THE THETA CORRESPONDENCE OVER ℝ WORKSHOP AT THE UNIVERSITY OF MARYLAND, MAY 1994

Jeffrey Adams

§1. Fock model: complex Lie algebra.

The Fock model is the algebraic oscillator representation: it is the (𝔤, K̃) module of the oscillator representation of S̃p(2n, ℝ). The basic references are [6,7,41]. See also [15], [10]. The presentation here is due to Steve Kudla (unpublished); a similar treatment is found in [42], as well as [44]. In this section we define the oscillator representation of the complex Heisenberg and symplectic Lie algebras.

Let ψ be a non–trivial linear form on ℂ, i.e. fix λ ∈ ℂ^{∗} and let ψ(z) = λz (z ∈ ℂ).

Definition 1.1.

Let V be a vector space with a symplectic form <, >, possibly degenerate. Let

Ω(V) = T (V)/I

where T is the tensor algebra of V and I is the two–sided ideal generated by elements of the form

v ⊗ w − w ⊗ v − ψ < v, w > v, w ∈ V

This is an associative algebra, sometimes referred to as the quantum algebra. For example if < v, w >= 0 for all v, w, then Ω(V) is isomorphic to the polyno-

mial algebra on V.

Let 𝔥 be the complex Heisenberg Lie algebra of dimension 2n + 1, so 𝔥 = W ⊕ L where W ≃ ℂ^{2n} is a vector space with a non–degenerate symplectic form <, >, and L ≃ ℂ is the center of 𝔥. Write L = ℂ11; the Lie algebra structure of 𝔥 is given by [v, w] =< v, w > 11 for v, w ∈ W. We consider ψ as an element of Hom(L, ℂ) by ψ(λ11) = ψ(λ) (λ ∈ ℂ). Let 𝔘(𝔥) be the universal enveloping algebra of 𝔥.

Lemma 1.2.

Ω(W) ≃ 𝔘(𝔥)/ < 11 − ψ(11) >

where < 11 − ψ(11) > is the two–sided ideal generated by the element 11 − ψ(11).

proof. Define a map 𝔥 → Ω(W) by w → \overline{w} ∈ T (W)/I and 11 → \overline{ψ(11)}. This extends to a map 𝔘(𝔥) → T (W)/I by the universal property of 𝔘(𝔥), and factors to the quotient; it is easily seen to be bijective.

Note: The definition of Ω(W) depends on ψ. Over ℂ the algebras defined by any ψ and ψ′ are isomorphic. The isomorphism takes v ∈ W to λv where λ^{2} = ψ(1)/ψ′(1). Over an arbitrary field F there are |F^{∗}/F^{∗2}| isomorphism classes of such algebras.

Since T (V) = T^{0}(V) ⊕ T^{1}(V) ⊕ … is graded, Ω(V) is filtered, i.e. Ω^{0}(V) ⊂ Ω^{1}(V) ⊂ … where Ω^{k}(V) consists of those elements which may be represented by some X ∈ T^{0}(V) ⊕ T^{1}(V) ⊕ ⋯ ⊕ T^{k}(V). Then if [A, B] = A ⊗ B − B ⊗ A is the anti–commutator bracket in Ω(V), it follows easily that

[Ω^{k}(V), Ω^{ℓ}(V)] ⊂ Ω^{k+ℓ−2}(V).

It follows that Ω^{2}(V) with the bracket [,] is a Lie algebra, and Ω^{1}(V) is an ideal in Ω^{2}(V). Similarly [Ω^{1}(V), Ω^{1}(V)] ⊂ Ω^{0}(V), and Ω^{1}(V) is a two–step nilpotent Lie algebra.

We therefore have the exact sequences of Lie algebras

0 → Ω^{0}(W) ↪ Ω^{1}(W) → Ω^{1}(W)/Ω^{0}(W) → 0 (1.3)(a) 0 → Ω^{1}(W) ↪ Ω^{2}(W) → Ω^{2}(W)/Ω^{1}(W) → 0 (1.3)(b)

In the case of V = W (i.e. <, > non–degenerate) it is easy to see Ω^{1}(W)/Ω^{0}(W) is isomorphic to W (as an abelian Lie algebra), Ω^{1}(W) is isomorphic to 𝔥, and the exact sequence (1.3)(a) splits. In fact something similar holds for (1.3)(b):

Lemma 1.4.

Let 𝔤 ≃ 𝔰𝔭(2n, ℂ) be the Lie algebra of Sp(W) ≃ Sp(2n, ℂ).

(a) Ω^{2}(W)/Ω^{1}(W) is isomorphic to 𝔤.

(b) The exact sequence (1.3)(b) splits, to give an isomorphism

Ω^{2}(W) ≃ 𝔥 ⊕ 𝔤

where the right–hand side is the semi–direct product of Lie algebras, in which 𝔤 acts on 𝔥 by

X · (w, t) = (Xw, t)

for X ∈ 𝔤, w ∈ W, t ∈ L.

proof.

Define an action of Ω^{2}(W) on Ω^{1}(W)/Ω^{0}(W) by X · Y Ω^{0}(W) = [X, Y]Ω^{0}(W). This is defined on Ω^{1}(W) since [Ω^{2}(W), Ω^{1}(W)] ⊂ Ω^{1}(W) and passes to the quotient since [Ω^{2}(W), Ω^{0}(W)] ⊂ Ω^{0}(W) (in fact this bracket is 0). By the isomorphism Ω^{1}(W)/Ω^{0}(W) ≃ W this defines a map Ω^{2}(W) → End(W). If v, w ∈ W write vw for the coset of v ⊗ w ∈ T (W), modulo(I). Explicitly for v_{1}, v_{2}, w ∈ W we have

v_{1}v_{2} · w = v_{1}v_{2}w − wv_{1}v_{2}
= v_{1}v_{2}w − v_{1}wv_{2} − ψ < w, v_{1} > v_{2}
= v_{1}v_{2}w − v_{1}v_{2}w − ψ < w, v_{2} > v_{1}
= ψ < v_{1}, w > v_{2} + ψ < v_{2}, w > v_{1}.

It follows readily that the image of Ω^{2}(W) is in 𝔰𝔭(W). Furthermore this map is injective, and by dimension counting it is an isomorphism, proving (a).

