GENUINE REPRESENTATIONS

OF THE METAPLECTIC GROUP

Jeffrey Adams and Dan Barbasch

§0. Introduction

Consider the dual pairs (O(p, q), Sp(2n, ℝ)) with p + q = 2n + 1. Let S̃p(2n, ℝ) be the metaplectic group, and Õ(p, q) the det^{\frac{1}{2}} cover of O(p, q) (we will be more precise in §1). For ψ a non–trivial additive character of ℝ, the oscillator representation ω(ψ) yields a bijection θ(ψ, p, q) between subsets of the irreducible representations of S̃p(2n, ℝ) and those of Õ(p, q) [5]. The representations of S̃p(2n, ℝ) which arise are all genuine, i.e. do not factor to the linear group Sp(2n, ℝ). The main result of this paper is an explicit description of this correspondence (Theorem 5.1).

Fix the discriminant δ = (−1)^{q} of the orthogonal space. An immediate corollary of Theorem 5.1 is a bijection, depending on ψ, between the set

S̃p(2n, ℝ)_{genuine}

of (equivalence classes of) genuine irreducible admissible representations of S̃p(2n, ℝ) and the union

⋃_{\begin{matrix}p+q=2n+1 \\ (−1)^{q}=δ \end{matrix}} SO(p, q)

of the irreducible admissible representations of the groups SO(p, q) (cf. Corollary 6.2 for details). This result confirms, in the real case, part of a conjecture of Kudla [8], which in turn is a generalization of a result of Waldspurger [22] in the case n = 1.

The bijection is one of similarity, rather than of duality, in that it takes small representations to small representations. For example it takes the trivial representations of SO(n + 1, n) and SO(n, n + 1) to the even halves of the oscillator representations of S̃p(2n, ℝ).

The metaplectic group is an example of a non–linear group, to which the machinery of the L–group does not apply. On the other hand it is of great importance in the theory of automorphic representations, so it is of interest to understand it in these terms. With this in mind notions such as L–packet, stability, etc. may be defined for S̃p(2n, ℝ) by carrying over the corresponding notions from SO(p, q). Even in the easiest examples it is clear that

Typeset by AMS-TEX

care must be taken in making such extensions. For example the representations in an L– packet defined in this manner may fail to have the same central character, a phenomenon which is forbidden for linear groups (and also for the larger L–packets and Arthur–packets of [4]).

This bijection is natural in terms of the Langlands classification. The Cartan subgroups of O(p, q) are isomorphic to those of Sp(2n, ℝ), and very roughly speaking the matching is given by the same characters. For example discrete series representations having the “same” Harish–Chandra parameter correspond. This naturality is expressed in the commutative diagram of Proposition 6.1. The correspondence of K–types on the space of joint harmonics also has nice properties; each K–type for S̃p(2n, ℝ) is harmonic for precisely one choice of p, q with given discriminant. Furthermore lowest K–types in the sense of Vogan are always of lowest degree in the sense of Howe [5].

These properties are special to the range in which the two groups are roughly the same size. Similar properties also hold for the dual pairs (O(p, q), Sp(2n, ℝ)) with p, q even and p+q = 2n, 2n+2 [11]. In fact our approach is quite close to that of [11], with the additional complications arising from the presence of non–trivial covering groups.

§1. Preliminaries

In this section we describe facts about the double covers, dual pairs and generalities about the metaplectic representation that we will need. The main reference for the double covers and the metaplectic representation are [10] and [15]. The setup is for any local field 𝔽, but we concentrate on the case 𝔽 = ℝ. We omit the details of many straightforward calculations.

For any positive integer m we equip W = ℝ^{2m} with the usual symplectic structure

given by J = ( \begin{matrix}0 & I_{m} \\ −I_{m} & 0 \end{matrix} ), and standard basis e_{1}, …, e_{m},f_{1}, …, f_{m}. Then Sp(2m, ℝ)

is the isometry group of this form, and the metaplectic cover S̃p(2m) is defined by the normalized cocycle c( , ) of [15] or [10]. Thus

S̃p(2m, ℝ) = Sp(2m, ℝ) × ℤ/2ℤ, (g, ϵ)(g′, ϵ′) = (gg′, ϵϵ′c(g, g′)).

If ψ is a non–trivial (unitary) additive character of ℝ, let ω(ψ) be the Harish–Chandra module of the oscillator representation of S̃p(2m, ℝ), ([15], section 4), ([10], part I). The character ψ may be written ψ_{a}(x) = e^{iax}; up to isomorphism ω(ψ_{a}) only depends on the image of a in ℝ^{∗}/ℝ^{∗^{2}}.

Let V be a real 2n + 1–dimensional vector space equipped with a non–degenerate symmetric bilinear form ( , ) of signature (p, q), and basis v_{1}, …, v_{p}, v′_{1}, … v′_{q} for which the matrix of ( , ) is diag(I_{p}, −I_{q}). We let O(p, q) denote the isometry group of ( , ). Now V ⊗ W has a symplectic structure with standard basis v_{1} ⊗ e_{1}, …, v′_{q} ⊗ f_{n}. The natural map α : O(p, q) × Sp(2m, ℝ) → Sp(2m(2n + 1), ℝ) makes (O(p, q), Sp(2n, ℝ)) into a dual pair. We write p_{0} = [\frac{p}{2}] and q_{0} = [\frac{q}{2}] for the ranks of O(p) and O(q).

The main result concerns the case m = n, but many secondary results hold with little or no restriction.

Let G̃L(m) be the two–fold cover of GL(m) defined by the cocyle

c(g, h) = (det(g), det(h))_{ℝ}

where (x, y)_{ℝ} is the Hilbert symbol [16] for ℝ. It is convenient to let Õ(p, q) be the two–fold cover of O(p, q) defined by c(g, h) = (det(g), det(h))_{ℝ}^{n} (p + q = 2n + 1). This is split over SO(p, q) and splits over O(p, q) if and only if n is even. For later use we let Õ(p, q)[k] be the cover defined by cocycle c(g, h)^{k}.

Now α lifts to a map α̃ : Õ(p, q) × S̃p(2n, ℝ) → S̃p(2n(2n + 1), ℝ). In particular for g ∈ O(p, q),

(1.1)(a)  α̃(g, ϵ) = (α(g), ϵ)

and for g ∈ Sp(2n, ℝ)

(1.1)(b)  α̃(g, ϵ) = (α(g), ϵ^{2n+1}Δ(g))

for a certain map Δ : Sp(2n, ℝ) → ±1.

The image of S̃p(2n, ℝ) in S̃p(2n(2n + 1), ℝ) is the inverse image of Sp(2n, ℝ), and the image of the center of S̃p(2n, ℝ) is the center of S̃p(2n(2n + 1), ℝ). Our choice of covering Õ(p, q) implies that the analogous statements hold for O(p, q). It also has the advantage that n odd and n even may be treated uniformly.

Given ψ, let

(1.2)(a)  χ(ψ)(x, ϵ) := γ(x, \frac{1}{2}ψ)ϵ,

where

(1.2)(b)  γ(a, ψ) := \frac{γ(aψ)}{γ(ψ)}

is the Weil index [15]. We have γ(ψ_{a}) = e^{\frac{2iπ}{8} sgna} and

(1.2)(c)  χ(ψ_{a})(x, ϵ) = sgn(a)e^{\frac{2iπ}{8} (sgnx−1)}ϵ.

This is a character of the ( , )_{ℝ} double cover of ℝ^{∗}, and so we can compose χ(ψ) with the determinant to get a character of G̃L(m). We denote the ensuing genuine character by the same symbol, χ(ψ)(g, ϵ) := χ(ψ)(det(g), ϵ).

This satisfies

(1.2)(d)  \overline{χ(ψ)}(x, ϵ) = sgn(x)χ(ψ)(x, ϵ),  χ(ψ)(x, ϵ)^{2} = sgn(x).

If V is an orthogonal space of dimension m and discriminant δ, we let (cf. [7], 2.5)

(1.2)(e)  χ(ψ, V)(g, ϵ) = γ(det(g), \frac{1}{2}ψ)^{−m}(δ, det(g))_{ℝ}ϵ^{m}.

In general, if G̃ is a double cover of G and π is a representation of G̃, we say π is of type k if π(ϵ) = ϵ^{k} for ϵ in the kernel of the covering. With this convention χ(ψ, V)(g, ϵ) is a character of G̃L(V) of type m = dim V.

If the signature of V is p, q with p + q odd, then

(1.2)(f)  χ(ψ, V) = χ(ψ)^{−p+q}.

This is the formula we will use most of the time. By (d) this may be thought of as sgn(det)^{\frac{−p+q}{2}} .

We fix a genuine character

(1.2)(g)  ξ(ψ)(g, ϵ) = \{ \begin{matrix}ϵ & n even \\ χ(ψ)(det(g), ϵ)^{−1} & n odd \end{matrix}

of Õ(p, q). The map π → π ⊗ ξ defines a bijection between the irreducible representations of O(p, q) and the irreducible genuine representations of Õ(p, q).

If ψ is fixed we drop it from the notation and write ω = ω(ψ), χ = χ(ψ), χ_{V} = χ(ψ, V) and ξ = ξ(ψ).

Pulling ω(ψ) back to Õ(p, q) × S̃p(2n, ℝ) via α̃ we obtain the representation correspondence for this dual pair [5]. This is a correspondence between certain irreducible Harish–Chandra modules.

By (1.1) the representations of Õ(p, q) and S̃p(2n, ℝ) in the image of the correspondence are genuine, i.e. of type 1. If π, π′ are genuine irreducible representations of S̃p(2n, ℝ) and Õ(p, q) respectively which correspond, we write

(1.3)  π′ = θ(ψ, p, q)(π), π = θ(ψ)(π′).

If ψ is fixed we write θ_{p,q} = θ(ψ, p, q) and θ = θ(ψ).

For an irreducible (admissible) representation π of a group G, we denote by π^{∗} its contragredient. In particular, for ξ as in (1.2)(g), we have

(1.4)(a)  ξ^{∗}(g, ϵ) = ξ ⊗ sgn(detg)^{n}.

Combined with the results on page 91 of [12], we find the following expressions for π^{∗}. In case G = S̃p, let τ = Adκ, where κ = diag(I_{m}, −I_{m}) (this is an outer automorphism of G). Then

(1.4)(b)  π^{∗} = \{ \begin{matrix}π ⊗ sgn^{n} & for G = Õ(V), \\ π ∘ τ, & for G = S̃p(W). \end{matrix}

From (1.2)(d) and the fact that ω(\overline{ψ}) ≃ ω(ψ)^{∗} we see

(1.4)(c)  θ(\overline{ψ}, p, q)(π) = θ(ψ, p, q)(π^{∗}) ⊗ sgn^{n}.

Let V′ denote the same space as V with form Q′ = −Q, of signature (q, p). Let δ be the tautological identification of O(V) and O(V′). Note that O(V′) ≅ O(q, p), and so we can identify representations of O(p, q) and O(q, p) by choosing such an isomorphism; however the θ–correspondences are different. The next lemma gives the relationship between θ_{p,q} and θ_{q,p}.

Lemma 1.5. For any irreducible representation π of S̃p(2n, ℝ),

θ(ψ, p, q)(π) = θ(ψ, q, p)(π^{∗}).

Proof. The map Id ⊗ κ is an isomorphism between V ⊗ W and V′ ⊗ W which interchanges Q · ⟨ , ⟩ and −Q · ⟨ , ⟩. Let Ψ be the ensuing isomorphism Sp(V ⊗ W) → Sp(V′ ⊗ W). The diagram

O(V) × Sp(W) ⟶^{α} Sp(V ⊗ W)
δ⊗τ↓  ↓^{Ψ}
O(V′) ⊗ Sp(W) ⟶^{α′} Sp(V′ ⊗ W)

is commutative and ω(ψ, V ⊗W) = ω(ψ, V′ ⊗W)∘Ψ. The lemma follows from the formula for π^{∗}. □

Note that (1.4)(c) and the Lemma give

(1.6)  θ(\overline{ψ}, p, q)(π) = θ(ψ, q, p)(π) ⊗ sgn^{n}.

Suppose W_{1} and W_{2} are symplectic spaces. Then W_{1} ⊕W_{2} inherits a natural symplectic structure and there is a canonical map S̃p(W_{1}) ⊗ S̃p(W_{2}) ⟶ S̃p(W_{1} ⊕ W_{2}). We will use this map in the special case S̃p(2m, ℝ) × S̃p(2m, ℝ) → S̃p(4m, ℝ). Similarly there is a canonical map Õ(p, q) × Õ(p′, q′) → Õ(p + p′, q + q′).

Lemma 1.7. Let ω_{n,p,q} be the oscillator representation of S̃p(2n(p + q), ℝ) restricted to the dual pair (S̃p(2n), Õ(p, q)).

(1) ω_{n,p,q} ⊗ω_{n,p,q} ≃ ω_{2n,p,q} ⊗sgn^{n} as representations of S̃p(2n, ℝ)×S̃p(2n, ℝ)×Õ(p, q) with Õ(p, q) acting diagonally on the left hand side.

(2) ω_{n,p,q} ⊗ ω_{n,p′,q′} ≃ ω_{n,p+p′,q+q′} as representations of S̃p(2n, ℝ) × Õ(p, q) × Õ(p′, q′), with S̃p(2n, ℝ) acting diagonally on the left hand side.

Proof. There are obvious isomorphisms between the polynomial Fock spaces ω_{n,p,q} in the statements. We need to check the equivariance. Assertion (1) for S̃p(2n, ℝ) × S̃p(2n, ℝ) and assertion (2) for Õ(p, q) × Õ(q, p) follow from the explicit descriptions of the actions in [15] on the smooth models.

If (X, Y) is a complete polarization of W then G̃L(X) acts on S(X) in the oscillator representation by

ω(ψ)(g, ϵ)(ϕ)(x) = |det(g)|^{− \frac{1}{2}} χ(ψ)(g, ϵ)^{−1}ϕ(g^{−1}x).

Since the action of Õ(p, q) is via a homomorphism to G̃L(X ⊗ V), it acts in ω_{n,p,q} by translation tensored with ξ. Assertion (1) for Õ(p, q) follows immediately, the twist if n is odd coming from χ(ψ)^{2} = sgn. The proof of assertion (2) is similar. □

The first part of the next Lemma is due to Rallis [14] and Przebinda [13]. The second is the result obtained by applying the same technique in the other direction. It says that the duality correspondence is a bijection when all O(p, q) with fixed discriminant (and p + q = 2n + 1) are considered at once. Thus we are reduced to proving occurence, and computing the correspondence explicitly.

Lemma 1.8.

(1) Suppose π is a representation of Õ(p, q), and θ(ψ)(π) ≠ 0. Then θ(ψ)(π⊗sgn) = 0.

(2) Let π be a genuine representation of S̃p(2m, ℝ), and suppose θ(ψ, p, q)(π) ≠ 0. Then θ(ψ)_{p′,q′}(π) = 0 for all (p′, q′) ≠ (p, q) with q′ ≡ q mod (2).

Proof. Suppose both π and π ⊗sgn are quotients of ω_{n,p,q} restricted to Õ(p, q). By Lemma 1.7(1) this implies π ⊗ π ⊗ sgn is a quotient of ω(ψ)_{2n,p,q} ⊗ sgn^{n}. Since π^{∗} ≃ π ⊗ sgn^{n}, and the trivial representation is a quotient of π ⊗ π^{∗}, this implies that sgn is a quotient of ω_{2n,p,q}. However this is impossible since (cf. Proposition 2.1) the sgn K–type of O(p, q) does not occur in the space of joint harmonics when paired with Sp(4n, ℝ). This proves (1).

Now suppose π is a quotient of both ω_{n,p,q} and ω_{n,p′,q′} restricted to S̃p(2m, ℝ). By Lemmas 1.5(1) and 1.7(2) this implies that π ⊗ π^{∗} is a quotient of ω_{n,p+q′,q+p′}. As in the proof of (1) this implies that the trivial representation is a quotient of ω_{n,p+q′,q+p′}. This can also be ruled out by K-types. By [6], cf. ([11],I.4), the trivial K–type for S̃p(2m, ℝ) occurs in this space only if O(p + q′, q + p′) is quasi–split, i.e. p + q′ − q − p′ = 0, ±1, ±2.

