corresponding to the hook diagram with the length of first low equal to m and the depth of the first column equal to a. λ p 0, λp det are trivial and determinant representations of O(p) repectively.

10 Remark 4.3. This decomposition and more general results are given by K. Nishiyama in [Ni]. In the following we discuss the Howe correspondence for (O(p, q), osp(, )). Although we could not prove the Howe correspondence completely, we can show the Howe correspondence is bijective for the representations of (O(p, q), osp(, )) with nonzero eigenvalue for the Casimir operators. (In Theorem 4.1, we showed the Casimir operators of (O(p, q) and osp(, )) coincide in the oscillator-spin representation.) This covers all but finitely many representations. Theorem 4.4. In the oscillator-spin representation ω p,q of (O(p, q), osp(, )), when the eigenvalue of Ω o(p,q) is nonzero, for each π R(O(p, q), ω p,q ), there is a unique π R(osp(, ), ω p,q ) such that π π R(O(p, q) osp(, ), ω p,q ). Sketch of proof: We give the proof in the case p, q even with p, q. The other cases can be handled by the similar arguments. We use the Schrödinger model to realize the oscillator-spin representation ω p,q on the space L (R p+q ) Λ (R p+q ). When we restrict ω p,q to O(p, q) SL(, R) SO, we find SL(, R) only acts on the function part and SO only acts on the exterior part. Therefore, ω p,q O(p,q) SL(,R) SO can be regarded as the extension from the oscillator representation ω p,q of (O(p, q), SL(, R)) such that O(p, q) acts both on the function part and exterior part. When ν / Z, we have π(ɛ ν 1) Λ k (R p+q ) = Ind G P (ɛ σ k ν 1) Ind G P (1 ɛ σ k 1 (ν + 1) 1) Ind G P (1 ɛ σ k 1 (ν 1) 1) Ind G P (ɛ σ k ν 1) Each summand is an irreducible degenerate principal series of O(p, q). Thus we see that Ind G P (ɛ σ k ν 1) only appears in π(ɛ ν 1) Λ k (R p+q ) π(1 ɛ (ν + 1) 1) Λ k+1 (R p+q ) π(ɛ ν 1) Λ k+ (R p+q ) π(1 ɛ (ν 1) 1) Λ k+1 (R p+q ) According to the results of the correspondence of (O(p, q), osp(, )) in the oscillator representation, we can see that exactly the following irreducible SL(, R) SO()- representations are in the isotypic component Ind G P (ɛ σ k ν 1) (ɛ p q (mod )) of O(p, q). U( ν + 1, ν + 1, k p + q ) U(ν , ν + 1, k + 1 p + q ) U( ν + 1, ν + 1 1, k + 1 p + q ) U(ν + 1, ν + 1, k + p + q ) (4.3) Similarly, the Ind G P (ɛ σ k ν 1)(ɛ p q + 1 (mod )) isotypic component of O(p, q) exactly contains U( ν +, ν, k p + q ) U(ν +, ν + 1, k + 1 p + q ) U( ν, ν, k + 1 p + q ) U(ν +, ν, k + p + q ) (4.4)

11 Since ν Z, the Casimir operator has nonzero eigenvalue. Therefore, these four SL SO representations in (4.3),(4.4) must be connected by the operators from the superpart and form irreducible osp(, )-representations W ( ν+1, ν 1 p+q, k +1 ) and W ( ν, ν p+q, k +1 ) respectively. This gives the correspondence when ν / Z. Now we fix k, (0 k p + q ), and consider the case ν Z +, ν ±( p+q k 1). We can show that W (ν, µ), F osp (ν, µ), W ( ν, µ) have the same eigenvalues for the Casimir operator Ω osp. In fact they are constituents of Ind osp(,) q U( ν+1, ν+1, µ) or Indq osp(,) U( ν+, ν, µ). From the computations in the proof of Theorem 3.1, we know that the infinitesimal characters of Ind G P (ɛ σ k ν 1)(ν C, 0 k p + q ) are pairwise distinct if ν ±(k + q p+q ). Since the Casimir operators of O(p, q) and osp(, ) are the same in the oscillator-spin representation, we conclude that the O(p) O(q)-types in the isotypic components of W (ν, k + 1 p+q ), F osp(ν, k + 1 p+q ) and W (ν, k + 1 p+q ) are all contained in Ind G P (ɛ σ k ( ν) 1) with ɛ ν + 1 p q (mod ). Our job now is to match the constituents of the induced representations of osp(, ) to constituents of Ind G P (ɛ σ k ( ν) 1) with ɛ ν + 1 p q (mod ). We first consider W (ν, k+1 p+q ). By Corollary.10, we can find the O(p) O(q)-types in the W (ν, k + 1 p+q )-isotypic component. We denote the set of these O(p) O(q)-types by W. What we still need to show is that these O(p) O(q)-types form a composition factor in Ind G P (ɛ σ k ( ν) 1). From the structures of irreducible osp(, )-modules in section, W (ν, k + 1 p + q ) sl so = V ν+1,k p+q V ν,k+1 p+q V ν+,k+1 p+q V ν,k+ p+q Let π D (ν) be the composition factor in Ind G P (ɛ ( ν) 1) which corresponds to V ν+1 in the oscillator representation of (O(p, q), SL(, R)). Then W is contained in π D (ν) Λ k (R p+q ). Since π D (ν) Λ k (R p+q ) is contained in a direct sum of degenerate principal series of O(p, q) with distinct infinitesimal character. Suppose ν < p+q k 1. By Theorem 3.1, when we tensor π D (ν) by Λ k (R p+q ), Zuckerman s translation functor φ u(ν) u(ν)+λ k π D (ν) gives a composition factor in Ind G P (ɛ σ k ( ν) 1) which we denote by π D (σ k ν). Since π D (ν) corresponds to V ν+1 in the oscillator representation, π D (σ k ν) contains all O(p) O(q)-types in the SL SO -isotypic component of V ν+1,k p+q This fact shows that W is the same as π D (σ k ν) in Ind G P (ɛ σ k ( ν) 1). Therefore, W (ν, k + 1 p+q ). π D (σ k ν) corresponds to W (ν, k + 1 p+q ). By similar reasoning, we also have the correspondence of W (ν, k + 1 p+q ) and π D(σ k ν) when ν > p+q k 1. The correspondence of F osp (ν, k + 1 p+q ) and W ( ν, k + 1 p+q ) can be proved analogously. Remark 4.5. The Howe correspondence of (O(p, q), osp(, )) can be depicted by the diagram below. We again consider the case p even, q even.