Choose a standard basis e_{1}, …, e_{n}, f_{1}, …, f_{n} of W, i.e. < e_{i}, f_{i} >= 1 and all other brackets are 0. Consider the subspace of Ω^{2}(W) spanned by the following elements. Note that in Ω(W) all the e_{j}s commute among themselves, similarly the f_{j}, and e_{i}f_{j} = f_{j}e_{i} + ψ(δ_{i,j}). Let

X_{i,j}^{+} = f_{i}f_{j} 1 ≤ i ≤ j ≤ n
(1.5)(a)  X_{i,j}^{−} = e_{i}e_{j} 1 ≤ i ≤ j ≤ n
Z_{i,j} = \frac{1}{2}(e_{i}f_{j} + f_{j}e_{i}) 1 ≤ i, j ≤ n

Note that

Z_{i,j} = \frac{1}{2}(f_{j}e_{i} + e_{i}f_{j}) = \frac{1}{2}(f_{j}e_{i} + e_{i}f_{j} − f_{j}e_{i} + f_{j}e_{i})

= f_{j}e_{i} + \frac{1}{2}[e_{i}, f_{j}]

= f_{j}e_{i} + \frac{1}{2}ψ(δ_{i,j}) (1.5)(b)

We claim the elements (1.5)(a) span a Lie subalgebra of Ω^{2} isomorphic to 𝔰𝔭(2n, ℂ) = 𝔤. To see this, recall (cf. [11], for example) 𝔤 has the decomposition 𝔤 = 𝔨 ⊕ 𝔭 = 𝔨 ⊕ 𝔭^{+} ⊕ 𝔭^{−} where 𝔨 ≃ 𝔤𝔩(n, ℂ) is the complexified Lie algebra of a maximal compact subgroup K of Sp(2n, ℝ). The Z_{i,j} span an algebra isomorphic to 𝔨, with a Cartan subalgebra 𝔱 the span of the Z_{i,i}. Let ϵ_{i} denote the element of 𝔱^{∗} satisfying ϵ_{i}(Z_{j,j}) = ψ(δ_{i,j}). Then Z_{j,k} (j ≠ k) is a root vector of weight ϵ_{j} − ϵ_{k}:

[Z_{i,i}, Z_{j,k}] = [f_{i}e_{i} + \frac{1}{2}ψ(1), f_{j}e_{k}] (by (1.5)(b))
= [f_{i}e_{i}, f_{j}e_{k}]
= f_{i}e_{i}f_{j}e_{k} − f_{j}e_{k}f_{i}e_{i}
= f_{i}f_{j}e_{i}e_{k} + f_{i}e_{k}ψ(δ_{i,j}) − f_{j}f_{i}e_{i}e_{k} − f_{j}e_{i}ψ(δ_{i,k})
= ψ(δ_{i,j} − δ_{i,k})f_{j}e_{k}
= (ϵ_{j} − ϵ_{k})(Z_{i,i})Z_{j,k}

Similarly X_{i,j}^{±} is a root vector of weight ±(ϵ_{i} + ϵ_{j}), so the X_{i,j}^{±} span 𝔭^{±}. It follows from similar calculations that [𝔨, 𝔭^{±}] = 𝔭^{±}, [𝔭^{+}, 𝔭^{−}] = [𝔨, 𝔨] = 𝔨, so this is a Lie algebra. From the description of the roots, the root system is of type C_{n}, proving the Lemma.

For later use we note that the Z_{i,j} with i < j together with the X_{i,j}^{+} are positive root vectors of 𝔱 in 𝔤.

Now let W = X⊕Y be a complete polarization of W (this is implicit in our choice of standard basis as in the proof of Lemma 1.4): Thus X and Y are Lagrangian planes, i.e. maximal isotropic subspaces of W. Since <, > restricted to Y is trivial, Ω(Y) ≃ ℂ[Y], the polynomial algebra on Y.

By the Poincare–Birkhoff–Witt theorem [21] Ω(W) = Ω(Y) ⊕ Ω(W)X. Let e_{1}, …, e_{n} be a basis of X, and f_{1}, … f_{n} be a basis of Y. Then this says that any element of Ω(W) may be written as a sum of monomials, where each monomial is in terms of f′_{i}s only, or has some e_{j} on the right. It follows immediately that

(1.6)  Ω(W)/Ω(W)X ≃ Ω(Y) ≃ ℂ[Y].

This carries a representation of Ω(W) by multiplication on the left. By Lemma 1.4 this defines by restriction a representation of 𝔥 and 𝔤, or equivalently 𝔘(𝔥) and 𝔘(𝔤), on ℂ[Y]. The center of 𝔥 acts by the character ψ.

Definition 1.7.

The Fock model of the oscillator representation of 𝔥 ⊕ 𝔤 is the vector space F = ℂ[Y], where Ω(W) acts by multiplication on the left and the isomorphism (1.6), and 𝔤 ⊕ 𝔥 acts via the embedding in Ω(W) (Lemma 1.4(b)).

This definition depends on the choice of ψ and complete polarization W = X⊕Y, and we will write F_{ψ} on occasion. The isomorphism class is independent of these choices. Over an arbitrary field F the number of isomorphism classes is |F^{∗}/F^{∗2}| (see the note following Lemma 1.2).

To be explicit, with the basis e_{i}, f_{j} above, let X (resp. Y) be the span of the e′_{i}s (resp. f′_{j}s). Then F = ℂ[f_{1}, …, f_{n}] is the space of polynomials in the f_{i}. The action of f_{i} ∈ 𝔥 is given by f_{i} · f_{j} = f_{i}f_{j}. To compute the action of e_{j}, in Ω(W) = T (W)/I write

e_{j}f_{i} = e_{j}f_{i} − f_{i}e_{j} + f_{i}e_{j}
= ψ(< e_{j}, f_{i} >) + f_{i}e_{j}
= ψ(δ_{i,j}) + f_{i}e_{j}

Passing to F by the isomorphism (1.6) the term f_{i}e_{j} is zero, and thus

e_{j} · f_{i} = ψ(δ_{i,j}).

Thus to compute the action of the e′_{i}s multiply on the left, then move all e′_{i}s terms past all f′_{j}s to the right using the defining relation of Ω, and finally all terms with e′_{i}s on the right are 0.

Under the isomorphism with a subspace of Ω^{2} the decomposition 𝔤 = 𝔭^{+} ⊕𝔨⊕𝔭^{−} becomes S^{2}(Y)⊕(X ⊗Y)⊕S^{2}(X). Here S^{2}(Y) acts by operators which raise degree by 2, S^{2}(X) lowers degree by 2, and X ⊗ Y acts by degree preserving operators.

This is a version of the canonical commutation relations. To see this we identify ℂ[Y] with the polynomial functions ℂ[z_{1}, …, z_{n}] on Y via the dual variables z_{i}(f_{j}) = δ_{i,j}. Write F = ∑ F^{n} where F^{n} is the polynomials of degree n. Then f_{i} ∈ W ⊂ 𝔥 acts by multiplication by z_{i}, e_{j} acts by ψ(1)\frac{d}{dz_{j}}, and L ⊂ 𝔥 acts by the character ψ. The image of 𝔘(𝔥) is the Weyl algebra of polynomial coefficient differential operators. More precisely, identify X and Y with abelian subalgebras of 𝔥. Then

X → \{\frac{d}{dz_{i}} \}
(1.8)(a)  Y → {z_{i}}

The action of 𝔤 is given by:

𝔭^{+} ≃ S^{2}(Y) → {z_{i}z_{j}}
𝔭^{−} ≃ S^{2}(X) → \{\frac{d^{2}}{dz_{i}dz_{j}} \}
(1.8)(b)  𝔨 ≃ X ⊗ Y → \{\frac{1}{2}(z_{i} \frac{d}{dz_{j}} + \frac{d}{dz_{j}} z_{i}) = z_{i} \frac{d}{dz_{j}} + \frac{1}{2}δ_{i,j}\}

Lemma 1.9.