This together with p + q = q + p gives q = q′ ± 0, 1, and since the discriminants are equal q = q′. Therefore p = p′, q = q′, proving (2). □

Groups will be denoted G, K, T …, their Lie algebras by 𝔤_{0}, 𝔨_{0}, 𝔱_{0} …, and their complexified Lie algebras by 𝔤, 𝔨, 𝔱 . . .. For G reductive, a Cartan involution will be denoted θ with fixed points K, 𝔨_{0} and 𝔨 respectively in G, 𝔤_{0} and 𝔤, and 𝔤 = 𝔨 ⊕ 𝔭 as usual. For 𝔥 a Cartan subalgebra of 𝔤 (always θ–stable) we will denote a system of roots by Δ = Δ(𝔥, 𝔤) with positive system Δ^{+}, and ρ = ρ(Δ^{+}) = \frac{1}{2} ∑_{α} α. This notation will be extended in various standard ways, for example ρ_{c} denotes one–half the sum of the positive compact imaginary roots, and ρ(𝔲) one–half the sum of the roots of a (nilpotent) subalgebra 𝔲. Unless otherwise stated G will denote Sp(2n, ℝ), G̃ will denote S̃p(2m, ℝ). These groups have maximal compact subgroups K and K̃ as chosen in §2. Similarly, G′ will denote O(p, q) with corresponding G̃′, K′ and K̃′.

We now describe the semisimple orbits and Cartan subgroups for Sp(2n, ℝ). We begin by choosing representatives for the conjugacy classes of Cartan subgroups as in [2].

For non-negative integers m, r, s with 2m + r + s = n we define a Cartan subgroup H_{Sp}^{m,r,s} of Sp(2n, ℝ) with Lie algebra 𝔥_{Sp0}^{m,r,s}. Write W = ℝ^{2n} = W_{1} ⊕ W_{2} ⊕ W_{3} where W_{1} is spanned by {e_{i}, f_{i}|1 ≤ i ≤ 2m}, W_{2} by {e_{i}, f_{i}|2m + 1 ≤ i < 2m + r} and W_{3} by {e_{i}, f_{j}|2m+r +1 ≤ i ≤ n}. We identify Sp(W_{i}) and 𝔰𝔭(W_{i}) with their images in Sp(2n, ℝ) and 𝔰𝔭(2n, ℝ). For z_{i} = x_{i} + iy_{i} ∈ ℂ, 1 ≤ i ≤ m let

(1.9)(a)  𝔥_{Sp}^{m,0,0}(z_{1}, …, z_{m}) = ( \begin{matrix}& X & Y \\ X & & & −Y \\ −Y & & & −X \\ & Y & −X \end{matrix} ) ∈ 𝔰𝔭(W_{1})

where X = diag(x_{1}, …, x_{m}) and Y = diag(y_{1}, …, y_{m}). For θ_{i} ∈ ℝ (1 ≤ i ≤ r) we let

(1.9)(b)  𝔥_{Sp}^{0,r,0}(θ_{1}, …, θ_{r}) = ( \begin{matrix}& X \\ −X \end{matrix} ) ∈ 𝔰𝔭(W_{2})

with X = diag(θ_{1}, …, θ_{r}), and for x_{i} ∈ ℝ (1 ≤ i ≤ s) let

(1.9)(c)  𝔥_{Sp}^{0,0,s}(x_{1}, …, x_{s}) = diag(x_{1}, …, x_{s}, −x_{1}, …, −x_{s}) ∈ 𝔰𝔭(W_{3}).

Taking the sum of these elements gives us an element

(1.9)(d)  𝔥_{Sp}^{m,r,s}(z_{1}, …, z_{m}, θ_{1}, …, θ_{r}, x_{1}, …, x_{s}) ∈ 𝔰𝔭(2n, ℝ)

and this defines the Cartan subalgebra 𝔥_{Sp0}^{m,r,s} of 𝔰𝔭(2n, ℝ), with complexification 𝔥_{Sp}^{m,r,s}. The compact Cartan subalgebra is 𝔱_{0} = 𝔥_{0}^{0,n,0}. Let

(1.9)(e)  H_{Sp}^{m,r,s} ≃ (ℂ^{∗})^{m} × (S^{1})^{r} × (ℝ^{∗})^{s}

be the Cartan subgroup of Sp(2n, ℝ) with Lie algebra 𝔥_{Sp0}^{m,r,s}. These are representatives for the conjugacy classes of Cartan subgroups of Sp(2n, ℝ). The compact Cartan subgroup is T = H_{Sp}^{0,n,0}. Write the elements of H_{Sp}^{m,r,s} accordingly as

(1.9)(f)  H_{Sp}^{m,r,s}(z_{1}, …, z_{m}, u_{1}, …, u_{r}, x_{1}, …, x_{2})

(z_{i} ∈ ℂ^{∗}, u_{i} ∈ S^{1}, x_{i} ∈ ℝ^{∗}).

The Weyl group of 𝔥_{Sp0}^{m,r,s} in 𝔰𝔭(2n, ℝ) is generated by all permuations of {z_{i}}, z_{i} → \overline{z}_{i}, −z_{i}, all permutations of \{θ_{i}\}, and all permuations and sign changes of \{x_{i}\}. This describes the semisimple orbits. Note that two semisimple elements 𝔥_{Sp}^{m,r,s}(z_{1}, …) and 𝔥_{Sp}^{m,r,s}(z′_{1}, …) are in the same orbit if and only if they have the same eigenvalues, and θ_{1}, …, θ_{r} and θ′_{1}, …, θ′_{r} are the same up to permutation.

We write H = T A with 𝔥 = 𝔱⊕𝔞, T = H ∩K and A = exp(𝔞_{0}) as usual. The centralizer of A is

(1.10)(a)  M ≅ GL(1, ℝ)^{s} × GL(2, ℝ)^{m} × Sp(2r, ℝ).

Let

(1.10)(b)  \overline{M} = G̃L(1, ℝ)^{s} × G̃L(2, ℝ)^{m} × S̃p(2r, ℝ)

with double covers of GL and Sp as at the beginning of this section. There is a natural surjection \overline{M} ↠ M̃.

Let H̃ (respectively \overline{H}) be the inverse image of H in M̃ (resp. \overline{M}). Then H̃, \overline{H} are Cartan subgroups of M̃, \overline{M}. Furthermore

(1.10)(c)  \overline{H} ≃ (ℝ̃^{∗})^{s} × (S̃^{1})^{r} × (ℂ^{∗} × ℤ/2ℤ)^{m}

where ℝ̃^{∗} is the two–fold cover of ℝ^{∗} defined by the Hilbert symbol, and S̃^{1} is the connected two–fold cover of S^{1} given by z → z^{2}, |z| = 1.

We now turn to a description of the Cartan subgroups and semisimple orbits for O(p, q). We follow [1].

Suppose 2m + s ≤ min(p, q). Write V = V_{1} ⊕ V_{2} ⊕ V_{3} where V_{1} = span {v_{i}, v′_{j} | 1 ≤ i, j ≤ 2m}, V_{2} = span {v_{i}, v′_{j} | 2m + 1 ≤ i, j ≤ 2m + s} and V_{3} = span {v_{i}, v′_{j} | 2m + s < i ≤ p, 2m + s < j ≤ q}. Then SO(V_{i}) is embedded naturally in SO(V) and we identify SO(V_{i}) and 𝔰𝔬(V_{i}) with their images in SO(V) and 𝔰𝔬(V). For w_{j} = x_{j} + iy_{j} ∈ ℂ let

(1.11)(a)  𝔥^{m,0,0}(w_{1}, …, w_{m}) = ( \begin{matrix}& Y & & X \\ −Y & & X \\ & X & & −Y \\ X & & Y \end{matrix} ) ∈ 𝔰𝔬(V_{1})

where X = diag(x_{1}, …, x_{m}), Y = diag(y_{1}, …, y_{m}). For c_{j} ∈ ℝ, let

(1.11)(b)  𝔥^{0,0,s}(c_{1}, …, c_{s}) = ( \begin{matrix}& X \\ X \end{matrix} ) ∈ 𝔰𝔬(V_{2})

where X = diag(c_{1}, …, c_{s}). Finally let r_{1} = [\frac{p−2m−s}{2}], r_{2} = [\frac{q−2m−s}{2}], and for θ_{i}, ϕ_{j} ∈ ℝ let

(1.11)(c) 𝔥^{0,r_{1}+r_{2},0}(θ_{1}, …, θ_{r_{1}}, ϕ_{1}, …, ϕ_{r_{2}}) = diag(θ̂_{1}, …, θ̂_{r_{1}}, ϕ̂_{1}, …, ϕ̂_{r_{2}}) ∈ 𝔰𝔬(V_{3})

with θ̂ = ( \begin{matrix}0 & θ \\ −θ & 0 \end{matrix} ).

Taking the sum of these elements gives us an element

(1.11)(d)  X = 𝔥_{p,q}^{m,r,s}(w_{1}, …, w_{m}, θ_{1}, …, θ_{r_{1}}, ϕ_{1}, …, ϕ_{r_{2}}, c_{1}, …, c_{s}) ∈ 𝔰𝔬(p, q)

and this defines a Cartan subalgebra 𝔥_{p,q0}^{m,r,s}, with complexification 𝔥_{p,q}^{m,r,s}. Let

(1.11)(e)  H_{p,q}^{m,r,s} ≃ (ℂ^{∗})^{m} × (S^{1})^{r} × (ℝ^{∗})^{s}

be the Cartan subgroup of SO(p, q) with Lie algebra 𝔥_{p,q0}^{m,r,s}. This gives a set of representatives of the conjugacy classes of Cartan subgroups of SO(p, q). The compact Cartan subgroup T is H_{p,q}^{0,n,0}. According to the decomposition (1.11)(e), we write elements of H_{p,q}^{m,r,s} as

(1.11)(f)  H_{p,q}^{m,r,s}(z_{1}, …, z_{m}, u_{1}, …, u_{r_{1}}, v_{1}, …, v_{r_{2}}, x_{1}, …, x_{s})

with z_{i} ∈ ℂ^{∗}, u_{i}, v_{i} ∈ S^{1}, x_{i} ∈ ℝ^{∗}.

The Weyl group of 𝔥_{p,q0}^{m,r,s} in 𝔬(p, q) is similar to the case of Sp. The only change is that on u_{1}, …, u_{r_{1}}, v_{1}, …, v_{r_{2}} it is of type B_{r_{1}} × B_{r_{2}} acting by permutation and sign changes on {u_{i}} and {v_{i}} separately.

The corresponding Cartan subgroup of O(p, q) is isomorphic to H_{p,q}^{m,r,s} × Z where Z is the center of O(p, q).

The centralizer of A is

(1.12)(a)  M′ ≅ GL(1, ℝ)^{s} × GL(2, ℝ)^{m} × O(p′, q′)

with p′ = p − s − 2m, q′ = q − s − 2m. The inverse image of M′ in Õ(p, q) is

(1.12)(b)  M̃′ ≅ GL(1, ℝ)^{s} × GL(2, ℝ)^{m} × Õ(p′, q′)[n].

It follows from the preceding discussion that there is a bijection (depending on the additive character ψ) between the regular semisimple adjoint orbits of Sp(2n, ℝ) and the union of the regular semisimple adjoint orbits of SO(p, q) with δ = (−1)^{q} fixed. This is explained in more detail in [1], where it is described geometrically in terms of the orbit correspondence; here we resort to a simple explicit description.

Fix ψ = ψ_{a} with a > 0. Let

(1.13)(a)  X = 𝔥_{Sp}^{m,r,s}(z_{1}, …, z_{m}, u_{1}, …, u_{r_{1}}, v_{1}, …, v_{r_{2}}, x_{1}, …, x_{s})

be a regular semisimple element, with u_{1} > ⋯ > u_{r_{1}} > 0 > v_{1} > ⋯ > v_{r_{2}}. Let p = 2m + r_{1} + s, q = 2m + r_{2} + s + 1 or p = 2m + r_{1} + s + 1, q = 2m + r_{2} + s, depending on δ. Then the orbit of X ∈ 𝔰𝔭(2n, ℝ) corresponds to the orbit of X′ ∈ 𝔰𝔬(p, q), where

(1.13)(b)  X′ = 𝔥_{p,q}^{m,r,s}(z_{1}, …, z_{m}, u_{1}, …, u_{r_{1}}, −v_{r_{2}}, …, −v_{1}, x_{1}, …, x_{s}).

If ψ = ψ_{a} with a < 0, then the same result holds, with u_{1}, …, u_{r_{1}}, −v_{r_{2}}, …, −v_{1} replaced by v_{1}, …, v_{r_{2}}, −u_{r_{1}}, …, −u_{1}.

By the preceding description of the semisimple orbits the following result is immediate. Let 𝔰𝔭(2n, ℝ)_{ss} be the regular semisimple orbits of 𝔰𝔭(2n, ℝ), and 𝔰𝔬(p, q)_{ss} similarly.

Lemma 1.14. Fix ψ. There is a bijection between

𝔰𝔭(2n, ℝ)_{ss}

and

⋃_{\begin{matrix}p+q=2n+1 \\ (−1)^{q}=δ \end{matrix}} 𝔬(p, q)_{ss}

We refer to this as the orbit correspondence.

We write X′ = O(ψ)(X) if the orbits of X and X′ correspond as in Lemma 1.14. Dualizing, we obtain a correspondence λ ↔ λ′ = O(ψ)(λ) of regular semisimple elements of the duals. Finally if X′ = O(ψ)(X), let 𝔥, 𝔥′ be the Cartan subalgebras centralizing X, X′ respectively. The correspondence gives rise naturally to a correspondence of systems of positive roots, which we write Δ^{+} ↔ Δ^{′+} = O(ψ)(Δ^{+}). As usual we drop ψ from the notation if it is fixed.

It is evident that every Cartan subgroup of SO(p, q) is isomorphic to a Cartan subgroup of Sp(2n, ℝ). This correspondence preserves conjugacy classes, and is a bijection on conjugacy classes if SO(p, q) is quasisplit. We use the correspondence of semisimple orbits to choose these isomorphisms as follows.

Fix δ = ±1 and ψ. Let (𝔥_{0}, Δ^{+}) be a pair consisting of a Cartan subalgebra of 𝔰𝔭(2n, ℝ) and a system of positive roots. Let ϕ : 𝔥_{0} → 𝔥′_{0} ⊂ 𝔰𝔬(p, q) be an isomorphism. By abuse of notation we write ϕ(Δ^{+}) for the natural system of positive roots of 𝔥′_{0}. More precisely, fix X ∈ 𝔥 so that Δ = {α | α(X) > 0}. Then ϕ(Δ^{+}) = {α′ | α′(ϕ(X)) > 0}.

Proposition 1.15. Given (𝔥_{0}, Δ^{+}), there exist p, q, and a pair (𝔥′_{0}, Δ^{′+}) such that Δ^{+′} = O(Δ^{+}) and 𝔥′_{0} is isomorphic to 𝔥_{0}. This determines p, q uniquely (subject to (−1)^{q} = δ). Furthermore the isomorphism ϕ : 𝔥_{0} → 𝔥′_{0} may be chosen so that ϕ(Δ^{+}) = Δ^{′+}. This determines ϕ up to conjugation by Sp(2n, ℝ) and O(p, q).

Furthermore ϕ lifts to an isomorphism ϕ : H → H′ ∩ SO(p, q). Write H = T A, M = Cent_{G}(A) as usual, and similarly for H′. Then ϕ extends to an isomorphism of the GL factors of M and M′ (cf. (1.10)(a) and (1.12)(a)).