12 6 θ( V (ν+1) ) V (ν+1) F (ν 1) V ν+1 θ - r? r Ind 6 θ(f (ν 1) ) θ(v ν+1 ) φ(θ( V (ν+1) )) φ = φ u(ν) u(ν)+λ k? - W ( ν, µ) F osp ( ν, µ) W (ν, µ) r µ = k + 1 p+q r Diagram:4.1 θ - φ(θ(f (ν 1) )) φ(θ(v ν+1 )) - When q = 1, based on the classification of the irreducible O(p, 1)-representations [Hi] and O(p)-spectrum analysis in the osp(, )-isotypic component, we can complete the correspondence of the oscillator-spin representation for (O(p, 1), osp(, )). Theorem 4.6 (Howe correspondence for (O(p, 1), osp(, ))). In the oscillator-spin representation ω p,1 of (O(p, 1), osp(, )), the set R(O(p, 1) osp(, ), ω p,1 ) is the graph of bijection between R(O(p, 1), ω p,1 ) and R(osp(, ), ω p,1 ). In other words, for each π R(O(p, 1), ω p,1 ), there is a unique π R(osp(, ), ω p,1 ) such that π π R(O(p, 1) osp(, ), ω p,1 ). References [Fo] G. Folland, Harmonic analysis in phase space, Annals of Mathematics studies 1, Princeton University Press. [GW] R. Goodmand and N. Wallach, Representations and Invariants of the classical Groups, Cambridge University Press. [Hi] T. Hirai, On infinitesimal Operators of Irreducible Representations of the Lorentz Group of the nth Order, Proc. Japan. Acad. 38 (196),

13 [Hi] T. Hirai, On irreducible Representations of the Lorentz Group of nth Order, Proc. Japan. Acad. 38 (196), [H1] R. Howe, Remarks on Classical Invariant Theory, Trans. Amer. Math. Soc. 313 (1989) [H] R. Howe, On some results of Strichartz and of Rallis and Schiffman, J. Funct. Anal. 3 (1979), [H3] R. Howe, Dual Pairs in Physics: Harmonic Oscillators, Photons, Electrons, and Singletons, Lectures in Applied Math 1, American Math. Soc., Providence, RI, 1985 [H4] R. Howe, Transcending classical invariant theory, J. Amer. Math. Soc, (1989), No.3, [HT] R. Howe and E.C. Tan, Homogeneous functions on light cones: the infinitesimal structure of some degenerate principal series, Bull. Amer. Math. Soc. 8 (1993), [HT] R. Howe and E.C. Tan, Non-abelian Harmonic Analysis, Applicants of SL (R), Universitext 96,1995 [K] V. G. Kac, Lie Superalgebra, Adv. in Math. 6, 8-96 (1997). [Kn1] A. Knapp, Lie Groups Beyond an Intrudction Progress in Mathematics, Birkhäuser. [Kn] A. Knapp, Representation Theory of Semisimple Groups, Priceton University Press. [LTZ] J.S. Li, E.C. Tan and C. Zhu, Tensor Product of Degenerate Principal Series and Local Theta Correspondence, J. Functional Analysis 186 (001), [LV] G.Lion and M.Vergne, The Weil representation Maslov Index and theta series, Progress in Mathematics 6. [Ni] K. Nishiyama, Super Dual Pairs and Highest Weight Modules of Orthosymplectic Algebras, Adv. in Math. 104 (1994), [Ni] K. Nishiyama, Oscillator Representations for Orthosymplectic Algebras, J. Alge. 19 (1990), [Se] A. Sergeev, The invariant polynomials on simple Lie superalgebras, Represent. Theory 3 (1999) [St] R. Strichartz, Harmonic Analysis on Hyperboloids, J. Functional analysis 1 (1973), [Zu] G. Zuckerman, Tensor products of finite and infinite dimensional Representaions of semisimple Lie groups, the Ann. of.math.,100(1977), No.., [ØZ1] B. ØRsted and G. Zhang, L -versions of the Howe correspondence I, Math. Scand. 80 (1997), [ØZ] B. ØRsted and G. Zhang, L -versions of the Howe correspondence II, J. Math. Pures Appl. 74, 1995 pp