(1) F is an irreducible representation of 𝔥, with central character ψ. The space of vectors annihilated by X ⊂ 𝔥 is the one–dimensional space of constants.

(2) As a representation of 𝔤, ℂ[Y] is the direct sum of two irreducible representations F_{±} of even and odd functions. The space of vectors in F_{+} (resp. F_{−}) annihilated by S^{2}(X) ⊂ 𝔤 is F^{0} (resp. F^{1}), of dimension 1 (resp. n).

The proof is immediate.

Remark: Suppose V is a vector space with a non–degenerate orthogonal form (,), and let Ω(V) = T (V)/ < v ⊗ w + w ⊗ v − ψ(v, w) > (cf. Definition 1.1). Then Ω is isomorphic to a quotient of the Cayley algebra [?] (compare Lemma 1.2) and this construction produces the spin module of 𝔰𝔬(V). There exists a theory of dual pairs in Spin(V) and the duality correspondence for the spin representation. See [19] [17], [1],.

Remark: The action of 𝔨 ≃ 𝔤𝔩(n, ℂ) on F ≃ ℂ[ℂ^{n}] is by the “standard” action on ℂ^{n}, twisted by a character. Up to a character this is the derived representation of the action of GL(n, ℂ) acting on polynomials by g · f(v) = f(g^{−1}v). This is the connection between the oscillator representation and classical invariant theory [19, 48].

For use in §3 we record a generalization of this construction. Formally the operators (1.8) can be made to act on spaces of functions other than polynomials. For example, it is immediate that for any γ ∈ ℂ the space

(1.10)  {P e^{γ ∑ z_{i}^{2}} | P ∈ ℂ[z_{1}, …, z_{n}]}

is invariant under this action, and so defines a representation of 𝔥 and 𝔤.

§2. Schrödinger model.

See the lectures of Brooks Roberts, and the references there; also see [15,16]. We now let W_{0} be a real vector space of dimension 2n with a non–degenerate

symplectic form <, >. Let Sp(W_{0}) ≃ Sp(2n, ℝ) be the isometry group of this form, and S̃p(2n, ℝ) its connected two–fold cover. Let K ≃ U(n) be a maximal compact subgroup of Sp(2n, ℝ), and K̃ its inverse image in S̃p(2n, ℝ). Then K̃ is connected, and is isomorphic to the det^{\frac{1}{2}} cover of K. In fact it is isomorphic to the subgroup of U(n)×ℂ^{∗} consisting of pairs (g, z) such that det(g) = z^{2}. This has a two–to–one map p : (g, z) → g onto U(n), and a genuine character det^{\frac{1}{2}} : (g, z) → z whose square is det.

Fix a non–trivial unitary additive character ψ of ℝ, and let ω = ω_{ψ} be the corresponding oscillator representation of S̃p(2n, ℝ) on a Hilbert space. If W_{0} = X_{0} ⊕ Y_{0} is a complete polarization of W_{0} we may realize ω on L^{2}(Y_{0}). This is the direct sum of two irreducible representations.

There is a vector Γ in ω_{ψ}, unique up to scalars, which is an eigenvector for the action of K̃. In the usual coordinates this is the Gaussian e^{− \frac{1}{2} ∑ x_{i}^{2}}.

For a ∈ ℝ write a·ψ(x) = ψ(ax). Then the oscillator representations defined by ψ and a^{2}·ψ are isomorphic. Up to isomorphism there are two oscillator representations of S̃p(2n, ℝ).

The K̃–finite vectors ωK̃ in ω are a dense subspace. The action of S̃p(2n, ℝ) lifts to an action of the real Lie algebra 𝔤_{0} = 𝔰𝔭(2n, ℝ) on ωK̃, and this extends by complex linearity to 𝔤 = 𝔰𝔭(2n, ℂ). Furthermore K̃ acts on this space in a way compatible with the action of 𝔤. The action of K̃ is locally finite: the subspace generated by any vector under the action of K̃ is finite dimensional. Finally the the multiplicity of any K̃–type is finite. (See §9, Example 2 for a further discussion of the K̃–structure of ω_{ψ}). This says that the K̃–finite vectors in ω are a (𝔤, K̃)– module [45] [23],.

§3. Fock model: real Lie algebra.

The discussion in section 1 made no reference to the real Lie algebra 𝔰𝔭(2n, ℝ), or (equivalently) the group K. In this section we describe the 𝔤 = 𝔰𝔭(2n, ℂ) module arising as the (𝔤, K̃)–module of the oscillator representation ω of S̃p(2n, ℝ).

Let W_{0} and Sp(W_{0}) ≃ Sp(2n, ℝ) be as in §2. Let 𝔤_{0} = 𝔰𝔭(W_{0}) ≃ 𝔰𝔭(2n, ℝ) be the real Lie algebra of G. Let W = W_{0} ⊗_{ℝ} ℂ, and extend <, > to W by linearity. We then obtain Sp(W) ≃ Sp(2n, ℂ) and 𝔤 = 𝔰𝔭(W) ≃ 𝔰𝔭(2n, ℂ) as in §1. Given a complete polarization W = X ⊕ Y of W and ψ : ℂ → ℂ we obtain the oscillator representation of 𝔤 on F = ℂ[Y]. The isomorphism class of F as a representation of 𝔤 is independent of the choice of complete polarization. However as a representation of 𝔤_{0} this depends on the position of this polarization with respect to the real structure. Totally complex polarization: We assume first that X is a totally complex subspace of W, i.e. X ∩ W_{0} = 0. See [7], also [32]. The choice of such X is equivalent to the choice of a compatible complex structure J on W_{0}: this means J ∈ Sp(W_{0}) satisfying J^{2} = −1. Given such J it extends by complex linearity to W, and we let X be the i–eigenspace of J. Conversely given X there exists unique J ∈ Sp(2n, ℝ) such that J|_{X} = iId. Note that Y = \overline{X} is the −i–eigenspace of J, and W = X ⊕ Y is a complete polarization of W.

Now J makes W_{0} a complex vector space of dimension n (with multiplication by i given by J). We denote this space W_{J}: this is a complex vector space of dimension n equal as a set to W_{0}. Furthermore {v, w} =< v, Jw > −i < v, w > is a non–degenerate Hermitian form on W_{J}. We assume this form is positive definite, i.e. of signature (n, 0).

To be explicit, if e_{1}, …, f_{n} is a standard basis of W_{0}, we can take X =< e_{1} +

if_{1}, …, e_{n} + if_{n} >, and then J = ( \begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix} ). That is Je_{i} = −f_{i}, Jf_{i} = e_{i}.

More generally we could take X spanned by some e_{i} + if_{i} and some e_{j} − if_{j}. The signature of {, } is then (p, q) where p (resp. q) is the number of plus (resp. minus) signs.

Let K = {g ∈ Sp(W_{0})|gJ = Jg}. Equivalently {gv, gw} = {v, w} for all v, w ∈ W_{J}, so K ≃ U(n, 0), and K is a maximal compact subgroup of Sp(W_{0}). Furthermore this is equivalent to the condition gX = X, gY = Y, i.e. K is the stabilizer of the complete polarization. The corresponding complexified Lie algebra 𝔨 is isomorphic to 𝔤𝔩(n, ℂ), and its intersection 𝔨_{0} with 𝔤_{0} is isomorphic to 𝔲(n).