§2. Maximal Compact Subgroups and Joint Harmonics

We first consider G = Sp(2n, ℝ), G̃ = S̃p(2n, ℝ). Recall W and J as in §1. Then

G := {g ∈ GL(W) | ^{t}gJg = J}.

We choose the maximal compact subgroup K of G to be

K := {g ∈ G | gJ = Jg }.

Since J^{2} = −Id, it defines a complex structure on W. Let W_{ℂ} ≃ ℂ^{n} denote the resulting complex space. Then W_{ℂ} admits a positive definite symmetric Hermitian form (v, w) =< Jv, w > +i < v, w > . This gives an isomorphism of K with the isometry group U(𝕎_{ℂ}, ( , )). We define the determinant character of K to be the pullback of the determinant character of this unitary group by the explicit isomorphism chosen.

The inverse image K̃ of K in G̃ is connected, and its representations may be studied by passing to the Lie algebra. To be explicit, K̃ is isomorphic to the det^{\frac{1}{2}} cover of K, i.e. to \overline{K} = \{(g, z)|g ∈ U(n), z ∈ ℂ^{∗}, det(g) = z^{2}\}. The character τ : (g, z) → z of \overline{K} satisfies τ^{2}(g) = det(g) and is denoted det^{1/2}. We choose an isomorphism, unique up to conjugation, of K̃ with \overline{K} so the character of K̃ acting on the the unique K̃-fixed line in ω(ψ) goes to det^{\frac{1}{2}} . (This line is spanned by the Gaussian in the Schroedinger model, or the constants in the Fock model.) We fix the Cartan subgroup T of K as in (1.9), with

the usual positive system Δ(𝔱, 𝔨). Then ˆ K̃_{genuine} is parametrized by certain dominant weights λ ∈ i𝔱_{0}^{∗}; in the usual coordinates λ = (a_{1}, …, a_{n}) with a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{n} and a_{i} ∈ ℤ + \frac{1}{2}. The distinguished character det^{1/2} corresponds to the weight (\frac{1}{2}, …, \frac{1}{2}). This parametrization depends on ψ.

Now consider G′ = O(p, q), G̃′ = Õ(p, q), with maximal compact subgroups K′ = O(p)× O(q), and K̃′. We fix the Cartan subgroup T of K′ as in (1.11), with the usual positive system. We identify an irreducible representations of O(p) with its “highest weight” λ = (λ_{0}; ϵ). Here λ_{0} = (a_{1}, …, a_{p_{0}}) ∈ i𝔱_{0}^{∗} is the usual highest weight of a finite dimensional representation of SO(p). We are following [23], where ϵ = 1 (resp. ϵ = −1) corresponds to the length of the first column less than (resp. greater than) or equal to p_{0}. If p is odd, then −Id acts by (−1)^{∑ a_{i}+∑ b_{i}}ϵ in this representation. If p is even and a_{p_{0}} ≠ 0 then ϵ = ±1 give the same representation; in all other cases they are distinct. Furthermore (0; −1) is the one–dimensional representation sgn(g) = sgn(det(g)) = det(g), and (a_{1}, …, a_{p_{0}}; ϵ) ⊗ sgn = (a_{1}, …, a_{p_{0}}; −ϵ).

A similar discussion holds for O(q), and the irreducible finite dimensional representations of K′ are parametrized by (a_{1}, …, a_{p_{0}}; ϵ) ⊗ (b_{1}, …, b_{q_{0}}; η). The irreducible genuine representations of K̃′ are also parametrized in the same way, by tensoring with the genuine character ξ of K̃′ as in (1.2)(g). (Here ξ is the character of Õ(p, q) given by (1.2)(g), restricted to K̃′.)

The action of K̃ × K̃′ on the space of joint harmonics gives a bijection between certain irreducible representations of K̃ and K̃′ [5]. If a K̃–type μ corresponds to a K̃′–type μ′, we write μ′ = H(ψ, p, q)(μ) and μ = H(ψ)(μ′). As usual we drop ψ from the notation if it has been fixed.

The next result follows from [6], as in ([11], I.4) and ([3], Proposition 1.4).

Proposition 2.1.

(1) The correspondence on the space of joint harmonics is as follows.

μ′ = (a_{1}, …, a_{p_{0}}; 1) ⊗ (b_{1}, …, b_{q_{0}}; 1) →

H(ψ)(μ′) = (a_{1}, …, a_{p_{0}}, −b_{q_{0}}, …, −b_{1}) + (\frac{p − q}{2} , …, \frac{p − q}{2} )

μ′ = (a_{1}, …, a_{k}, 0, …, 0; −1) ⊗ (b_{1}, …, b_{ℓ}, 0, …, 0; 1) →

H(ψ)(μ′) = (a_{1}, …, a_{k}, \overbrace{1, …, 1}^{p−2k}, 0, …, 0, −b_{ℓ}, …, −b_{1}) + (\frac{p − q}{2} , …, \frac{p − q}{2} )

with p − k + ℓ ≤ n,

μ′ = (a_{1}, …, a_{k}, 0, …, 0; 1) ⊗ (b_{1}, …, b_{ℓ}, 0, …, 0; −1) →

H(ψ)(μ′) = (a_{1}, …, a_{k}, 0, …, 0, \overbrace{−1, …, −1}^{q−2ℓ}, −b_{ℓ}, …, −b_{1}) + (\frac{p − q}{2} , …, \frac{p − q}{2} )

with q + k − ℓ ≤ n.

(2) The p, q–degree of μ = (a_{1}, …, a_{n}) is ∑_{i=1}^{n} |a_{i} − \frac{p−q}{2} |. The degree of

μ′ = (a_{1}, …, a_{k}, 0, …, 0; ϵ) ⊗ (b_{1}, …, b_{ℓ}, 0, …, 0; η)

is equal to ∑_{i} a_{i} + \frac{1−ϵ}{2}(p − 2k) + ∑ b_{i} + \frac{1−η}{2} (q − 2ℓ).

Note that the dependence on ψ is via the dependence of the parametrization of K̃′-types on ψ.

The images of H(ψ, p, q) and H(ψ) are described by the next Proposition.

Proposition 2.2.

(1) Let μ be any (genuine) K̃–type for S̃p(2n). Consider the groups O(p, q) with fixed discriminant. Then there is a unique choice of p and q such that μ is p, q–harmonic.

(2) Let μ′ = (a_{1}, …, a_{k}, 0, ..., 0; ϵ) ⊗ (b_{1}, …, b_{ℓ}, 0, …, 0; η) be a (genuine) K̃′–type for Õ(p, q). Then μ′ is in the space of joint harmonics if and only if k + \frac{1−ϵ}{2}(p − 2k) + ℓ + \frac{1−η}{2} (q − 2ℓ) ≤ n. This holds for precisely one of μ′ and μ′ ⊗ sgn.

Proof. Part (2) is an immediate consequence of Proposition 2.1, and we omit the details. For part (1), we claim that we may write μ uniquely in the form

(2.3)(a)  (a_{1}, …, a_{r}, μ_{0}, b_{1}, …, b_{s})

where

(2.3)(b)  μ_{0} = (r − s, …, r − s) + (\overbrace{\frac{1}{2}, …, \frac{1}{2}}^{x}, \overbrace{−\frac{1}{2}, …, −\frac{1}{2}}^{y})

with a_{r} ≥ r − s + \frac{1}{2}, r − s − \frac{1}{2} ≥ b_{1}, and at least one of these inequalities is strict. The algorithm in chapter 6 of [18] (see §6 for more detailed calculations) attaches to μ an element λ^{G}(μ) ∈ 𝔱_{c}^{∗}. It is of the form

(2.4)  λ^{G}(μ) = (α_{1}, …, α_{r}, \overbrace{0, …, 0}^{t}, β_{1}, …, β_{s})

with α_{1} ≥ ⋯ ≥ α_{r} > 0 > β_{1} ≥ ⋯ ≥ β_{s}. Thus r, s if they exist, are uniquely determined. The same holds for x, y from the particular form of μ. Running the algorithm in reverse on the λ’s as in (2.4), we see that every μ must be of the form (2.3)(a) with some choice of (x, y) (essentially chapter 6 in [18]).

Given this form, μ corresponds to μ′ in the space of joint harmonics for O(2r + t + 1, 2s + t), with

μ′ = (a_{1} − r + s − \frac{1}{2}, …, a_{r} − r + s − \frac{1}{2}, 0, …, 0; 1)⊗

(−b_{s} + r − s + \frac{1}{2}, …, −b_{1} + r − s + \frac{1}{2}, \overbrace{1, …, 1}^{j}, 0, …, 0; ϵ),

where

(j, ϵ) = \{ \begin{matrix}(y, +1) & 0 ≤ y ≤ [\frac{t}{2}] \\ (x, −1) & [\frac{t+1}{2}] ≤ y ≤ s (⇔ 0 ≤ x ≤ [\frac{t}{2}]). \end{matrix}

Similarly μ corresponds to μ′ in the space of joint harmonics for O(2r + t, 2s + t + 1), with

μ′ = (a_{1} − r + s + \frac{1}{2}, …, a_{r} − r + s + \frac{1}{2}, \overbrace{1, …, 1}^{j}, 0, …, 0; ϵ)⊗

(−b_{s} + r − s − \frac{1}{2}, …, −b_{1} + r − s − \frac{1}{2}, 0, …, 0; +1),

where

(j, ϵ) = \{ \begin{matrix}(x, +1) & 0 ≤ x ≤ [\frac{t}{2}] \\ (y, −1) & [\frac{t+1}{2}] ≤ x ≤ s (⇔ 0 ≤ y ≤ [\frac{t}{2}]). \end{matrix}

Note that for t even and x = y = [\frac{t}{2}] the two cases agree, since for even orthogonal groups (a_{1}, …, 1; +1) ≅ (a_{1}, …, 1; −1).

It remains to show μ is p, q–harmonic for at most one choice of q with (−1)^{q} = δ. Given δ, to determine (p, q) it is enough to find p_{0}. Assume δ = 1, the other case is similar. Suppose μ is p, q–harmonic corresponding to a μ′ as in Proposition 2.1. Then in the expression μ = (μ_{1}, …, μ_{n}), we must have μ_{p_{0}} = \frac{p−q}{2} if ϵ = 1, or μ_{p_{0}} = \frac{p−q}{2} + 1 if ϵ = −1. This is the same as μ_{p_{0}} + (n + 1 − 2p_{0}) = \frac{3}{2} or \frac{5}{2}. But the sequence r_{i} = μ_{i} + (n + 1 − 2i) decreases monotonically by at least 2 each consecutive term, so there is at most one i such that r_{i} = 5/2 or 3/2, never both. If all r_{i} > \frac{5}{2} then p_{0} = n, if all r_{i} < \frac{1}{2}, then p_{0} = 0. This proves (1). □

Remark. This Proposition also follows naturally from the calculations in the proof of Proposition 6.1. Namely if μ and μ′ correspond, then λ^{G}(μ) and λ^{G}(μ′) correspond in a simple fashion, implying in addition that r, s, t given by formula (2.4) for μ, and the (r′, s′, t′) coming from formula (6.6) coincide. Thus p = 2r + t + 1 or p = 2r + t according to the parity of δ.

Example 2.4. The example of small weights ([18], Definition 5.3.24) is important. These are weights of the form

μ = (\overbrace{\frac{1}{2}, …, \frac{1}{2}}^{x}, \overbrace{−\frac{1}{2}, …, −\frac{1}{2}}^{y}).

Then μ is small, i.e. μ_{0} = μ and it corresponds to a μ′ in the split group O(n + 1, n) as well as O(n, n + 1). Specifically μ corresponds to μ′ for O(n + 1, n) with

μ′ = (0, …, 0; 1) ⊗ (\overbrace{1, …, 1}^{r}, 0, …, 0; ϵ)

with

(r, ϵ) = \{ \begin{matrix}(y, 1) & 0 ≤ y ≤ [\frac{n}{2}] \\ (x, −1) & [\frac{n+1}{2}] ≤ y ≤ n. \end{matrix}

On the other hand it goes to μ′ for O(n, n + 1) with

μ′ = (\overbrace{1, …, 1}^{r}, 0, …, 0; ϵ) ⊗ (0, …, 0; 1)

with

(r, ϵ) = \{ \begin{matrix}(x, 1) & 0 ≤ x ≤ [\frac{n}{2}] \\ (y, −1) & [\frac{n+1}{2}] ≤ x ≤ n. \end{matrix}

§3. Discrete Series

A genuine discrete series representation π of S̃p(2n) is determined by its Harish–Chandra parameter λ. In coordinates we write

λ = (a_{1}, …, a_{k}, b_{1}, …, b_{ℓ}) ∈ i𝔱_{0}^{∗}, (𝔱 as in 1.9)

with a_{1} > ⋯ > a_{k} > 0 > b_{1} > ⋯ > b_{ℓ}, a_{i}, b_{j} ∈ ℤ + \frac{1}{2} and a_{i} + b_{j} ≠ 0 for all i, j. Then π has lowest K–type λ + ρ(Δ^{+}) − 2ρ_{c}(Δ^{+}) where Δ^{+} is the system for which λ is dominant.

The genuine limits of discrete series for S̃p(2n, ℝ) are obtained by allowing λ to be singular with respect to a set of simple noncompact roots. Explicitly these representations are parametrized by pairs (Δ^{+}, λ) where λ is dominant with respect to the roots in Δ^{+}. In coordinates it is of the form

(\overbrace{a_{1} …, a_{1}}^{m_{1}}, \overbrace{a_{2} …, a_{2}}^{m_{2}}, …, \overbrace{a_{r} …, a_{r}}^{m_{r}}, \overbrace{−a_{r} …, −a_{r}}^{n_{r}}, …, \overbrace{−a_{2} …, −a_{2}}^{n_{2}}, \overbrace{−a_{1} …, −a_{1}}^{n_{1}})

with a_{i} ∈ ℤ + \frac{1}{2}, a_{1} > ⋯ > a_{r} > 0, and |m_{i} − n_{i}| ≤ 1 for all i. The lowest K–type of π has the same form as for discrete series, λ + ρ(Δ^{+}) − 2ρ_{c}(Δ^{+}).

Similarly a discrete series representation of SO(p, q) is given by its Harish–Chandra parameter λ = (a_{1}, …, a_{p_{0}}, b_{1}, …, b_{q_{0}}) with a_{i}, b_{j} ∈ ℤ + \frac{1}{2} satisfying a_{1} > ⋯ > a_{p_{0}} > 0, b_{1} > ⋯ > b_{q_{0}} > 0, a_{i} ≠ b_{j} ∀i, j. Assume for the moment that p is odd and q is even. The lowest K′–type μ = λ + ρ(Δ^{+}) − 2ρ_{c}(Δ^{+}) is of the form

μ = (x_{1}, …, x_{k}, 0, …, 0; ±1) ⊗ (y_{1}, …, y_{q_{0}}; ±1) x_{k}, y_{j} ∈ ℤ > 0.

The second ±1 has no effect since y_{q_{0}} > 0. The two representations given by the first ±1 have the same restriction to S(O(p) × O(q)) since sgn ⊗ sgn of O(p) × O(q) is trivial on this subgroup. Passing to O(p, q) we obtain the following Lemma.

Lemma 3.1. The discrete series representations π of O(p, q) are parametrized by

λ = (λ_{0}; ϵ) = (a_{1}, …, a_{p_{1}}, b_{1}, …, b_{q_{0}}; ϵ)

with a_{1} > ⋯ > a_{p_{0}} > 0; b_{1} > ⋯ > b_{q_{0}} > 0; a_{i}, b_{j} ∈ ℤ + \frac{1}{2}, and a_{i} − b_{j} ≠ 0 for all i, j. Here π is determined by its Harish–Chandra parameter λ_{0} and its lowest K′–type μ which is of the form

(3.2)  μ = \{ \begin{matrix}(x_{1}, …, x_{p_{0}}, 1) ⊗ (y_{1}, …, y_{ℓ}, 0, …, 0; ϵ) (x_{p_{0}} > 0) & p even, q odd \\ (x_{1}, …, x_{k}, 0, …, 0; ϵ) ⊗ (y_{1}, …, y_{q_{0}}; +1) (y_{q_{0}} > 0) & p odd, q even. \end{matrix}

The genuine discrete series of Õ(p, q) are obtained by tensoring the discrete series of O(p, q) with ξ, so we use the same parameters.