As in §1 consider the Fock model of the oscillator representation on ℂ[Y]. Then 𝔨 ≃ X ⊗ Y acts by operators which preserve degree (cf. §1), and therefore 𝔨 acts locally finitely. The action of 𝔨_{0} exponentiates to a certain two–fold cover K̃ of K (see §2). It follows that ℂ[Y] is a (𝔤, K̃)–module. We have 𝔭^{+} ≃ S^{2}(Y) acts by operators of degree 2, and 𝔭^{−} ≃ S^{2}(X) by operators of degree −2. Real polarization: At the opposite extreme we could take a complete polarization of W to be the complexification of one of W_{0}. To distinguish this case from the previous one we write W_{0} = X_{0}⊕Y_{0} and W = X ⊕Y, so X = X_{0}⊗_{ℝ}ℂ, X_{0} = X ∩W_{0}, and similarly for Y. We can then define the representation of 𝔤 on F = ℂ[Y]. The subalgebra of 𝔤, i.e. X ⊗Y which acts locally finitely is again isomorphic to 𝔤𝔩(n, ℂ), but the intersection with 𝔤_{0} is then 𝔤𝔩(n, ℝ). Assume exp(ψ) : ℝ → ℂ^{∗} is unitary, i.e. ψ(z) = λz for λ ∈ iℝ.

This is essentially the Schrödinger model. Recall (§2) in the Schrödinger model the Gaussian Γ = e^{− \frac{1}{2} ∑ x_{i}^{2}} is an eigenvector for the action of K̃, or equivalently 𝔨.

In the Fock model on ℂ[Y], the constants are the unique eigenspace for 𝔨 ≃ X⊗Y. The constants are also characterized as the kernel of the action of X ⊂ 𝔥. Therefore we look for a vector in the Schrödinger model on ℂ[Y] annihilated by X. Choose a basis of Y with dual coordinates z_{1}, …, z_{n} so that ℂ[Y] ≃ ℂ[z_{1}, …, z_{n}]. Recall (1.10) we can extend the action on ℂ[z_{1}, …, z_{n}] to polynomials times a Gaussian.

Lemma 3.1.

(1) There is a unique value of γ for which the Gaussian

Γ = e^{−γ \frac{1}{2} ∑ z_{i}^{2}}

is annihilated by X ⊂ 𝔥.

(2) The space

ℂ[Y]Γ = {P e^{γ ∑ z_{i}^{2}} | P ∈ ℂ[z_{1}, …, z_{n}]}

is a (𝔤, K̃)–module,

(3) ℂ[Y] and ℂ[Y]Γ are isomorphic as 𝔥 and as (𝔤, K̃)–modules.

(4) Given Y we can choose Y (totally complex) so that

Γ = e^{− \frac{1}{2} ∑ z_{i}^{2}}.

Then ℂ[Y] is isomorphic to the space of K̃–finite vectors L^{2}(Y_{0})K̃ in the Schrödinger model L^{2}(Y_{0}) of the oscillator representation.

sketch of proof. This essentially reduces to SL(2). Let e, f be a standard basis of W_{0}, and let X = ℂ < e >, Y = ℂ < f >. Let X = ℂ < e+iνf >, Y = ℂ < e−iνf > for some 0 ≠ ν ∈ ℂ − iℝ. Let λ = ψ(1). With dual coordinate z on Y satisfying z(gf) = 1 we have e ∈ X acts on ℂ[Y] by λ\frac{d}{dz}, and f ∈ Y acts by multiplication by z. Therefore e + νif acts by τ = iνz + λ\frac{d}{dz}, and τ · e^{γz^{2}} = (2γλ + iν)ze^{γz^{2}}. Thus taking

γ = \frac{−iν}{2λ}

gives τ · e^{γz^{2}} = 0. It is immediate that γ is determined uniquely. This proves (1), and (2) follows readily from this.

It is an easy argument by induction on degree that ℂ[Y]Γ is irreducible as a representation of 𝔥. Since the same holds for ℂ[Y], and the central characters agree, by the Stone–von Neumann theorem these are isomorphic as representations of 𝔥, and (3) follows immediately from this.

For (4) choose ν = −iλ, so γ = −\frac{1}{2} and Γ has the stated form. Finally restricting the functions in ℂ[Y]Γ to Y_{0} gives functions in L^{2}(Y_{0}), and this gives the isomorphism of (4).

Remark: Suppose X =< e_{1}, …, e_{n} >, Y =< f_{1}, …, f_{n} > as usual, and X =< e_{1} + if_{1}, …, e_{n} + if_{n} >, Y =< e_{1} − if_{1}, …, e_{n} − if_{n} >. Also assume ψ(1) = i and ν = 1. Then K is the usual maximal compact subgroup of Sp(2n, ℝ), and this is how the Fock/Schrödinger models are usually stated.

Remark: As far as the actions of the Lie algebras are concerned, we see that both the Fock and Schrödinger models are examples of the construction of §1, extended to polynomials times the Gaussian in the latter case. However the full Hilbert spaces in the two cases look somewhat different. In the Fock model this consists of holomorphic functions on Y which are square integrable with respect to a certain measure [7], whereas in the Schrödinger model this is L^{2}(Y_{0}). The existence of these two isomorphic models accounts for the simultaneously analytic and algebraic nature of the oscillator representations.

This duality is nicely illustrated by looking at the operators in the two models in the case of SL(2). We use the notation of the proof of Lemma 3.1, with λ = i and ν = 1. We write 𝔰𝔩(2, ℂ) = 𝔨 ⊕ 𝔭^{+} ⊕ 𝔭^{−}, or alternatively as = 𝔪 ⊕ 𝔫 ⊕ \overline{𝔫}.

Fock Model:

Here the space of K̃–finite vectors in the oscillator representation is F = ℂ[Y] = ℂ[z].

𝔨 ⊕ 𝔭 acts by:

( \begin{matrix}0 & i \\ −i & 0 \end{matrix} ) ∈ 𝔨 → −z \frac{d}{dz} − \frac{1}{2}
( \begin{matrix}1 & i \\ i & −1 \end{matrix} ) ∈ 𝔭^{+} → z^{2}
( \begin{matrix}1 & −i \\ −i & −1 \end{matrix} ) ∈ 𝔭^{−} → −\frac{d^{2}}{dz^{2}}

And the operators from 𝔪 ⊕ 𝔫 ⊕ \overline{𝔫}:

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ∈ 𝔪 → \frac{1}{2}(z^{2} − \frac{d^{2}}{dz^{2}} )
( \begin{matrix}0 & 4i \\ 0 & 0 \end{matrix} ) ∈ 𝔫 → −2z \frac{d}{dz} − 1 + z^{2} + \frac{d^{2}}{dz^{2}}
\binom{0}{−4i 0} ∈ \overline{𝔫} → −2z \frac{d}{dz} − 1 − z^{2} − \frac{d^{2}}{dz^{2}}

Schrödinger Model:

Now the K̃–finite vectors are polynomials in x times the Gaussian e^{− \frac{1}{2} x^{2}}. Here the formulas for 𝔪 ⊕ 𝔫 ⊕ \overline{𝔫} are easy:

( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) ∈ 𝔪 → x\frac{d}{dx} + \frac{1}{2}
( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ∈ 𝔫 → \frac{i}{2}x^{2}
( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) ∈ \overline{𝔫} → \frac{i}{2} \frac{d}{dx^{2}}

whereas 𝔨 ⊕ 𝔭 acts by:

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) ∈ 𝔨 → \frac{i}{2}(x^{2} − \frac{d^{2}}{dx^{2}} )
( \begin{matrix}1 & i \\ i & −1 \end{matrix} ) ∈ 𝔭^{+} → (x\frac{d}{dx} + \frac{1}{2}) − \frac{1}{2}x^{2} − \frac{1}{2} \frac{d^{2}}{dx^{2}}
( \begin{matrix}1 & −i \\ −i & −1 \end{matrix} ) ∈ 𝔭^{−} → (x\frac{d}{dx} + \frac{1}{2}) + \frac{1}{2}x^{2} + \frac{1}{2} \frac{d^{2}}{dx^{2}}

We look at the isomorphism between the Fock and Schrödinger models more closely. It takes the constants in the Fock model to the Gaussian in the Schrödinger model. It intertwines the actions of Y, for which these vectors are cyclic, and the isomorphism is determined by these conditions.

Consider the example of SL(2) in the proof of Lemma 3.1, and the action of e − νif ∈ Y on ℂ[Y] and ℂ[Y]Γ; the isomorphism intertwines these actions. With ν = −iλ let z dual to e − λf be the coordinate on Y and x dual to f on Y. We conclude multiplication by z goes to λ(\frac{d}{dx} − x) on ℂ[Y]Γ. Thus the isomorphism takes

z^{n} → λ^{n}(\frac{d}{dx} − x)^{n} · e^{− \frac{1}{2} x^{2}}.

Note that (\frac{d}{dx} − x)(x^{n}Γ) = (−2x^{n+1} + lower order terms)Γ. It follows that this isomorphism takes

z^{n} → p_{n}(x)e^{− \frac{1}{2} x^{2}}

where p_{n}(z) = ((−2λz)^{n} + lower order terms)e^{− \frac{1}{2} z^{2}}. The following Lemma follows readily from this discussion.

Lemma 3.2.

Given ψ and W_{0} = X_{0} ⊕ Y_{0}, let L^{2}(Y_{0}) be the corresponding oscillator representation, with K̃–finite vectors L^{2}(Y_{0})K̃. Let W = X ⊕ Y be as in Lemma 3.1(4) so that there is an isomorphism of 𝔥 and (𝔤, K̃)–modules

ϕ : ℂ[Y] → L^{2}(Y_{0})K̃

Then we can choose coordinates z_{1}, …, z_{n} on Y and x_{1}, …, x_{n} on Y_{0} such that if P (z_{1}, …, z_{n}) is a polynomial of degree d then

ϕ(P) = (c_{d}P (x_{1}, …, x_{n}) + Q)Γ

where Q is a polynomial of degree strictly less than d, and c_{d} is a constant depending only on d.

In fact ϕ(z_{1}^{a_{1}} … z_{n}^{a_{n}}) = (cx_{1}^{a_{1}} … x_{n}^{a_{n}} + Q)Γ and up to a constant cx_{1}^{a_{1}} … x_{n}^{a_{n}} + Q is a Hermite polynomial [20].

§4. Duality.

The results in this section are due to R. Howe [18].

We begin by listing the irreducible reductive dual pairs over ℝ. This follows the general scheme of [14], and is spelled out in this case in [18]. See the lectures of Brooks Roberts for more details.

The irreducible pairs of type I, together with the corresponding division algebra and involution are:

(1) (O(p, q), Sp(2n, ℝ)) ⊂ Sp(2n(p + q), ℝ), (ℝ, 1)

(2) (U(p, q), U(r, s)) ⊂ Sp(2(p + q)(r + s), ℝ) (ℂ, )

(3) (Sp(p, q), O^{∗}(2n)) ⊂ Sp(4n(p + q), ℝ) (ℍ, )

(4) (O(m, ℂ), Sp(2n, ℂ)) ⊂ Sp(4mn, ℝ) (ℂ, 1)

The type II dual pairs are

(1) (GL(m, ℝ), GL(n, ℝ)) ⊂ Sp(2mn, ℝ)

(2) (GL(m, ℂ), GL(n, ℂ)) ⊂ Sp(4mn, ℝ)

(3) (GL(m, ℍ), GL(n, ℍ)) ⊂ Sp(8mn, ℝ)

Notation is as in [11]. The group GL(m, ℍ) is sometimes called U^{∗}(2m) and is a real form of GL(2m, ℂ).

Now let (G, G′) be a reductive dual pair in Sp(2n, ℝ). Fix a non–trivial unitary additive character ψ of ℝ, and let ω_{ψ} be the corresponding oscillator representation. Fix a maximal compact subgroup 𝕂 ≃ U(n) of Sp(2n, ℝ) and let F_{ψ} ≃ ℂ[z_{1}, …, z_{n}] be the corresponding Fock model.

We may assume K = G ∩ 𝕂 and K′ = G′ ∩ 𝕂 are maximal compact subgroups of G, G′. Then F is a (𝔤 ⊕ 𝔤′, K̃ · K̃′) module.

Remark 4.1 For H a subgroup of Sp(W), H̃ will always denote the inverse image of H in S̃p(W), unless otherwise stated. For (G, G′) a dual pair G̃ commutes with

G̃′, and so { \atop G · G′} = G̃ · G̃′.

We denote an outer tensor product of representations π_{i} of G_{i} as π_{1} ˇ⊗π_{2}. This is a representation of G_{1} × G_{2}. In the case G = G_{1} = G_{2} the restriction to the diagonal subgroup G is the ordinary tensor product.

We will often abuse notation and write π ˇ⊗π′ for a representation of G̃ · G̃′. Here π (resp. π′) is a genuine representation of G̃ (resp. G̃′) and π ˇ⊗π′ is a representation of G̃ × G̃′. There is a surjective map G̃ × G̃′ → G̃ · G̃′, and π ˇ⊗π′ is trivial on the kernel, so it factors to a genuine representation of G̃ · G̃′. If G ∩ G′ = 1 then this map is two–to–one.

Definition 4.2.

R(𝔤, K̃, ψ) is the set of isomorphism classes of irreducible (𝔤, K̃) modules π such that there exists a non–zero (𝔤, K̃)–module map (necessarily a surjection) F → π.

Theorem 4.3. If π ∈ R(𝔤, K̃, ψ), there exists a unique π′ ∈ R(𝔤′, K̃′, ψ) such that

Hom_{𝔤⊕𝔤′,K̃·K̃′}(F, π ˇ⊗π′) ≠ 0.

In fact this Hom space is one–dimensional.