We will refer to λ = (λ_{0}; ϵ) as a Harish–Chandra parameter for O(p, q) or Õ(p, q). The limits of discrete series are parametrized as for S̃p(2n, ℝ), by pairs (Δ^{+}, λ) where λ_{0} is Δ^{+}–dominant and λ_{0} of the form

(\overbrace{a_{1} …, a_{1}}^{m_{1}}, \overbrace{a_{2} …, a_{2}}^{m_{2}}, …, \overbrace{a_{r} …, a_{r}}^{m_{r}}, \overbrace{a_{1} …, a_{1}}^{n_{1}}, \overbrace{a_{2} …, a_{2}}^{n_{2}}, …, \overbrace{a_{r} …, a_{r}}^{n_{r}})

with a_{i} ∈ ℤ + \frac{1}{2}, a_{1} > ⋯ > a_{r} > 0, and |m_{i} − n_{i}| ≤ 1 for all i. Again μ is of the form (3.2).

Theorem 3.3. Fix ψ and δ = ±1. (1) Let π be a genuine discrete series representation of S̃p(2n) with Harish–Chandra parameter λ. Choose p, q so that λ occurs in the orbit correspondence for the dual pair (Sp(2n, ℝ), O(p, q)). Recall (Lemma 1.14) p, q is uniquely determined, subject to (−1)^{q} = δ.

Let λ′ = O(λ) be a corresponding element of 𝔰𝔬(p, q)^{∗}. Then π occurs in the representation correspondence with Õ(p, q), and θ_{p,q}(π) is the discrete series representation with Harish Chandra parameter (λ′; +1). Furthermore π does not occur in the correspondence for any other Õ(r, s) (with (−1)^{s} = δ).

If μ is the lowest K̃–type of π, then μ is of lowest p, q–degree, and H(μ) is the lowest K̃′–type of π′.

Conversely every discrete series of Õ(p, q) with Harish-Chandra parameter (∗; +1) corresponds to a discrete series of S̃p(2n), and those of the form (∗; −1) do not occur in the correspondence.

(2) The same results as in (a) holds for limits of discrete series, where if π is given by data (Δ^{+}, λ) then θ_{p,q}(π) is given by (O(Δ^{+}), (O(λ); 1)).

Explicitly (cf. 1.13) let ψ = ψ_{a} with a > 0, and suppose λ = (a_{1}, …, a_{p_{0}}, b_{1}, …, b_{q_{0}}) with a_{1} > ⋯ > a_{p_{0}} > 0 > b_{1} > ⋯ > b_{q_{0}}. Then p = 2p_{0} +1, q = 2q_{0} or p = 2p_{0}, q = 2q_{0} +1 and

λ′ = ((a_{1}, …, a_{p_{0}}, −b_{q_{0}}, …, −b_{1}); +1).

Theorem 3.3 will be proved in §9.

Note: The minimal K–type of a discrete series representation or a limit of discrete series is unique, and such a representation is determined by its minimal K–type. This follows from [6] or in our case from the results of §6.

§4. Standard Modules

We use the version of the Langlands classification of [21], which is valid for disconnected and non–linear groups of Harish–Chandra’s class. Throughout this section G will denote O(p, q) (p = 2n+1) or Sp(2n, ℝ), with maximal compact subgroup K and covering groups G̃ and K̃.

We first consider G = O(p, q). Let H = T A be a θ–stable Cartan subgroup of G. Recall from §1 that H is isomorphic to

(4.1)(a)  (ℂ^{∗})^{m} × (S^{1})^{r} × (ℝ^{∗})^{s} × Z

with 2m + s ≤ min(p, q), Z the center of O(p, q). The centralizer M of A is isomorphic to

(4.1)(b)  GL(1, ℝ)^{s} × GL(2, ℝ)^{m} × O(p′, q′), (p′ = p − s − 2m, q′ = q − s − 2m)

and the inverse image M̃ of M in Õ(p, q) is isomorphic to

(4.1)(c)  GL(1, ℝ)^{s} × GL(2, ℝ)^{m} × Õ(p′, q′)[n].

For inducing data on M̃ we take an irreducible representation σ = β ⊗ γ ⊗ τ of M̃. Here β = β_{1} ⊗ ⋯ ⊗ β_{s} is a product of characters, γ = γ_{1} ⊗ ⋯ ⊗ γ_{m} is a product of relative limits of discrete series representations and τ is a limit of discrete series representation of Õ(p′, q′). The restriction of σ to A is a multiple of a character ν; choose a parabolic subgroup P̃ = M̃N so that Re < α, ν >≥ 0 for all roots α of 𝔞 in 𝔫.

The standard module for Õ(p, q) associated to this data is

(4.1)(d)  X(P̃, σ) = Ind_{M̃N}^{Õ(p,q)}(σ)

(here and elsewhere we extend σ to M̃N trivially on N). This has the same type as does τ. If (M̃, σ) also satisfy condition (F-2) of [21], which we make explicit in Lemma 4.3, then this module has a unique irreducible quotient, and every irreducible representation is obtained this way. The data (M̃, σ) are unique up to conjugation by K̃ and will be called inducing data for π.

We next describe standard modules for genuine representations of S̃p(2n, ℝ). Recall from §1 that a Cartan subgroup H of Sp(2n, ℝ) is isomorphic to

(4.2)(a)  (ℂ^{∗})^{m} × (S^{1})^{r} × (ℝ^{∗})^{s} (r + s + 2m = n)

in which case M is isomorphic to

(4.2)(b)  GL(1, ℝ)^{s} × GL(2, ℝ)^{m} × Sp(2s, ℝ).

Let

(4.2)(c)  \overline{M} = G̃L(1, ℝ)^{s} × G̃L(2, ℝ)^{m} × S̃p(2s, ℝ).

For inducing data we take \overline{σ} = β ⊗ γ ⊗ τ with β = β_{1} ⊗ ⋯ ⊗ β_{s} a product of genuine characters, γ = γ_{1} ⊗⋯⊗γ_{m} a product of genuine relative limits of discrete series representations, and τ a genuine limit discrete series representation. Then \overline{σ} factors to a genuine representation σ of M̃, and every genuine representation σ of M̃ comes from a unique such σ̃. Choosing N satisfying the positivity condition as above, the standard module associated to the data (\overline{M}, \overline{σ}) or (M̃, σ) is

(4.2)(d)  X(P̃, σ) = Ind_{M̃N}^{S̃p(2n,ℝ)}(σ).

It has the same properties as in the case of Õ(p, q). We freely pass back and forth between (\overline{M}, \overline{σ}) and (M̃, σ) without further comment.

Condition (F-2) of [21] for Õ(p, q) and S̃p(2n, ℝ) is made explicit as follows. Fix a genuine character χ of G̃L(1), and write

(4.2)(e)  β_{i}(x, ϵ) = |x|^{ν_{i}}sgn(x)^{δ_{i}} · \{ \begin{matrix}1 & Õ(p, q) \\ χ(x, ϵ) & S̃p(2n, ℝ) \end{matrix}

A limit of discrete series representation of GL(2, ℝ) is parametrized by (k, μ) with k ∈ ℕ and μ ∈ ℂ; the lowest K–type of this representation has highest weight k + 1 for O(2). The genuine limit of discrete series representations of G̃L(2, ℝ) are parametrized the same way by tensoring with χ(det); this is independent of χ since for such a representation π ⊗ sgn ≃ π.

Lemma 4.3. Let G = Õ(p, q) or S̃p(2n, ℝ) as before. The data (M̃, σ) satisfy condition (F-2) of [21] if and only if:

(1) For each GL(2, ℝ)–factor, μ = 0 implies k ∈ ℤ,

(2) ν_{i} = ±ν_{j} implies δ_{i} = δ_{j}.

In this case, X(P̃, σ) has a unique irreducible quotient.

We will prove this in §7.

We also use character data for these groups as described in [21], which refers to [18,20]. Unexplained notation is as in [21].

A limit character for G̃ is a pair (H̃, γ). Here H̃ is a Cartan subgroup of G̃, and γ is a triple

(4.4)(a)  (Ψ, Γ, \overline{γ})

consisting of a positive system Ψ for the imaginary roots of 𝔥 in 𝔤, a character Γ of H̃, and an element \overline{γ} of 𝔥^{∗}. These must satisfy two conditions. First of all < α, \overline{γ} >≥ 0 for all α ∈ Ψ, and dΓ = \overline{γ} + ρ(Ψ) − 2ρ_{c}(Ψ).

A limit character is called final if in addition it satisfies the following two conditions. First of all if α is a simple root of Ψ then

(4.4)(b)  < α, \overline{γ} >= 0 implies α is non–compact.

Secondly if α is a real root of 𝔥 in 𝔤 then

(4.4)(c)  < α, \overline{γ} >= 0 implies Γ(m_{α}) ≠ ϵ_{α}

for m_{α} ∈ H and ϵ_{α} = ±1 as in [18,8.3.11], i.e. α does not satisfy the parity condition. We will make condition (4.4)(c) explicit and relate this data to inducing data in §6.

Attached to a final limit character γ is a standard module X(γ) which has an irreducible quotient \overline{X}(γ), and the K̃–conjugacy classes of final limit characters thereby parametrize the admissible dual of G̃. The central character of X(γ) is the restriction of Γ to the center of G; in particular X(γ) is genuine if and only if Γ is genuine.

§5. Main Results

We consider the dual pairs (Õ(p, q), S̃p(2n, ℝ)) with p + q = 2n + 1. Throughout this section we fix δ = ±1, and a non–trivial additive character ψ of ℝ. Recall (1.2) χ = χ(ψ) is a genuine character of G̃L(m, ℝ) for any m. Also recall for V an orthogonal space of signature (p, q), the genuine character χ_{V} = χ(ψ, V) of G̃L(m, ℝ) satisfies χ_{V} = χ(ψ)^{−p+q}. We write θ_{p,q} = θ(ψ, p, q) for the θ-correspondence as in (1.3).

Theorem 5.1. Let π be a genuine irreducible representation of S̃p(2n), with inducing data (cf. §4)

\overline{M} = G̃L(1, ℝ)^{s} × G̃L(2, ℝ)^{m} × S̃p(2r, ℝ) σ = α ⊗ β ⊗ τ.

By Theorem 3.3, there exist p′, q′ satisfying p′ + q′ = 2c + 1 and (−1)^{s} = δ(−1)^{m}, such that τ is in the domain of θ_{p′,q′}. Let η = θ_{p′,q′}(τ).

Let p = a + 2b + p′, q = a + 2b + q′. Then p + q = 2n + 1, (−1)^{q} = δ, θ_{p,q}(π) ≠ 0 and π is in the domain of θ_{p,q}. The inducing data for θ_{p,q}(π) are given by

M̃′ = GL(1, ℝ)^{s} × GL(2, ℝ)^{m} × Õ(p′, q′)[n] σ′ = α^{∗}χ_{V} ⊗ β^{∗}χ_{V} ⊗ η′

In these formulas, α^{∗}χ_{V} means α_{1}^{∗}χ_{V} ⊗ ⋯ ⊗ α_{k}^{∗}χ_{V} , similarly for β, and η′ is given by

η′ = \{ \begin{matrix}η & a even, \\ ηχ & a odd, n even, \\ ηχ^{−1} & a odd, n odd. \end{matrix}

Note. To define σ′ in Theorem 5.1 we have identified the GL(1) and GL(2) factors of M and M′ as in §1.

We summarize some useful properties of this correspondence which follow immediately from Theorem 5.1 and its proof.

Corollary 5.2.

(1) Let μ be a lowest K̃–type of π. Then μ is of lowest p, q–degree in π.

(2) p, q are the unique choice with (−1)^{q} = δ and H_{p,q}(μ) ≠ 0.

(3) Let μ be a minimal K̃–type of π, and write the element λ^{G}(μ) of 𝔱 associated to μ by the Vogan algorithm (cf. §6) as

λ^{G}(μ) = (λ_{1}, …, α_{r}, \overbrace{0, …, 0}^{t}, β_{1}, …, β_{s})

with α_{1} ≥ ⋯ ≥ α_{r} > 0 > β_{1} ≥ ⋯ ≥ β_{s}. Then p, q = 2r + t + 1, 2s + t or 2r + t, 2s + t + 1.

(4) μ′ = H_{p,q}(μ) is a lowest K̃′–type of π′. In particular μ′ has multiplicity one in π′ and the standard module of π′.

(5) H_{p,q} defines a bijection between the lowest K̃–types of π and the lowest K̃′–types of π′.

(6) A representation π of S̃p(2n, ℝ) occurs in the correspondence for the dual pair (Sp(2n, ℝ), O(p, q)) if and only if some (equivalently every) minimal K̃–type μ is p, q–harmonic.

(7) A representation π′ of Õ(p, q) occurs in the correspondence if and only if some (equivalently every) minimal K̃′-type is harmonic.

A comment is also in order due to our choice of coverings of orthogonal groups (§1). The group Õ(p′, q′) in M̃′ is Õ(p′, q′)[n], while η is defined on Õ(p′, q′)[c]. Since n − c ≡ a mod (2), there is an identification in the definition of η′ if a is odd. Strictly speaking it should read η′ = (ηχ)ξ (n even) or η′ = (ηξ)χ^{−1} (n odd) where ξ is the genuine character 1 ⊗ sgn of the trivial cover O(r, s) × ℤ/2ℤ of O(r, s) (cf. 1.2).

Let SO(p, q) be the admissible dual of SO(p, q), i.e. the set of equivalence classes of irreducible admissible Harish–Chandra modules for SO(p, q), and let S̃p(2n, ℝ)_{genuine} be the genuine admissible dual of S̃p(2n, ℝ).

Corollary 5.3. Fix δ and ψ. Then the representation correspondence gives a bijection

S̃p(2n, ℝ)_{genuine} ⟷^{1−1} ⋃_{\begin{matrix}p+q=2n+1 \\ (−1)^{q}=δ \end{matrix}} SO(p, q) .

More precisely, if π is a genuine irreducible representation of S̃p(2n), let π′_{0} = θ(ψ, p, q)(π) be the θ–lift of π to Õ(p, q) for the unique choice of p, q for which this is non–zero. Then π′_{0} ⊗ ξ^{−1} factors to O(p, q), and let π′ be the restriction to SO(p, q). Then π → π′ gives one direction of the bijection.

Conversely if π′ is an irreducible representation of SO(p, q), extend π′ to an irreducible representation of O(p, q) (there are two such choices), and tensor with ξ. Precisely one such choice of representation π′_{0} is in the domain of θ(ψ); let π = θ(ψ)(π′_{0}).

§6. Some Calculations

In this section we do some calculations involving K–types. The main results are Propositions 6.1, 6.18, 6.21 and 6.29. Throughout this section we fix ψ = ψ_{a} with a > 0 (cf. §1).

Let μ be a K–type for a group G. Proposition 5.3.3 of [18] produces an element λ ∈ 𝔱^{∗} where 𝔱 is a fundamental Cartan subalgebra of 𝔤. We refer to this map as the Vogan algorithm, and denote it μ → V(μ) = λ.

Proposition 6.1. Let μ be a K̃–type for S̃p(2n), and suppose μ is p, q–harmonic. Then the following diagram is commutative:

Sp(2n) O(p, q)
μ ⟶^{H} μ′
V↓  ↓^{V}
λ ⟶^{O} λ′

A small but useful observation is that for the purposes of computation it is better to compute the inverse of V, i.e. the multi–valued map λ → μ. With this in mind we summarize some standard theory [18], [6].