Definition 4.4. We define a correspondence

R(𝔤, K̃, ψ) ↔ R(𝔤′, K̃′, ψ)

by the condition π ↔ π′ if

Hom_{𝔤⊕𝔤′,K̃·K̃′}(ω_{ψ}, π ˇ⊗π′) ≠ 0.

By Theorem 4.3 this is a bijection. We refer to this as the duality correspondence or (local) theta–correspondence, and we write π′ = θ(ψ)(π) and π = θ(ψ)(π′).

The duality correspondence is a bijection between certain (𝔤, K̃) and (𝔤′, K̃′) modules. By the standard theory of (𝔤, K)–modules a module π in R(𝔤, K̃, ψ) corresponds to a genuine representation of G̃. Let G̃_{genuine} denote the irreducible admissible genuine representations of G̃. Then the duality correspondence defines a bijection between subsets of G̃_{genuine} and G̃′_{genuine}.

Given π ∈ R(𝔤, K̃, ψ) let N (π) be the intersection of the kernels of all ϕ ∈ Hom_{(𝔤,K̃)}(F_{ψ}, π). Then N (π) is stable under the action of (𝔤 ⊕ 𝔤′, K̃ · K̃′), and F_{ψ}/N (π) is isomorphic to π ˇ⊗π′_{1} for some (𝔤′, K̃′) module π′_{1}. The preceding Theorem follows immediately from

Theorem 4.5. The module π′_{1} is of finite length, and has a unique irreducible quotient.

Then π′ is the unique irreducible quotient of π′_{1}.

Theorem 4.3 and 4.5 have smooth and unitary analogues, i.e. with F replaced by the unitary representation of S̃p(2n, ℝ) on a Hilbert space H, or on the smooth vectors H^{∞} of H. See [18].

Remark We say a two–fold covering p : G̃ → G splits if the exact sequence

(4.6)  1 → ℤ/2ℤ → G̃ → G → 1

splits. All of our extensions will be central, i.e. the kernel of the covering map is contained in the center of G̃, so equivalently G̃ ≃ G × ℤ/2ℤ. The choice of splitting is equivalent to the choice of this isomorphism. Given such a choice there is a bijection between the irreducible genuine representations of G̃ and the irreducible representations of G. This bijection is realized by tensoring with the distinguished character ζ = 11 ˇ⊗sgn of G × ℤ/2ℤ.

More generally even if (4.6) is not split, there may exist a genuine character ζ of G̃. For example this is the case if G̃ is the det^{\frac{1}{2}} cover of GL(n) or a subgroup such as O(n) or U(n) (cf. §2). In fact this is the case for all the groups under consideration except the metaplectic group S̃p(2n, ℝ). Tensoring with ζ then determines a bijection as in the split case.

Thus, given some choices, in the duality correspondence between G̃_{genuine} and G̃′_{genuine} in some cases we can replace G̃_{genuine} by G , and similarly G̃′.

Open problem: Explicitly compute the duality correspondence of Definition 4.4.

Here “explicitly” has several possible meanings. In some sense the best solution is to compute the bijection in terms of some parameterization of representations, such as the Langlands classification or variations of it. However other descriptions are possible.

Question: How does the duality correspondence behave with respect to unitarity?

In the stable range θ(ψ) preserves unitarity going from the smaller to the larger group: if π is unitary then so is θ(ψ)(π) [27]. Very little is known in general. It takes Hermitian representations to Hermitian representations [34].

§5. Compact dual pairs.

In this section we consider the duality correspondence when one member G or G′ of the dual pair (G, G′) is compact. We refer to such a dual pair as a compact dual pair. For the general dual pair (G, G′) the maximal compact subgroup K is a member of a compact dual pair (K, M′), and the duality correspondence for (K, M′) plays a fundamental role in determining that for (G, G′).

The duality correspondence for compact dual pairs is substantially simpler than the general case. The representations of the non–compact member of the dual pair are all (unitary) highest weight modules, and hence determined by their highest weight. (In fact all unitary highest weight modules of the classical groups arise this way [8].) An explicit computation of most cases is found in [22], and a closely related discussion is in [19]. In fact a number of such cases, treated in an ad hoc manner, provided some of the initial evidence for Howe’s formalism and duality conjecture [9,11,26,39].

Let (K, G′) be a dual pair in Sp(2n, ℝ) with K compact. Let 𝕂, ψ and F be as in §4. We assume K, K′ ⊂ 𝕂 where K′ is a maximal compact subgroup of G. Recall (§1) the complexified Lie algebra of Sp(2n, ℝ) has the decomposition 𝔭^{−} ⊕ 𝔨 ⊕ 𝔭^{+}.

The complexified Lie algebra 𝔤′ of G′ has the decomposition 𝔤′ = 𝔭^{′,+} ⊕ 𝔨′ ⊕ 𝔭^{′,−} given by intersection with the corresponding decomposition of 𝔰𝔭(2n, ℂ).

Definition 5.1. Let

H(K) = {P ∈ F | X · P = 0 ∀X ∈ 𝔭^{′,−}},

the space of K–harmonics.

Since the action of 𝔭^{′,−} commutes with the action of K̃, H(K) is clearly K̃– invariant.

Recall that K̃ preserves the degree of polynomials in F, so the polynomials F^{n} of degree n are a K̃–stable subspace.

Definition 5.2.

Let σ be an irreducible representation of K̃. The degree d(σ) of σ is defined to be the minimal degree d such that the σ isotypic component F_{σ}^{d} of F^{d} is non–zero, or ∞ if F_{σ} is empty.

If d(σ) ≠ ∞, consider F_{σ}, which is invariant by K̃ and 𝔤′.

Theorem 5.3. [19]

(1) F_{σ} is irreducible as a representation of K̃ × 𝔤′. In particular F_{σ} ≃ σ ⊗ π′ for some irreducible (𝔤′, K̃′)–module π′.

(2) The correspondence σ ↔ π′ is a bijection.

(3) The polynomials of lowest degree (=d(σ)) in F_{σ} are precisely H(K)_{σ}.

(4) F_{σ} = 𝔘(𝔭^{′,+}) · H(K)_{σ}

(5) H(K)_{σ} is irreducible as a representation of K̃ × 𝔨′, so H(K)_{σ} ≃ σ ⊗ τ′ for some irreducible representation τ′ of 𝔨′.

(6) The correspondence σ ↔ τ′ is a bijection.

In other words

F ≃ ∑ σ ⊗ π′

as a representation of K̃ × 𝔤′, and σ ↔ π′ is a bijection. This is a special case of Theorem 4.3. Furthermore each π′ contains a unique K̃′–type τ′ of lowest degree, which generates π′ under the action of 𝔘(𝔭^{′,+}), and finally the correspondence σ ↔ τ′ is also a bijection.

Thus π′ is a highest weight module, in that it has a cyclic vector under the action of the Borel subalgebra 𝔟 = 𝔟_{𝔨′} ⊕ 𝔭^{′,+}, where 𝔟_{𝔨′} is a Borel subalgebra of 𝔨′ [21]. Such a module is completely determined by its highest weight, which is the highest weight of τ′ as a representation of 𝔨′.