Let G be a reductive group with a compact Cartan subgroup T_{c}. We use 𝔱_{c} as a fixed complex Cartan subalgebra of 𝔤 (an “abstract” Cartan subalgebra in the sense of [19]). Let α_{1}, …, α_{k} be a set of strongly orthogonal non–compact roots of 𝔱_{c} in 𝔤. Associated to this set is a G–conjugacy class of Cartan subgroups of G. We choose H in this conjugacy class, and writing H = T A as usual we may and do assume 𝔱 ⊂ 𝔱_{c}, and 𝔱^{∗} ⊂ 𝔱_{c}^{∗}.

The Cartan involution of 𝔤 carried back to 𝔱_{c} via a Cayley transform gives an involution σ of 𝔱_{c}. Let \overline{γ} = (λ, ν) ∈ i𝔱_{c,0}^{∗} × 𝔱_{c}^{∗} satisfying σ(λ) = λ and σ(ν) = −ν. Then the Cayley transform identifies λ (resp. ν) with an element of i𝔱_{0}^{∗} (resp. 𝔞^{∗}), and \overline{γ} with an element of 𝔥^{∗}.

Let π be an irreducible representation of G with character data (H, γ) = (Ψ, Γ, \overline{γ}) (cf. §4). Write \overline{γ} = (λ, ν) with λ ∈ 𝔱_{c}^{∗}, and let 𝔮 = 𝔮(λ) = 𝔩 ⊕ 𝔲 be the θ–stable parabolic subalgebra of 𝔤 defined by λ ([18],Definition 5.2.1). The normalizer L of 𝔮 in G is quasi– split. The minimal K–types of π are of the form

(6.2)  μ = λ + ρ(𝔲 ∩ 𝔭) − ρ(𝔲 ∩ 𝔨) + μ_{L}

for some fine L ∩ K–type μ_{L}.

Proof of Proposition 6.1. Let G̃ = S̃p(2n, ℝ), G̃′ = Õ(p, q), etc. , with maximal compact subgroups and compact Cartan subgroups chosen as in §2. Given μ, let π = \overline{X}(γ) be an irreducible representation with lowest K̃–type μ. Then \overline{γ} = (λ, ν) with λ = V(μ) ∈ 𝔱_{c}^{∗}. By the above discussion μ = λ + ρ(𝔲 ∩ 𝔭) − ρ(𝔲 ∩ 𝔨) + μ_{L}. To avoid covering groups we work on the Lie algebras whenever possible.

After conjugating by W (𝔱_{c}, 𝔤) we may write

(6.3)  λ = (\overbrace{λ_{1}, …, λ_{1}}^{x_{1}}, …, \overbrace{λ_{r}, …, λ_{r}}^{x_{r}}, \overbrace{0, …, 0}^{m_{0}}, \overbrace{−λ_{r}, …, −λ_{r}}^{y_{r}}, …, \overbrace{−λ_{1}, …, −λ_{1}}^{y_{1}})

with λ_{1} > ⋯ > λ_{r} > 0; x_{i}, y_{i} ≥ 0.

The fine 𝔨 ∩ 𝔩_{0}–types μ_{L} for 𝔩_{0} ≃ ∏_{i=1}^{r} 𝔲(x_{i}, y_{i}) × 𝔰𝔭(2m_{0}, ℝ) are described as follows. If x_{i} ≠ y_{i} then μ_{L} is trivial on this factor. If x_{i} = y_{i}, then μ_{L} is trivial, or has highest weight

(6.4)(a)  ±(\overbrace{\frac{1}{2}, …, \frac{1}{2}}^{x_{i}}, \overbrace{\frac{1}{2}, …, \frac{1}{2}}^{y_{i}}),

on this factor. Finally on 𝔰𝔭(2m_{0}, ℝ), μ_{L} has highest weight of the form

(6.4)(b)  (1, …, 1, 0, …, 0) or (0, …, 0, −1, …, −1)

or

(6.4)(c)  (\overbrace{\frac{1}{2}, …, \frac{1}{2}}^{u}, \overbrace{−\frac{1}{2}, …, −\frac{1}{2}}^{v}).

In the case of a genuine representation π of S̃p(2n, ℝ), μ_{L} will have form (6.4)(c) on this factor.

A straightforward computation now gives:

(6.5)(a) μ = (\overbrace{α_{1}, …, α_{1}}^{x_{1}}, …, \overbrace{α_{r}, …, α_{r}}^{x_{r}}, \overbrace{x̃ − ỹ + \frac{1}{2}, …, x̃ − ỹ + \frac{1}{2}}^{m_{1}},

\overbrace{x̃ − ỹ − \frac{1}{2}, …, x̃ − ỹ − \frac{1}{2}}^{m_{2}}, \overbrace{β_{r}, …, β_{r}}^{y_{r}}, …, \overbrace{β_{1}, …, β_{1}}^{y_{1}}).

Here

x̃_{k} = ∑_{i=1}^{k} x_{i}, ỹ_{k} = ∑_{i=1}^{k} y_{i},

(6.5)(b)  α_{i} = λ_{i} + (x̃_{i−1} − ỹ_{i−1}) + \frac{1}{2}(x_{i} − y_{i}) + \frac{1}{2} + \frac{ϵ_{i}}{2} (ϵ_{i} = 0, ±1),
β_{i} = −λ_{i} + (x̃_{i−1} − ỹ_{i−1}) + \frac{1}{2}(x_{i} − y_{i}) − \frac{1}{2} + \frac{ϵ_{i}}{2} ,
ϵ_{i} = \{ \begin{matrix}±1 & x_{i} = y_{i} and λ_{i} ∈ ℤ + \frac{1}{2} \\ 0 & otherwise, \end{matrix}

and m_{1} and m_{2} are any non–negative integers with m_{1} + m_{2} = m_{0}. We set x̃ = x̃_{r}, ỹ = ỹ_{r}.

We now let λ′ = O(λ), and do the corresponding calculation on the orthogonal group. It follows from Proposition 2.2 that μ is p, q–harmonic with p = 2x + m_{0} + 1, q = 2y + m_{0} or p = 2x + m_{0}, q = 2y + m_{0} + 1. We consider only the first case, the second is similar.

From 1.13 we have

(6.6)  λ′ = (\overbrace{λ_{1}, …, λ_{1}}^{x_{1}}, …, \overbrace{λ_{r}, …, λ_{r}}^{x_{r}}, \overbrace{0, …, 0}^{m_{0}^{+}}; \overbrace{λ_{1}, …, λ_{1}}^{y_{1}}, …, \overbrace{λ_{r}, …, λ_{r}}^{y_{r}}, \overbrace{0, …, 0}^{m_{0}^{−}})

with m_{0}^{+} = [\frac{m_{0}+1}{2}] and m_{0}^{−} = [\frac{m_{0}}{2} ].

As before with 𝔮′ = 𝔮′(λ′) = 𝔩′ ⊕ 𝔲′

(6.7)  μ′ = λ′ + ρ(𝔲′ ∩ 𝔭′) − ρ(𝔲′ ∩ 𝔨′) + μ_{L′}.

We assume first that m_{0} is even, and compute (6.8)(a)

μ′ = (\overbrace{α′_{1}, …, α′_{1}}^{x_{1}}, …, \overbrace{α′_{r}, …, α′_{r}}^{x_{r}}, \overbrace{0, …, 0}^{m_{1}}) ⊗ (\overbrace{β′_{1}, …, β′_{1}}^{x_{1}}, …, \overbrace{β′_{r}, …, β′_{r}}^{x_{r}}, \overbrace{0, …, 0}^{m_{2}}) + μ_{L′}.

Here

α′_{i} = λ_{i} − \frac{p − q}{2} + (x̃_{i−1} − ỹ_{i−1}) + \frac{1}{2}(x_{i} − y_{i}) + \frac{1}{2}
(6.8)(b)  β′_{i} = λ_{i} + \frac{p − q}{2} + (ỹ_{i−1} − x̃_{i−1}) + \frac{1}{2}(y_{i} − x_{i}) + \frac{1}{2}.

Now 𝔩′_{0} ≃ ∏_{i=1}^{r} 𝔲(x_{i}, y_{i}) ⊕ 𝔬(m_{0} + 1, m_{0}), and on the unitary group factors fine 𝔨′ ∩ 𝔩′_{0}–types are as in (6.4)(a). On O(m_{0} + 1, m_{0}), the fine L′ ∩ K′ type μ_{L′} is of the form

(6.9)  (0, …, 0; δ) ⊗ (\overbrace{1, …, 1}^{s}, 0, …, 0; ϵ).

A similar statement holds for Õ(m_{0} + 1, m_{0}), upon tensoring with ξ.

It follows from Proposition 2.1 that if μ is any K̃–type for S̃p(2n, ℝ) of the form (6.2), then μ′ = H(μ) is of the form (6.7). In (6.7) we take μ_{L′} to be the same as μ_{L} on the unitary group factors, and on O(m_{0} + 1, m_{0}) it is given by (6.9) with

(δ, s, ϵ) = \{ \begin{matrix}(+1, m_{2}, +1) & 0 ≤ 2m_{2} ≤ m_{0} \\ (+1, m_{1}, −1) & m_{0} < 2m_{2} ≤ 2m_{0}. \end{matrix}

The other case (p = 2x + m_{0}) is similar. This completes the proof of Proposition 6.1. □

Proposition 6.10.

(a) Let π = \overline{X}(γ) be an irreducible genuine representation of S̃p(2n, ℝ). Write \overline{γ} = (λ, ν) and λ as in (6.3), and let 𝔮 = 𝔮(λ) = 𝔩 ⊕ 𝔲. Then the lowest K–types of X(γ) are of the form

μ = λ + ρ(𝔲 ∩ 𝔭) − ρ(𝔲 ∩ 𝔨) + μ_{L}

such that all the possible μ_{L} have the same restriction to 𝔰𝔭(2m_{0}, ℝ). Thus μ_{L} is

trivial except on 𝔰𝔭 and on factors 𝔲(x_{i}, y_{i}) of 𝔩_{0} with x_{i} = y_{i} and λ_{i} ∈ ℤ + \frac{1}{2}.

(b) The analogous statement holds for Õ(p, q).

Proof. This follows from the preceding discussion, and the following Lemma.

Lemma 6.11. Let π be a genuine principal series representation of S̃p(2n, ℝ) or Õ(n + 1, n). Then π contains a unique fine K–type.

Proof. For S̃p(2n, ℝ), let \overline{A} = (ℝ̃^{∗})^{n} as in §1. We consider Γ as a character of \overline{A}. Write

Γ = |x|^{ν_{i}}sgn(x)^{δ_{i}}χ^{−1}

on the i^{th} factor (ν_{i} ∈ ℂ, δ_{i} = 0, 1). Let n_{1} = ∑_{i} δ_{i}, and n_{0} = n − n_{1}. Then by Frobenius reciprocity

(\overbrace{\frac{1}{2}, …, \frac{1}{2}}^{n_{0}}, \overbrace{−\frac{1}{2}, …, −\frac{1}{2}}^{n_{1}})

is the unique fine K–type in the corresponding induced representation.

The proof for Õ(p, q) is similar. We omit the details. This completes the proof of Proposition 6.10. □

We now describe character data in more detail, and relate this to inducing data (cf. §4). Let (H̃, γ) be a limit character for G = S̃p(2n, ℝ). As in (6.3) write (6.12)(a)

λ = (\overbrace{λ_{1}, …, λ_{1}}^{x_{1}}, …, \overbrace{λ_{r}, …, λ_{r}}^{x_{r}}, \overbrace{0, …, 0}^{m_{0}}, \overbrace{−λ_{r}, …, −λ_{r}}^{y_{r}}, …, \overbrace{−λ_{1}, …, −λ_{1}}^{y_{1}}), 2λ_{i} ∈ ℤ.

and corresponding to this write

(6.12)(b) ν = (α_{1}^{1}, …, α_{1}^{x_{1}}, …, α_{r}^{1}, …, α_{r}^{x_{r}}, ν_{1}, …, ν_{m_{0}}, β_{r}^{1}, …, β_{r}^{y_{r}}, …, β_{1}^{1}, …, β_{1}^{y_{1}}).

For any i, (because we may conjugate by the stabilizer of λ in K) we may assume β_{i}^{j} = α_{i}^{j}

for all j ≤ min(x_{i}, y_{i}). For the parameter to be genuine, we also need x_{i} ≠ y_{i} ⇒ λ_{i} ∈ ℤ+ \frac{1}{2},

x_{i} > y_{i} ⇒ α_{i}^{x_{i}} = 0, and y_{i} > x_{i} ⇒ β_{i}^{y_{i}} = 0.

For each i let

ℓ_{i} = \{ \begin{matrix}x_{i} = y_{i} & λ_{i} ∈ ℤ \\ |\{j ≤ min(x_{i}, y_{i})|α_{i}^{j} ≠ 0\}| & λ_{i} ∈ ℤ + \frac{1}{2}, \end{matrix}

x′_{i} = x_{i} − ℓ_{i} (6.13)(a)

y′_{i} = y_{i} − ℓ_{i}
t_{i} = x′_{i} + y′_{i}.

Then set

(6.13)(b)  ℓ = ∑_{i} ℓ_{i}, t = ∑_{i} t_{i}.

Let \overline{H} be the covering group of H defined in §1, and let M, M̃ and \overline{M} be as in §4. In fact \overline{H} and \overline{M} are determined by \overline{γ}: \overline{H} is isomorphic to

˜(ℝ^{∗})^{m_{0}} × (ℂ^{∗})^{ℓ} × { \atop U(1)}^{t}

and \overline{M} is isomorphic to

G̃L(1)^{m_{0}} × G̃L(2)^{ℓ} × S̃p(2t, ℝ).

(That \overline{H} is so determined is due to the condition that γ satisfies condition (4.4)(b); see the proof of Proposition 6.15.)

Now γ determines a (relative) discrete series representation \overline{σ} of \overline{M}, explicitly described as follows. For each i there are ℓ_{i} limits of (relative) discrete series representations of G̃L(2), all with lowest K–type (2λ_{i} + 1)χ, and the center acting by |det|^{α_{i}^{j}}.

On S̃p(2t, ℝ) there is a limit of discrete series representation with Harish–Chandra parameter

(6.14)(a)  (\overbrace{λ_{1}, …, λ_{1}}^{x′_{1}}, …, \overbrace{λ_{r}, …, λ_{r}}^{x′_{r}}, \overbrace{−λ_{r}, …, −λ_{r}}^{y′_{r}}, …, \overbrace{−λ_{1}, …, −λ_{1}}^{y′_{1}}).

The positive imaginary roots on this factor are the corresponding restriction of Ψ.

The character Γ of H̃ satisfies dΓ = \overline{γ} + ρ(Ψ) − 2ρ_{c}(Ψ). Thus Γ is determined by \overline{γ} and Ψ except on the cover of the ℝ^{∗} factors. We consider Γ as a character of \overline{H}, genuine on each factor for which the cover is non–trivial (so Γ factors to a genuine character of H̃). For i = 1, …, m_{0} write Γ on the corresponding factor of ℝ̃^{∗} as

(6.14)(b)  Γ(x, ϵ) = |x|^{ν_{i}}sgn(x)^{δ_{i}}χ(x, ϵ)^{−1}.

Proposition 6.15. (H̃, γ) satisfies condition (4.4)(b) if and only if ν_{i} = ±ν_{j} ⇒ δ_{i} = δ_{j}.

Proof. Suppose < α, \overline{γ} >= 0. If α is a long root and γ is data for a genuine representation, then Γ(m_{α}) = ±i, and (4.4)(b) is immediate for these roots. If α is a short root on a factor of ℂ^{∗} then a straightforward calculation shows that (4.4)(b) holds if and only if λ_{i} ∈ ℤ. This is taken care of by our choice of H: the short real roots for which α(\overline{γ}) = 0 and λ_{i} ∈ ℤ + \frac{1}{2} are imaginary. If α is a short real root on the factors of ℝ^{∗} then a similar calculation shows (4.4)(b) is equivalent to the condition stated in the Proposition.

The preceding steps may be reversed to express character data in terms of inducing data. □

We turn next to an orthogonal group O(p, q) = O(2p_{0}+1, 2q_{0}). Let γ be a limit character, and write

(6.16)(a) λ = (\overbrace{λ_{1}, …, λ_{1}}^{x_{1}}, …, \overbrace{λ_{r}, …, λ_{r}}^{x_{r}}, \overbrace{0, …, 0}^{m_{0}^{+}}) ⊗ (\overbrace{λ_{1}, …, λ_{1}}^{y_{1}}, …, \overbrace{λ_{r}, …, λ_{r}}^{y_{r}}, \overbrace{0, …, 0}^{m_{0}^{−}})

as in (6.6). Then write the real part of the parameter as

(6.16)(b) ν = (α_{1}^{1}, …, α_{1}^{x_{1}}, …, α_{r}^{1}, …, α_{r}^{x_{r}}, ν_{1}, …, ν_{m_{0}^{+}})⊗

(β_{1}^{1}, …, β_{1}^{y_{1}}, …, β_{r}^{1}, …, β_{r}^{y_{r}}, ν_{m_{0}^{+}+1}, …, ν_{m_{0}^{+}+m_{0}^{−}}).

The corresponding Cartan subgroup H of SO(p, q) is isomorphic to

(ℝ^{∗})^{m_{0}^{+}+m_{0}^{−}} × (ℂ^{∗})^{ℓ} × U(1)^{t}

where ℓ and t are defined as in the previous case (cf. (6.12)(c)). Again Γ is determined by \overline{γ} except on the copies of ℝ^{∗}. Write Γ(x) = |x|^{ν_{i}}sgn(x)^{δ_{i}} on these terms.

Lemma 6.17. γ satisfies condition (4.4)(b) if and only if ν_{i} = ±ν_{j} ⇒ δ_{i} = δ_{j}.

Proof. If α is a short root then m_{α} = 1 and ϵ_{α} = −1 [18, 8.3.8 and 8.3.11] The first fact comes down to the isomorphism SO(2, 1) ≃ P GL(2, ℝ), and the second from a straightforward calculation that the integers d_{i} of [18,8.3.9] are even. Thus (4.4)(b) is automatic for these roots. The proof for the other roots is the same as for the symplectic group. We omit the details. □

This result extends in the obvious way to O(p, q) and Õ(p, q).

Proposition 6.18.

Let G̃ = S̃p(2n, ℝ) or Õ(p, q) and let

(6.18)(a)  IndG̃ ; M̃N(σ)

be a standard module for G̃ (cf. §4). Let μ be a minimal K̃–type of (6.18)(a), and suppose μ is p, q–harmonic. Let μ_{M} be the (unique) minimal { \atop K ∩ M}–type of σ. Then deg_{p,q}(μ) = deg_{p,q}(μ_{M}), and μ_{M} is contained in the restriction of μ to { \atop K ∩ M}.

Proof. We first consider G̃ = S̃p(2n, ℝ). Write V(μ) = λ as in (6.3), μ as in (6.5)(a), and other notation as in (6.5)(b). By the proof of Lemma 6.11 we have m_{1} = ∑_{i} δ_{i} with Γ written as in (6.13).

Then p = 2x̃ + m_{0} + 1, q = 2ỹ + m_{0} or p = 2x̃ + m_{0}, q = 2ỹ + m_{0} + 1. We consider only the first case, the second is similar. Let z = \frac{p−q}{2} = x̃ − ỹ + \frac{1}{2}.

By Proposition 2.1, deg_{p,q}(μ) = ∑_{i} x_{i}|α_{i} − z| + ∑_{i} y_{i}|β_{i} − z| + m_{2}.

It is not hard to see that α_{i} − z ≥ 0 ≥ β_{i} − z. This implies the degree of μ is the sum of the following terms:

= ∑_{i} ℓ_{i}(2λ_{i} + 1) (6.19)(a)

+ ∑_{i} x′_{i}(α_{i} − z) (6.19)(b)

− ∑_{i} y′_{i}(β_{i} − z) (6.19)(c)

+ m_{2} (6.19)(d)

On the other hand with σ described preceding Proposition 6.15 we compute the lowest { \atop K ∩ M}–type μ_{M} of σ. We pull this back to the group \overline{M}. The degree of \overline{μ}_{M} is the sum of the degrees of the factors. With notation as in (6.13), on each of the ℓ_{i} factors of type G̃L(2), the degree of μ_{M} is 2λ_{i} + 1. This contributes

(6.20)(a)  ∑_{i} ℓ_{i}(2λ_{i} + 1)  (independent of ϵ_{i})

to the degree of \overline{μ}_{M}.

On the S̃p(2t, ℝ) factor, σ is a limit of discrete series representation with Harish– Chandra parameter λ given by (6.14)(a). Define x̃′, ỹ′, α′_{i} etc. by applying (6.5)(b). Then μ_{M} on this factor is (2x̃′ + 1, 2ỹ′)–harmonic. With z′ = (2x̃′ + 1 − 2ỹ′)/2, we see that the degree of μ_{M} is the sum of

∑_{i} x′_{i}|α′_{i} − z′|, (6.20)(b)

∑_{i} y′_{i}|β′_{i} − z′|. (6.20)(c)

By (6.13)(a) x′_{i} − y′_{i} = x_{i} − y_{i} for all i. Therefore by (6.5), x̃′_{i} − ỹ′_{i} = x̃_{i} − ỹ_{i}, α′_{i} = α_{i}, and β′_{i} = β_{i} for all i. Also x̃′ − ỹ′ = x̃ − ỹ and so z′ = z. It follows that (6.20)(b) (resp. (c)) equals (6.16)(b) (resp. (c)).

Finally on each G̃L(1) factor, the degree is δ_{i}, which gives a contribution of

(6.20)(d)  m_{2}

to the degree of \overline{μ}_{M}. Comparing (6.16) to (6.20) we conclude that deg(μ) = deg(μ_{M}). This calculation also shows that the highest weight of μ_{M} is the same as the highest weight of μ (independent of the choice of μ), proving the last claim of the Proposition in this case.

The proof for Õ(p, q) is similar. We won’t use this fact, and so we leave the details to the reader. □

Proposition 6.21. Let μ be a lowest K̃–type of an irreducible genuine representation π of S̃p(2n, ℝ), and choose p, q so that μ is p, q–harmonic. Then μ is of lowest p, q–degree in π, and in the standard module of π.

Proof. Obviously it is enough to prove the second claim. We use notation as in the proof of Proposition 6.18. In particular write π = X(γ), λ as in (6.3) and μ as in (6.5)(a). Let p, q and z be as in the proof of Proposition 6.18, and write deg = deg_{p,q}. For any k–tuple write deg_{z}(x_{1}, …, x_{k}) = ∑_{i} |x_{i} − z|. With x̃, ỹ, m_{0} as in (6.5) (computed for μ), write any K̃–type γ as

(6.22)  γ = (γ_{+}, γ_{0}, γ_{−}) = (a_{1}, …, ax̃, c_{1}, …, c_{m_{0}}, bỹ, …, b_{1}).

Then deg(γ) = deg_{z}(γ_{+}) + deg_{z}(γ_{0}) + deg_{z}(γ_{−}). By (6.5)(a) μ satisfies

(6.23)  α_{1}, …, αx̃ ≥ z ≥ βỹ, …, β_{1}.

The standard module X(γ) may be realized as a derived functor module from the parabolic subalgebra 𝔮 = 𝔮(λ) of 𝔤 [18]. By the generalized Blattner formula ([18], Theorem 6.3.12) the highest weight of any K̃–type μ′ of π may be written

(6.24)  μ′ = μ + ∑_{α} m_{α}α (m_{α} ≥ 0)

where the sum runs over the roots of 𝔱_{c} in 𝔩 and 𝔲 ∩ 𝔭, and μ is a lowest K̃–type of π. Furthermore if the sum is restricted to roots in 𝔩, then the resulting weight is the highest weight of a { \atop K ∩ L}–type in the corresponding principal series representation X_{L}(γ) of L̃.

In our coordinates these roots are those in the following table.

(6.25)(a)  ϵ_{i} ± ϵ_{j} 1 ≤ i, j ≤ x̃
(6.25)(b)  −ϵ_{i} ± ϵ_{j} x̃ + m_{0} < i, j ≤ n
(6.25)(c)  ±(ϵ_{i} + ϵ_{j}) 1 ≤ i ≤ x̃, x̃ + m_{0} < j ≤ n
(6.25)(d)  ϵ_{i} + ϵ_{j}, 2ϵ_{i} 1 ≤ i ≤ x̃ < j ≤ x̃ + m_{0}
(6.25)(e)  −ϵ_{i} − ϵ_{j}, −2ϵ_{j} x̃ < i ≤ x̃ + m_{0} < j ≤ n
(6.25)(f)  ±ϵ_{i} ± ϵ_{j}, ±2ϵ_{j} x̃ < i, j ≤ x̃ + m_{0}

These roots also satisfy

< α, λ >≥ 0, and if < α, λ >> 0 then α is not of the form ϵ_{i} − ϵ_{j}.

Lemma 6.26. If γ is any k–tuple satisfying (6.23), then

deg(γ + ∑_{a,b,c} m_{α}α) ≥ deg(γ)

for any m_{α} ≥ 0. Here ∑_{a,b,c} denotes a sum over roots of the form (6.25)(a,b,c).

Proof. Adding roots of the form ϵ_{i} + ϵ_{j} and −ϵ_{i} − ϵ_{j} of type (6.25)(a–b) changes γ to a γ′ satisfying (6.23) and such that deg(γ′) ≥ deg(γ), with equality if and only if all m_{α} = 0. So we may as well assume that no such roots occur. Then

γ + ∑_{a,b,c} m_{α}α = (…, a_{i} + k_{i} + α_{i}, …, …, c_{1}, …, c_{m_{0}}, … b_{j} + ℓ_{j} + β_{j}, …)

where ∑ k_{i} = ∑ l_{j} and ∑ α_{i} = ∑ β_{j} = 0. Then

∑ |a_{i} + α_{i} + k_{i} − z| + ∑ |b_{j} + β_{j} + ℓ_{j} − z| ≥

∑(a_{i} − z + α_{i} + k_{i}) + ∑(z − b_{j} − β_{j} − ℓ_{j}) = ∑(a_{i} − z) + ∑(z − b_{j}).

The claim follows. □

Lemma 6.26 applies to μ + ∑_{d,e,f} m_{α}α, and gives

(6.27)(a)  deg(μ′) ≥ deg(μ + ∑_{d,e,f} m_{α}α).

The right hand side of (6.27)(a) equals

(6.27)(b) deg_{z}([μ + ∑_{d} m_{α}α]_{+}) + deg_{z}([μ + ∑_{d,e,f} m_{α}α]_{0}) + deg_{z}([μ + ∑_{e} m_{α}α]_{−})
(6.27)(c) ≥ deg_{z}(μ_{+}) + deg_{z}([μ + ∑_{d,e,f} m_{α}α]_{0}) + ∑_{d,e} m_{α} + deg_{z}(μ_{−})
(6.27)(d) ≥ deg_{z}(μ_{+}) + deg_{z}([μ + ∑_{f} m_{α}α]_{0}) + deg_{z}(μ_{−}).

Here (d) follows from repeated applications of the inequality deg_{z}([γ + α]_{0}) + 1 ≥ deg_{z}(γ_{0}) for any weight γ and α in (6.25)(d,e).

Separating the sum (6.24) into roots of 𝔩 and 𝔲 ∩ 𝔭 gives [μ + ∑_{f} m_{α}α]_{0} = [μ + ∑_{α∈Δ(𝔱_{c},𝔩)} m_{α}α]_{0}, so by the discussion following (6.24) this is a { \atop K ∩ L}–type of a principal series representation of S̃p(2m_{0}, ℝ). This has (unique) lowest { \atop K ∩ L}–type

(\overbrace{\frac{1}{2}, …, \frac{1}{2}}^{m_{1}}, \overbrace{−\frac{1}{2}, …, −\frac{1}{2}}^{m_{2}})

(cf. (6.5)(a)). Lemma 6.28. Let π be a minimal principal series representation of S̃p(2n, ℝ), containing

the (unique) fine K–type μ = (\overbrace{\frac{1}{2}, …, \frac{1}{2}}^{m_{1}}, \overbrace{−\frac{1}{2}, …, −\frac{1}{2}}^{m_{2}}) (cf. Lemma 6.11). Then μ is of lowest n + 1, n–degree in π. Proof. This follows easily from Frobenius reciprocity. We omit the details. □

Proposition 6.21 follows from (6.27)(d) and Lemma 6.28. □

We only need part (1) of the next Proposition for the proof of the main results in §5. Proposition 6.29.

Let μ be a lowest K̃′–type of an irreducible genuine representation π of Õ(p, q).

(1) If π is a discrete series representation then μ is of lowest degree in π.

(2) For any π, assume μ occurs in the space of joint harmonics. Then μ is of lowest degree in π.

Proof. We may safely ignore the covering groups, and for the remainder of this section we let G = O(p, q) = O(2p_{0} + 1, 2q_{0}), K = O(p) × O(q), etc.

Let π be a discrete series representation of G with Harish–Chandra parameter λ = (λ_{0}; ϵ) (cf. §3) and lowest K–type μ. Let μ′ be any K–type of π.

Suppose μ is of the form

(6.30)(a)  μ = (a_{1}, …, a_{r}, 0, …, 0; ϵ) ⊗ (b_{1}, …, b_{q_{0}}; η)

for some 0 ≤ r ≤ p_{0} (cf. §3). Write any K–type γ as γ = (γ_{+}, γ_{0}, γ_{−}) with γ_{+} = (a_{1}, …, a_{r}), γ_{−} = (b_{1}, …, b_{q_{0}}; η) and γ_{0} = (a_{r+1}, …, a_{p_{0}}; ϵ). Then deg(γ) = deg_{0}(γ_{+}) + deg(γ_{0}) + deg(γ_{−}) where the second and third terms are for the groups O(p − 2r) and O(q) respectively.

It follows from the formula μ = λ_{0} + ρ(𝔲 ∩ 𝔭) − ρ(𝔲 ∩ 𝔭) that λ_{0} may be written

(6.30)(b)  λ_{0} = (α_{1}, …, α_{r}, p_{0} − r − \frac{1}{2}, …, \frac{3}{2}, \frac{1}{2}; β_{1}, …, β_{q_{0}}).

By induction by stages ([18], Corollary 6.3.10) it follows that π may be realized as a derived functor module for 𝔮 = 𝔩 ⊕ 𝔲 with L ≃ U(1)^{q_{0}+r} × O(p − 2r), from a one–dimensional representation π_{L} of L. The O(p−2r) component of π_{L} is the one-dimensional representation (0, …, 0; ϵ) realized on the space ℂ_{ϵ}. By the Blattner formula (6.24) it follows that

(6.31)  μ′ = μ + ∑_{α} m_{α}α

with α ∈ 𝔲 ∩ 𝔭.

The roots of 𝔲 ∩ 𝔭 are (among those) of the form

(6.32)(a)  ±(e_{i} − e_{j}), e_{i} + e_{j}, e_{j} 1 ≤ i ≤ r, p_{0} + 1 ≤ j ≤ n
(6.32)(b)  ±ϵ_{i} + ϵ_{j}, r + 1 ≤ i ≤ p_{0}, p_{0} + 1 ≤ j ≤ n

As in (6.27)(a–c) it is immediate that

(6.33)(a) deg(μ′) ≥ deg(μ + ∑_{b} m_{α}α)

(6.33)(b)  ≥ deg(μ_{+}) + deg([μ + ∑_{b} m_{α}α]_{0}) + deg(μ_{−}) + ∑_{b} m_{α}deg(μ_{−})

where the subscripts denote the roots of (6.32)(a,b). Noting that [μ + ∑_{b} m_{α}α]_{0} = μ′_{0}, it is enough to show

(6.34)  deg(μ′_{0}) + ∑_{b} m_{α} ≥ deg(μ_{0}).