Note that K̃ may be a non–trivial covering group of K, and K̃ as well as K may be disconnected. However except for these technicalities, the correspondence σ ↔ π′ is explicitly described by the correspondence of highest weights of σ for K̃ and τ′ for 𝔨′. We proceed to do this for all irreducible compact dual pairs.

The irreducible compact dual pairs are the following.

(1) (O(p), Sp(2n, ℝ)) ⊂ Sp(2pn, ℝ),

(2) (U(p), U(m, n)) ⊂ Sp(2p(m + n), ℝ),

(3) (Sp(p), O^{∗}(2n)) ⊂ Sp(4pn, ℝ)

(An exceptional case is (Sp(p, q), O^{∗}(2)) with pq ≠ 0. Here O^{∗}(2) ≃ U(1) is compact, but Theorem 5.3 does not apply, because the decomposition preceding Definition 5.1 does not hold.)

I. (O(p), Sp(2n, ℝ)).

The irreducible representations of SO(p) are parameterized by highest weights (a_{1}, …, a_{[ \frac{p}{2} ]}) with a_{i} ∈ ℤ and a_{1} ≥ a_{2} ⋯ ≥ a_{[ \frac{p}{2} ]−1} ≥ |a_{[ \frac{p}{2} ]}|. Furthermore if p is odd we may assume a_{[ \frac{p}{2} ]} ≥ 0.

Following Weyl [48] we parameterize representations of O(p) by restriction from U(p). Consider the irreducible representations of U(p) given by Young’s diagrams with p rows of length a_{1}, …, a_{p}, or equivalently with highest weight (a_{1}, …, a_{p}) with a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{p} ≥ 0. We embed O(p) in U(p) in the usual way, i.e O(p) = U(p) ∩ GL(p, ℝ).

Lemma 5.4. [48]

The irreducible representations of O(p) are parametrized by Young diagrams Y with rows of length a_{1} ≥ a_{2} ≥ … a_{p} ≥ 0 such that the sum of the lengths of the first two columns is less than or equal to p. The representation of O(p) defined by Y is defined to be the irreducible summand of the representation of U(p) defined by Y which contains the highest weight vector.

Equivalently the highest weights of the representations of U(p) are of the form

(5.5)(a)  (a_{1}, …, a_{k}, 0, …, 0) a_{k} > 0, k ≤ [\frac{p}{2}]

or

(5.5)(b)  (a_{1}, …, a_{k}, \overbrace{1, …, 1}^{ℓ}, 0, …, 0), a_{k} ≥ 1, 2k + ℓ ≤ p.

In (a) the length of the first column is less than or equal to [\frac{p}{2}], and we say this representation has “highest weight” (a_{1}, …, a_{k}, 0, …, 0; 1). In (b) the length of the first column is greater than [\frac{p}{2}] and we denote the “highest weight” of this representation (a_{1}, …, a_{k}, 0, …, 0; −1)).

For example the sgn representation of O(p) is the restriction of the determinant representation of U(p), which has highest weight (1, …, 1); the corresponding “highest weight” of O(p) is (0, …, 0; −1).

The representations given by (a_{1}, …, a_{k}, 0, …, ϵ) with ϵ = ±1 differ by tensoring with sgn. These two representations are not isomorphic to one another unless p is even and a_{\frac{p}{2}} ≠ 0.

The genuine irreducible representations of the det^{\frac{1}{2}} cover of U(n) are parametrized by highest weights (a_{1}, …, a_{n}) with a_{i} ∈ ℤ + \frac{1}{2} and a_{1} ≥ a_{2} … a_{n}.

Now the inverse image K̃ of K = O(p) in the metaplectic group is isomorphic to the cover defined by det^{\frac{n}{2}} , and is therefore trivial over the connected component SO(p), and also over O(p) if and only if n is even. In any event Õ(p) has a genuine character ζ, and we identify genuine representations of Õ(p) with representations of O(p) by tensoring with ζ. There are two choices of ζ if n is odd. The covering S̃p(2n, ℝ) of Sp(2n, ℝ) is split if and only if p is even, otherwise it is the metaplectic group. The maximal compact subgroup K′ of Sp(2n, ℝ) is isomorphic to U(n). There is a choice of this isomorphism, which will be specified below. Then K̃′ is the split cover of U(n) if p is even, or is isomorphic to the det^{\frac{1}{2}} cover Ũ(n) if n is odd. Since this is connected it is equivalent to work on the Lie algebra 𝔨′.

The duality correspondence in this case is a correspondence between genuine irreducible representations of K̃ and K̃′, or equivalently 𝔨′. We identify genuine representations of K̃ with representations of O(p), and those of K̃′ with Ũ(n) or 𝔤𝔩(n, ℂ). The choices involved are determined by the following convention.

Normalization: We choose an isomorphism of K′ with U(n) so that the constants F^{0} of F are considered the irreducible representation of Ũ(n) or 𝔤𝔩(n, ℂ) with highest weight (\frac{p}{2}, …, \frac{p}{2}).

Let ζ be the character by which K̃ acts on F^{0}. The correspondence σ → σ ⊗ ζ defines a bijection between the irreducible representations of O(p) and the genuine irreducible representations of K̃.

We thus describe the duality correspondence σ ↔ τ′ in terms of representations of O(p) and Ũ(n) or 𝔤𝔩(n, ℂ).

This normalization depends on ψ. For example it says that the highest weights of the oscillator representation defined by ψ are (\frac{1}{2}, …, \frac{1}{2}) and (\frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}).

Proposition 5.5. The duality correspondence for the dual pair (O(p), Sp(2n, ℝ)) is given by the correspondence of highest weights σ ↔ τ′:

O(p) : σ = (a_{1}, …, a_{k}, 0, …, 0; ϵ)
Sp(2n, ℝ) : τ′ = (a_{1} + \frac{p}{2}, …, a_{k} + \frac{p}{2}, \overbrace{\frac{p}{2} + 1, …, \frac{p}{2} + 1}^{\frac{1−ϵ}{2} (p−2k)}, \frac{p}{2}, …, \frac{p}{2})

All such highest weights occur, subject to the constraints k ≤ [\frac{p}{2}] and k + \frac{1−ϵ}{2}(p − 2k) ≤ n.

This means that the weight σ for O(p) is the highest weight of the irreducible representation σ, and the weight for Sp(2n, ℝ) is the highest weight of the K̃′–type of τ′ of lowest degree in π′.

II. (U(p), U(m, n))

The inverse image K̃ of U(p) in S̃p(2p(m + n), ℝ) is isomorphic to the cover defined by the character det^{\frac{m+n}{2}} . The cover of U(m, n) is the det^{\frac{p}{2}} cover. Therefore the correspondence σ ↔ τ′ is described in terms of highest weights for unitary groups or their two–fold covers, given by integers or half–integers respectively.

We assume the isomorphisms of the covers of U(p) and U(m, n) with the det^{∗} covers have been chosen so the constants F^{0} have weight (\frac{m−n}{2}, …, \frac{m−n}{2}) for K̃, and weight (\frac{p}{2}, …, \frac{p}{2}) ⊗ (−\frac{p}{2}, …, −\frac{p}{2}) for K̃′.