This requires a refinement of the Blattner formula. We only consider the factor O(p − 2r) of L. Let τ, τ′ be the O(p − 2r) factor of H^{0}(𝔲 ∩ 𝔨, V_{μ}) and H^{0}(𝔲 ∩ 𝔨, V_{μ′}) respectively. These are the finite dimensional L ∩ K–modules with the same highest weight as μ and μ′ respectively. It follows from [18,6.3.12] (a sharpening of (6.31)) that (with m = ∑_{b} m_{α}),

(6.35)  mult[τ′ : τ ⊗ S^{m}(𝔲 ∩ 𝔭)|_{O(p−2r)}] > 0.

Recall from §1 that we realized each representation of O(p − 2r) as the highest weight factor in a representation of U(p − 2r). As a module for O(p − 2r), 𝔲 ∩ 𝔭 is isomorphic to a direct sum of copies of the standard module; thus it is a module for U(p − 2r) as well. So we can decompose S^{m}(𝔲 ∩ 𝔭) with respect to U(p − 2r) and then restrict to O(p − 2r).

It follows that the highest weight of any irreducible summand of S^{m}(𝔲 ∩ 𝔭), is of the form

(c_{1}, …, c_{p−2r}) with ∑ c_{i} = m, c_{i} ≥ 0 for all i,

when written as a weight for U(p − 2r).

Lemma 6.36.

Let τ, τ′ be irreducible representations of O(n), with τ one–dimensional. Let γ be an irreducible representation of U(n) with highest weight (c_{1}, …, c_{n}), c_{i} ≥ 0. Suppose

mult[τ′ : τ ⊗ (γ|_{O(n)})] > 0.

Then deg(τ′) + ∑_{i=1}^{n} c_{i} ≥ deg(τ).

Proof. This is obvious if τ is trivial, so assume τ = sgn. Replacing τ′ with τ′ ⊗ sgn it is enough to show

(6.37)  mult[τ′ : γ|_{O(n)}] > 0 ⇒ deg(τ′ ⊗ sgn) + ∑ c_{i} ≥ n.

Write τ′ = (a_{1}, …, a_{k}, 0, …, 0; ϵ), so deg(τ′) = ∑ a_{i} + \frac{1−ϵ}{2}(n − 2k). If the multiplicity is greater than zero, then γ must contain a vector of weight (d_{1}, …, d_{n}) which is a highest weight vector for τ′. It follows that: (6.38)

(a) d_{i} − d_{n+1−i} = a_{i} i ≤ k,

(b) d_{i} − d_{n+1−i} = 0, k + 1 ≤ i ≤ [\frac{n}{2}],

(c) d_{i} ≡ d_{n+1−i} ≡ \frac{1−ϵ}{2} mod (2), k + 1 ≤ i ≤ [\frac{n}{2}],

(d) d_{\frac{n+1}{2}} ≡ \frac{1−ϵ}{2} mod (2) if n is odd.

In addition, the relations ∑_{i=1}^{n} d_{i} = ∑_{i=1}^{n} c_{i} and d_{i} ≥ 0 for all i, hold.

By (6.38)(a), d_{i} ≥ a_{i} ≥ 1 for 1 ≤ i ≤ k. Thus

(6.39)  ∑_{1}^{k}(a_{i} + d_{i}) ≥ 2k.

If ϵ = 1, then (6.37) becomes ∑_{1}^{k} a_{i}+∑_{1}^{n} d_{i} ≥ 2k,, which is immediate from (6.38). Assume ϵ = −1. We need to show ∑_{1}^{k}(a_{i} + d_{i}) + ∑_{k+1}^{n} d_{i} ≥ n. By (6.38)(c) and (6.38)(d), we get d_{i} ≥ 1 for k + 1 ≤ i ≤ n − k. The assertion follows from this together with (6.39). □

This also completes the proof of Proposition 6.29(1). □

Part (2) of the Proposition may be proved similarly, using an extensionion of Lemma 6.36 to general τ, and a version of Lemma 6.28 for O(p, q). Since we won’t need it we omit the details, but we note that it is also an immediate consequence of Theorem 5.1.

§7. Occurence of the Discrete Series

In this section we prove that the entire genuine discrete series of S̃p(2n, ℝ), and half of the genuine discrete series of Õ(p, q), occur in the correspondence. We assume p+q = 2n+1 throughout, and fix ψ. The arguments hold for p + q = 2n as well, recovering some of the results of [11]. We depart from our convention of §1 and let Õ(p, q) = O(p, q) if n is even, and we let S̃p(2n, ℝ) = Sp(2n, ℝ) when considering a dual pair (Sp(2n, ℝ), O(p, q)) with p + q even.

Proposition 7.1.

(1) Let π be a genuine discrete series representation of S̃p(2n, ℝ). Then, for any δ = ±1, π occurs in the correspondence with some Õ(p, q), (−1)^{q} = δ.

(2) Let π be a genuine discrete series representation of Õ(p, q). Then precisely one of π and π ⊗ sgn occurs in the correspondence with S̃p(2n, ℝ).

This follows from a doubling of variables argument due to Kudla and Rallis. The proof is divided into a series of Lemmas.

For an irreducible representation π of S̃p(2n, ℝ) (resp. Õ(p, q)), let R_{n,p,q} denote the maximal quotient of ω_{n,p,q} on which S̃p(2n, ℝ) (resp. Õ(p, q)) acts by a multiple of π [5].

We embed Sp(2n, ℝ) × Sp(2n, ℝ) diagonally in Sp(4n, ℝ). This induces a natural map S̃p(2n, ℝ) × S̃p(2n, ℝ) → S̃p(4n, ℝ). Similarly Õ(p, q) × Õ(q, p) maps to Õ(p + q, p + q).

Lemma 7.2.

(1) For any irreducible representation π of S̃p(2n, ℝ),

(7.3)(a)  θ_{p,q}(π) ≠ 0 ⇔ Hom_{S̃p(2n,ℝ)×S̃p(2n,ℝ)}(R_{2n,p,q}(11), π ⊗ π) ≠ 0.

(2) For any irreducible representation π of Õ(p, q),

(7.3)(b)  θ(π) ≠ 0 ⇔ Hom_{Õ(p,q)×Õ(q,p)}(R_{n,p+q,q+p)}(11), π ⊗ π) ≠ 0.

Proof. We prove (1), the proof of (2) is similar. To conserve notation let G = S̃p(2n, ℝ), G′ = Õ(p, q), and let ω_{n} be the oscillator representation for the dual pair (G, G′). Then according to ([11], I.8),

(7.4)(a)  Hom_{G}(ω_{n}, π) ≠ 0 ⇔ Hom_{G×G′}(ω_{n}, π ⊗ π′) ≠ 0 for some π′.

Thus if Hom_{G}(ω_{n}, π) ≠ 0, then Hom_{G×G×G′×G′}(ω_{n} ⊗ ω_{n}, π ⊗ π ⊗ π′ ⊗ π′) ≠ 0. Since π′ is a genuine representation of Õ(p, q), (π′)^{∗} ≃ π′ ⊗ sgn^{n}. Therefore sgn^{n} is a quotient of π′ ⊗ π′, which gives

(7.4)(b)  Hom_{G}(ω_{n}, π) ≠ 0 ⇒ Hom_{G×G×Δ(G′)}(ω_{n} ⊗ ω_{n}, π ⊗ π ⊗ sgn^{n}) ≠ 0,

where Δ(G′) is the diagonal subgroup of G′ × G′. By Lemma 1.7 we may replace ω_{n} ⊗ ω_{n} with ω_{2n} ⊗ sgn^{n}. Thus the right hand side of (7.4)(b) is equivalent to

(7.4)(c)  Hom_{G×G×Δ(G′)}(ω_{2n}, π ⊗ π ⊗ 11) ≠ 0 ⇔ Hom_{G×G}(R_{2n,p,q}(11), π ⊗ π) ≠ 0.

Thius proves one direction of the statement. On the other hand, if (7.4)(c) holds (i.e. the right hand side is nonzero), the same is true for (7.4)(b), and ignoring the Δ(G′) action, we see that Hom_{G×G}(ω_{n} ⊗ ω_{n}, π ⊗ π) ≠ 0. This is easily seen to imply Hom_{G}(ω_{n}, π) ≠ 0, proving the Lemma. □

Let P = MN be the stabilizer of the Lagrangian subspace L_{0} =< e_{1}, …, e_{2n} >, and P̃ = M̃N its inverse image in S̃p(4n, ℝ). For α ∈ ℤ/4ℤ we consider the Harish–Chandra module of the induced representation

(7.5)(a)  IndS̃p(4n,ℝ) ; P̃(χ^{α}).

We are using normalized induction, so this representation is unitarily induced and completely reducible. Similarly, for G′ = Õ(p, q), we consider the induced representation

(7.5)(b)  IndÕ(2n+1,2n+1) ; P̃(χ),

where the Levi component of P = MN is isomorphic to GL(m), M̃ ≃ M × ℤ/2ℤ, and χ = 11 ⊗ sgn.

Lemma 7.6.

(1) For α = ±1

IndS̃p(2n,ℝ) ; P̃(χ^{α}) ≃ ⨁_{\begin{matrix}p+q=2n+1 \\ p−q=α \end{matrix}} R_{2n,p,q}(11).

(2)

IndÕ(2n+1,2n+1) ; P̃(χ ⊗ 1) ≃ R_{n,2n+1,2n+1}(11) ⊕ (R_{n,2n+1,2n+1}(11) ⊗ sgn).

Proof. Part (1) is proved in [9], and both (1) and (2) are in [24]. □

Lemma 7.7.

(1) Let π be a genuine irreducible representation of S̃p(2n, ℝ). Then θ_{p,q}(π) ≠ 0 for some p, q with (−1)^{q} = δ if and only if

(7.7)(a)  Hom_{S̃p(2n,ℝ)×S̃p(2n,ℝ)}[IndS̃p(2n,ℝ) ; P̃(χ^{α}), π ⊗ π] ≠ 0, for α = δ(−1)^{n}.

(2) For π an irreducible representation of Õ(p, q), θ(π) ≠ 0 or θ(π ⊗ sgn) ≠ 0 if and only if

(7.7)(b)  Hom_{Õ(p,q)×Õ(q,p)}[IndÕ(2n+1,2n+1) ; P̃(χ), π ⊗ π] ≠ 0.

Proof. This follows immediately from Lemmas 7.2 and 7.6.

Lemma 7.8. Let π be the Harish–Chandra module of a genuine discrete series representation of S̃p(2n, ℝ) (resp. Õ(p, q)). Then the space (7.7)(a) (resp. (b)) is non–zero.

Proof. The two cases are similar, so we treat only S̃p(2n, ℝ). Let X be the variety of Lagrangian subspaces of ℝ^{4n}. Then X ≃ S̃p(2n, ℝ)/P̃. Let L^{2} − I_{2n}(α) be the L^{2}–induced version of (7.5)(a). This is realized on L^{2} sections of the induced bundle B = S̃p(4n, ℝ)×P̃ (χ^{α}) over X.

Let L =< e_{1} + e_{n+1}, …, e_{n} + e_{2n}, f_{1} + f_{n+1}, …, f_{n} + f_{2n} >. Then the orbit O of L by G = Sp(2n, ℝ) × Sp(2n, ℝ) is open in X. Let H be the stabilizer of L in G so O ≃ G/H. Then H ≃ Sp(2n, ℝ) is embedded in G ≃ Sp(2n, ℝ) × Sp(2n, ℝ) via g → (g, τ(g)) where τ is the outer automorphism of Sp(2n, ℝ) of 1.4. Let Z ≃ ℤ/2ℤ be the kernel of the covering map S̃p(2n, ℝ) → Sp(2n, R). Passing to the coverings we see that the map

(7.9)(a)  O ≃ G̃/H̃ → S̃p(2n, ℝ)/Z

given by (g, h, ϵ)H̃ → (gτ(h^{−1}), ϵ)Z is an isomorphism, and induces an isomorphism between the restriction of B to O and

(7.9)(b)  G̃ ×_{Z} χ|_{Z}.

Under this isomorphism the action of G̃ on O becomes

(g, h, ϵ) · (x, δ)Z = (gxτ(h^{−1}), ϵδ)Z

(g, h, x ∈ Sp(2n, ℝ)). Thus sections of the bundle (7.9)(b) are identified with L_{χ}^{2}(S̃p(2n, ℝ)), i.e. L^{2} functions on S̃p(2n, ℝ) transforming by χ under Z, with S̃p(2n, ℝ) × S̃p(2n, ℝ) acting by conjugation twisted by τ. Since τ takes π to π^{∗} it follows that the discrete spectrum of this space is precisely the sum of τ ⊗ τ where τ runs over the genuine discrete series representations of S̃p(2n, ℝ).

Therefore there is a non–zero map

ϕ : L^{2} − I_{2n}(α) ⟶^{restriction} L_{χ}^{2}(S̃p(2n, ℝ)) → π ⊗ π

intertwining the action of S̃p(2n, ℝ) × S̃p(2n, ℝ), where the first map is restriction of sections to O.

To complete the proof we need to replace L^{2} − I_{2n}(α) and π by their Harish-Chandra modules. Let 𝕂̃ (resp. K̃) be the maximal compact subgroup of S̃p(4n, ℝ) (resp. S̃p(2n, ℝ)). The restriction of the 𝕂̃–finite functions I_{2n}(χ) of L^{2} − I_{2n}(α) is a dense subspace of the K̃ × K̃–finite functions on O. Therefore ϕ restricted to I_{2n}(α) is non–zero. □

Proof of Proposition 7.1. Part (1), and the occurence of either π or π ⊗ sgn in the correspondence in (2), is an immediate consequence of Lemma 7.8. The fact that both π and π ⊗ sgn cannot occur in (2) was proved in Lemma 1.5. □

§8. Induction Principle

In this section we turn to a more general setting and let V be an orthogonal space of signature (p, q) and W be a symplectic space of dimension 2n, with no further restrictions. Throughout this section we fix ψ and let ω be the corresponding oscillator representation for the dual pair (O(V), Sp(W)).

Suppose we are given a decomposition

(8.1)(a)  W = W_{1}^{+} ⊕ ⋯ ⊕ W_{r}^{+} ⊕ W^{0} ⊕ W_{1}^{−} ⊕ ⋯ ⊕ W_{r}^{−}

where all W_{j}^{±} are isotropic, W_{j}^{+} and W_{j}^{−} are in duality, and W^{0} is a non–degenerate symplectic space or 0. Let P = MN be the stablizer in Sp(W) of the flags 0 ⊂ W_{1}^{−} ⊂ W_{1}^{−} ⊕ W_{2}^{−} ⊂ ⋯ ⊂ W^{−} = ∑_{i} W_{i}^{−}. Let P̃ = M̃N be the inverse image of P in S̃p(W). There is a surjective map

(8.1)(b)  \overline{M} = G̃L(W_{1}) × ⋯ × G̃L(W_{r}) × S̃p(W^{0}) ↠ M̃.

Let

(8.1)(c)  V = V_{1}^{+} ⊕ ⋯ ⊕ V_{r}^{+} ⊕ V^{0} ⊕ V_{1}^{−} ⊕ ⋯ ⊕ V_{r}^{−}

be a decomposition of V, and define P′ = M′N′ and P̃′ = M̃′N′ in an analogous manner as for the symplectic group. In this case M̃′ ≃ GL(V_{1}) × ⋯ × GL(V_{r}) × Õ(V^{0}).

Let ω_{M} denote the oscillator representation for (M, M′). This is the product of the oscillator representations for the dual pairs (GL(W_{j}), GL(V_{j})) and (Sp(W^{0}), O(V^{0})). It gives a correspondence between representations of \overline{M} and

(8.1)(d)  \overline{M}′ = G̃L(V_{1}) × ⋯ × G̃L(V_{r}) × Õ(V^{0}).

If one member of a dual pair is the trivial group, then we take the trivial representation for the oscillator representation for this pair.

Set χ = χ(ψ) and χ_{V} = χ(ψ, V) as in (1.2).

Definition 8.2 ([7], 1.1.1). For a, b ∈ ℤ define the character ζ(a, b) of G̃L(1) by

ζ(a, b)(x, ϵ) = χ(x^{a}, ϵ^{a})χ(x^{b}, ϵ^{b})^{−1}

= γ(x^{a}, \frac{1}{2}ψ)γ(x^{b}, \frac{1}{2}ψ)^{−1}ϵ^{a−b}

= \{ \begin{matrix}1 & a ≡ b(2) \\ χ^{−1}(x, ϵ) & a even, b odd \\ χ(x, ϵ) & a odd, b even. \end{matrix}

For (g, ϵ) ∈ G̃L(m) let ζ(a, b)(g, ϵ) = ζ(a, b)(det(g), ϵ).

Note that ζ(a, b) is of type a + b.

Let k_{i} = dim(V_{i}), k̃_{i} = ∑_{j=1}^{i} k_{j}, k = k̃_{r}, ℓ_{i} = dim(W_{i}), and ℓ̃_{i} = ∑_{j=1}^{i} ℓ_{j}, ℓ = ℓ̃_{r}.

Definition 8.3. Let m = p + q, and define the character ξ of \overline{M} by:

ξ = \{ \begin{matrix}|det|^{n− \frac{m}{2} −ℓ̃_{j}+k̃_{j}+ \frac{1}{2} ℓ_{j}− \frac{1}{2} k_{j}+ \frac{1}{2}} ζ(k_{j}, k)χ(det^{k}, ϵ^{k})χ_{V} & G̃L(W_{j}) \\ 1 & S̃p(W^{0}) \end{matrix}

and ξ′ of \overline{M}′ by

ξ′ = \{ \begin{matrix}|det|^{−n+ \frac{m}{2} +ℓ̃_{j}−k̃_{j}− \frac{1}{2} ℓ_{j}+ \frac{1}{2} k_{j}− \frac{1}{2}} ζ(ℓ_{j}, ℓ)χ(det^{ℓ}, ϵ^{ℓ}) & G̃L(V_{j}) \\ χ(det^{−ℓ}, ϵ)(−1, det)_{ℝ}^{ℓ(n−ℓ)}ϵ & Õ(V^{0}). \end{matrix}

Theorem 8.4: Induction Principle I. Let σ be a representation of \overline{M} and σ′ a representation of \overline{M}′. Suppose there is a non–zero \overline{M} × \overline{M}′ equivariant map

(8.5)(a)  ω_{M} → σ ⊗ σ′.

Then there is a non–zero Õ(V) × S̃p(W) equivariant map

(8.5)(b)  ϕ : ω → IndS̃p(W) ; P̃(σξ) ⊗ IndÕ(V) ; P̃′(σ′ξ′).

Here σξ factors to M̃, and extends to P̃ trivially on N, and σ′ξ′ factors to M̃′ and extends to P̃′ trivially on N′.

Note: The cover Õ(V^{0}) of O(V^{0}) in (8.1)(d) (resp. (8.3)) is Õ(V^{0})[n − ℓ] (resp. Õ(V^{0})[ℓ]). Then σ′ξ′ and the representation in (8.5)(b) are naturally representations of the covers of O(V^{0})[n] and O(V)[n].

Proof. The proof is essentially the same as the proofs of ([7], Theorem 2.5) and ([3], Corollary 3.21). It follows from Frobenius reciprocity and the following two Lemmas.

Lemma 8.5. In the setting of Theorem 8.4, suppose r = 1 so that V = V^{+} ⊕ V^{0} ⊕ V^{−}, P_{V} = M_{V} N_{V} is the stabilizer of V^{−}, M_{V} ≃ GL(V^{+}) × O(V^{0}), and similarly for Sp(W). Then there is a surjective M̃_{V} × M̃_{W} equivariant map

ω ↠ ω_{M}ν

where ν is the following character of \overline{M}_{V} × \overline{M}_{W}.

ν = \{ \begin{matrix}|det|^{−n+ \frac{ℓ}{2}} γ(det^{ℓ}, \frac{1}{2}ψ)ϵ^{ℓ} & (g, ϵ) ∈ G̃L(V^{+}) \\ |det|^{− \frac{m}{2} + \frac{k}{2}} γ(det^{k}, \frac{1}{2}ψ)χ_{V} & (g, ϵ) ∈ G̃L(W^{+}) \\ γ(det^{ℓ}, \frac{1}{2}ψ)(−1, det)_{ℝ}^{nℓ}ϵ^{ℓ} & (g, ϵ) ∈ Õ(V^{0}) \\ 1 & (g, ϵ) ∈ S̃p(W^{0}) \end{matrix}

Proof. The proof is the same as the proof of ([3], Proposition 3.13). □

Lemma 8.6. Let (GL(V), GL(W)) be a dual pair. Suppose V = V_{1} ⊕ V_{2}, and let P_{V} = M_{V} N_{V} be the stabilizer of V_{1}, so M_{V} ≃ GL(V_{1}) × GL(V_{2}). Let \overline{M} = G̃L(V_{1}) × G̃L(V_{2}), and let P̃ = M̃N be the inverse image of P in G̃L(V). Similarly let W = W_{1} ⊕ W_{2}, etc. Let ω_{M} be the oscillator representation for the dual pair (M_{V} , M_{W}). Set k_{i} = dim(V_{i}) and ℓ_{i} = dim(W_{i}). Then there is a surjective P̃ × P̃ equivariant map ω ↠ ω_{M}ν, where ν is the following character:

ν = \{ \begin{matrix}|det|^{− \frac{1}{2} ℓ_{2}}ζ(ℓ_{1}, ℓ) & G̃L(V_{1}) \\ |det|^{\frac{1}{2} ℓ_{1}}ζ(ℓ_{2}, ℓ) & G̃L(V_{2}) \\ |det|^{− \frac{1}{2} k_{2}}ζ(k_{1}, k) & G̃L(W_{1}) \\ |det|^{\frac{1}{2} k_{1}}ζ(k_{2}, k) & G̃L(W_{2}) \end{matrix}

Proof. See the proof of ([3], Proposition 3.13). We omit the details. □

In the setting of Theorem 8.4, let \overline{K}\overline{M} be a maximal compact subgroup of \overline{M}. There is a surjective map from \overline{K}\overline{M} to the maximal compact subgroup { \atop K ∩ M} of M̃. The \overline{K}\overline{M}–type in the next result factors to, and is identified with, a representation of { \atop K ∩ M}.

Theorem 8.7: Induction Principle II. In the setting of Theorem 8.4, suppose μ is a K̃–type for S̃p(2n, ℝ), and μ\overline{M} is a \overline{K}\overline{M}–type for \overline{M} satisfying the following properties.

(1) μ_{M} is of minimal degree in σ,

(2) deg(μ) = deg(μ_{M}), μ contains μ_{M} in its restriction to { \atop K ∩ M} and is of minimal degree and multiplicity one in

IndÕ(V) ; P̃(σα),

(3) There exist characters α and α′ of \overline{M} and \overline{M}′, trivial on { \atop K ∩ M} and { \atop K′ ∩ M′} such

that σα ⊗ σ′α′ is also a quotient of ω_{M}, and IndS̃p(𝕎)(W) ; P̃(σξα) is irreducible.

Then μ ⊗ H(μ) is in the image of ϕ.

Proof. The proof is the same as the proof of [3], Proposition 3.25. □

Theorem 8.8. In the setting of Theorem 8.4, assume dim(V_{i}^{+}) = dim(W_{i}^{+}) = k_{i} for all 1 ≤ i ≤ r; so with k = ∑_{i} k_{i},

\overline{M} ≃ G̃L(k_{1}) × ⋯ × G̃L(k_{r}) × S̃p(2n − 2k, ℝ)
\overline{M}′ ≃ GL(k_{1}) × ⋯ × GL(k_{r}) × Õ(p − k, q − k).

Let σ_{i} be any irreducible representation of GL(k_{i}), and suppose σ_{0} corresponds to τ_{0} for the dual pair (O(p − k, q − k), Sp(2n − 2k, ℝ)). Then there is a non-zero map Φ from ω to the tensor product of

IndS̃p(2n,ℝ) ; P̃(σ_{1} ⊗ ⋯ ⊗ σ_{r} ⊗ σ_{0})

and

IndÕ(p,q) ; P̃(σ_{1}^{∗}χ_{V} ⊗ ⋯ × σ_{r}^{∗}χ_{V} ⊗ τ_{0}ζ(n, k + n)).

Proof. The existence of Φ follows from Theorem 8.4 and the fact that for the dual pair (GL(m), GL(m)) the correspondence is π → π ⊗ (−1, det)_{ℝ}^{m} for all π. □

Note: The oscillator representation for the dual pair GL(m), GL(m) may be normalized so that the action of the dual pair factors to the linear groups. We are using the unnormalized oscillator representation, which accounts for the term (−1, det)_{ℝ}^{m}.

§9. Proof of the Main Results

We prove the four results in §5, and Theorem 3.3. Throughout this section we fix ψ and δ = ±1.

The most natural way to prove Theorem 5.1 would be to prove it first for the discrete series, and then in general using the induction principle (§8). Unfortunately, the results in §7 are not sharp enough to compute the correspondence of the discrete series.

Instead we proceed by induction on n. Given the result for S̃p(2n − 2, ℝ), the induction principle computes the result for all representations but the discrete series of S̃p(2n, ℝ) (and Õ(p, q)). This implies that the representation correspondence can only map discrete series to discrete series; since these are in the domain of the correspondence, it remains to match up parameters. This is a relatively simple matter using the results on harmonic K–types in section 6. Proof of Theorem 5.1 n = 0:

This is not quite empty, but an exercise in the definitions and covering groups. Consider the dual pair (O(1, 0), Sp(0, ℝ)). The group Õ(1, 0) ≃ ℤ/2ℤ is isomorphic to O(1, 0) × ℤ/2ℤ, and S̃p(0, ℝ) ≃ ℤ/2ℤ. This dual pair is mapped to S̃p(0, ℝ) ≃ ℤ/2ℤ, and the correspondence is obtained by restricting the non–trivial character of this group to the dual pair. This takes the non–trivial character of S̃p(0, ℝ) to ξ = 11 ⊗ sgn of Õ(1, 0). This is as predicted by Theorem 3.3, and Theorem 5.1 is immediate. The case of O(0, 1) is the same. This completes the proof in this case. Inductive Step: Induced Representations

Assume Theorems 3.3 and 5.1 for S̃p(2n − 2, ℝ), and let π be a genuine irreducible representation of S̃p(2n, ℝ), which is not a discrete series or a limit of discrete series.

Let M̃ and σ be inducing data for π as in Theorem 5.1. We choose P̃ = M̃N to be of the form in Theorem 8.8 (cf. 8.1), and so π is the (unique) irreducible quotient of the standard module

(9.1)(a)  IndS̃p(2n,ℝ) ; P̃(σ).

By Theorem 8.8 there is a non–zero map Φ from ω to the tensor product of (9.1)(a) and

(9.1)(b)  IndÕ(p,q) ; P̃′(σ′).

Here p′, q′, M̃′ and σ′ are as in Theorem 5.1, the twist by η coming from ζ(n, k + n) of Theorem 8.8.

Recall from (§4) that π is the unique irreducible constituent of (9.1)(a) containing a minimal K̃–type μ, and similarly π′. It is enough to show

(9.2)  μ is in the image of Φ.

Let π⊗π′′ be any irreducible quotient of the image of Φ. By [5] π′′ contains H(μ). Write the standard module (9.1)(a) as X(γ), with \overline{γ} = (λ, ν), and similarly (9.1)(b). It is immediate from the calculations in §6 that O(λ) = λ′ (cf. the proof of Theorem 5.5 below). By Proposition 6.1, H(μ) is a minimal K̃′–type of (9.1)(b), and it follows that π′′ = π′, as we needed to show.

To see (9.2) we apply Theorem 8.7. Let μ_{M} be the minimal { \atop K ∩ M}–type of σ. By Corollary 5.2 applied to S̃p(2r, ℝ) the S̃p(2r, ℝ) component of μ_{M} is p′, q′–harmonic. By Proposition 6.21, and ([11],III.9) for the GL terms, μ_{M} is of lowest p′, q′–degree in σ, so condition (1) of Theorem 8.7 holds. By Proposition 6.18, deg_{p,q}(μ) = deg_{p,q}(μ_{M}), and since p − q = p′ − q′ this equals deg_{p′,q′}(μ_{M}). Also by Proposition 6.18, the restriction of μ to { \atop K ∩ M} contains μ_{M}. By Proposition 6.21, μ is of lowest degree in (9.1)(a), and also of multiplicity one (this is a general fact about standard modules). This verifies (8.7)(2).

Take α to be a generic character of \overline{M} given by a power of |det| on each of the GL terms, and let α′ = α^{∗}. Then σα ⊗ σ′α′ is a quotient of ω_{M} (cf. the proof of Theorem 8.8) and (9.1)(a) is irreducible by the usual argument. Thus (3) for Theorem 8.7 holds, and applying the theorem we conclude (9.2). Inductive Step: Limits of Discrete Series

Let π be a genuine limit of discrete series representation of S̃p(2n, ℝ) not in the discrete series. Then π may be realized as the unique irreducible quotient of

(9.3)(a)  IndS̃p(2n,ℝ) ; P̃(σ)

containing the (unique) minimal K̃–type μ. Here M̃ ≃ G̃L(2)^{ℓ} × S̃p(2t, ℝ) and σ is a discrete series representation of M̃. In the notation of (6.13), ℓ = ∑_{i} min(x_{i}, y_{i}) and t = ∑_{i} |x_{i} − y_{i}|. This follows from the discussion in §6. As for (9.1)(a), by Theorem 8.8 there is a non–zero map from the tensor product of (9.3)(a) and

(9.3)(b)  IndÕ(p,q) ; P̃′(σ′)

where σ′ is now a discrete series representation of M̃′. The same argument applied with (9.3)(a,b) in place of (9.1)(a,b) proves that the lowest K̃ and K̃′–types of (9.3)(a) and (9.3)(b) correspond. Theorem 5.1 reduces to Theorem 3.3(2) in this case, and holds from the calculations of §6. Inductive Step: Discrete Series.

Theorem 5.1 reduces to Theorem 3.3(1) in this case, so it is enough to prove Theorem 3.3(1). Proof of Theorem 3.3(1).

It is convenient to start on the orthogonal group. So let π′ be in the discrete series representation of Õ(p, q). By Proposition 6.29 and Proposition 2.1, π′ occurs in the representation correspondence, while π′ ⊗ sgn does not. The corresponding representation π of S̃p(2n, ℝ) is also in the discrete series (the representation correspondence is the graph of a bijection, π′ is in the domain and all but the discrete series in the range are accounted for).

Let μ′_{0} be the lowest K̃′–type of π′, and μ_{0} the lowest K̃–type of π. By Proposition 6.28(2), μ′_{0} is of lowest degree and occurs in the space of joint harmonics; let μ = H(μ′). It is enough to show μ = μ_{0}.

We calculate the length of V(μ), the element defined by the Vogan algorithm applied to μ (cf. §6). If λ is the Harish–Chandra parameter of π, then V(μ_{0}) = λ, and (the Weyl group orbit of) λ is the infinitesimal character. The relation between infinitesimal characters is given by the orbit map which preserves lengths (cf. §1), so |λ| = |λ′|, where λ′ is the Harish–Chandra parameter for π′. Therefore

|V(μ_{0})| = |λ|
= |λ′|
= |V(μ′_{0})|

= |O(V(μ′_{0}))| (9.2)

= |V(H(μ′_{0}))| by Proposition 6.1
= |V(μ)|.

Thus μ_{0} and μ are both lambda–lowest K̃–types of π [18,Definition 5.4.1]. Therefore [17,Lemma 8.8] μ and μ_{0} are both lowest K̃–types, and therefore equal. This accounts for all genuine discrete series representations of S̃p(2n, ℝ). This completes the proof of Theorem 3.3(1) as well as Theorem 5.1. □

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