Proposition 5.6. The duality correspondence for (U(p), U(m, n)) is given by

U(p) : σ = (a_{1}, …, a_{k}, 0, …, 0, b_{1}, …, b_{ℓ}) + (\frac{m − n}{2}, …, \frac{m − n}{2}) →
U(m, n) : τ′ = (a_{1}, …, a_{k}, 0, …, 0) ⊗ (0, …, 0, b_{1}, …, b_{ℓ})

+ (\frac{p}{2}, …, \frac{p}{2}) ⊗ (−\frac{p}{2}, …, −\frac{p}{2}).

All such weights occur, subject to the obvious constraints k + ℓ ≤ p, k ≤ m, ℓ ≤ n.

III. (Sp(p), O^{∗}(2n)).

Both Sp(p) and O^{∗}(2n) are connected and simply connected, so the coverings are all split, and we consider only the linear groups. The maximal compact subgroup of O^{∗}(2n) is isomorphic to U(n).

Sp(p) : σ = (a_{1}, …, a_{k}, 0, …, 0) →
O^{∗}(2n)) : τ′ = (a_{1} + p, …, a_{k} + p, p, …, p)

All such weights occur, subject to the obvious constraints k ≤ p, n.

§6. Joint harmonics.

The proof of the results in §4 depends heavily on information about K–types. This information is also important for computing the correspondence. The results in this section are all from [18].

We continue with the setup of §4. The maximal compact subgroup K of G is itself a member of a dual pair (K, M′) with K ⊂ G and M′ ⊃ G′ (this is an example of a see–saw dual pair [24]) and we may apply the machinery of §5. The same applies mutatis mutandis to (M, K′).

Definition 6.1.

The joint harmonics H is the space of polynomials which are both K–harmonic and K′–harmonic. That is

H = H(K) ∩ H(K′).

This is a K̃ · K̃′ invariant subspace of F.

Let R(K, H) be the isomorphism classes of irreducible K̃–modules σ such that H_{σ} ≠ 0, and define R(K′, H) similarly. As in the definition of the duality correspondence, we say σ ∈ R(K, H) corresponds to σ′ ∈ R(K′, H) if σ ˇ⊗σ′ is a direct summand of H.

Definition 6.2. Suppose π ∈ R(𝔤, K, ψ) and σ is a K̃–type occurring in π, i.e. π_{σ} ≠ 0. Then we say σ is of minimal degree in π if d(σ) is minimal among the degrees of all K̃–types of π.

Since d(σ) ≥ 0, it is immediate that the set of K̃–types of π of minimal degree is non–empty. The notion of K̃–type of π of minimal degree depends on the dual pair, and so is not an intrinsic notion to 𝔤. The relationship between the K̃–types of minimal degree and those which are minimal in the sense of Vogan [45] is a subtle and important one.

The following structural result is a key to the proof of Theorem 4.3, as well as to an explicit understanding of the duality correspondence.

Theorem 6.3.

(1) The correspondence σ → σ′ defined on H is a bijection

R(K, H) ↔ R(K′, H).

We write σ′ = θ(ψ, H)(σ).

(2) H generates F as a representation of 𝔤 ⊕ 𝔤′, i.e. 𝔘(𝔤 ⊕ 𝔤′)H = F.

(3) Suppose π ∈ R(𝔤, K̃, ψ) and σ is a K̃–type of π of minimal degree (Definition 6.2). Then σ ∈ R(K, H).

(4) In the setting of (3), let π′ = θ(ψ)(π), and let σ′ = θ(ψ, H)(σ). Then σ′ is a K̃′–type of π′ of lowest degree.

Thus the duality correspondence θ comes equipped with a correspondence θ(ψ, H) of K̃, K̃′–types of lowest degree. It is therefore important to understand θ(ψ, H). From the results of §5 this may be computed explicitly, as we now describe.

We continue to work with our dual pair (G, G′) together with two auxiliary pairs (K, M′) and (M, K′). We also consider the maximal compact subgroups K_{M} and K_{M′} of M and M′; (K_{M}, K_{M′}) is a dual pair with both members compact. We have the diagram:

(6.4)  K_{M} \begin{matrix}↗ \\ ↖ \end{matrix} \begin{matrix}M \\ ↑ \\ G \\ ↑ \\ K \end{matrix} \begin{matrix}⟷ \\ ⟷ \\ ⟷ \end{matrix} \begin{matrix}K′ \\ ↓ \\ G′ \\ ↓ \\ M′ \end{matrix} \begin{matrix}↘ \\ ↙ \end{matrix} K_{M′}

In the presence of several dual pairs we will write θ(ψ, K, M′) for the duality correspondence between representations of K̃ and M̃′ via the dual pair (K, M′), and others similarly.

Lemma 6.5.

Let σ be a K̃–type of degree d. Define:

(1) π′ = θ(ψ, K, M′)(σ),

(2) τ′ the unique K̃_{M′}–type of π′ of degree d,

(3) τ = θ(ψ, K_{M′}, K_{M})(τ′).

Then

(1) Suppose τ is the K̃_{M}–type of lowest degree in a representation π of M with σ′ = θ(ψ, M, K′)(π) ≠ 0. Then σ ˇ⊗σ′ occurs in H, i.e. θ(ψ, H)(σ) = σ′. The degrees of σ, σ′, τ and τ′ are all d. Furthermore all other K̃′–types of τ′ restricted to K̃′ have degree strictly less than d.

(2) Suppose condition (1) fails. Then σ does not occur in H, i.e. θ(ψ, σ) = 0, and all K̃′–types of τ′ restricted to K̃′ have degree strictly less than d.

This is illustrated by the following diagram.

\begin{matrix}K \\ σ \end{matrix} \underrightarrow{⟶^{θ(K,M′)}} \begin{matrix}M′ \\ π′ \end{matrix} \underrightarrow{⟶^{min. deg.}} \begin{matrix}K_{M′} \\ τ′ \end{matrix} \underrightarrow{⟶^{θ(K_{M′} ,K_{M})}} \begin{matrix}K_{M} \\ τ \end{matrix} \underrightarrow{⟶^{min. deg.}} \begin{matrix}M \\ π \end{matrix} \underrightarrow{⟶^{θ(M,K′)}} \begin{matrix}K′ \\ σ′ \end{matrix}

The following explicit computation of the correspondence of joint harmonics is an immediate consequence of this Lemma and the formulas of §4. Normalizations and parameterizations of representations in terms of highest weights are as in §4. Proposition 6.7. Explicit correspondence of joint harmonics for irreducible dual pairs.

θ(ψ, H)(σ) = σ′ for σ and σ′ as follows. I. (G, G′) = (O(p, q), Sp(2n, ℝ)), (K, K′) = (O(p) × O(q), U(n).

σ = (a_{1}, …, a_{k}, 0, …, 0; δ) ˇ⊗(b_{1}, …, b_{ℓ}, 0, …, 0; ϵ) →
σ′ = (a_{1} + \frac{p − q}{2} , …, a_{2} + \frac{p − q}{2} , \overbrace{1 + \frac{p − q}{2} , …, 1 + \frac{p − q}{2}}^{\frac{1−δ}{2} (p−2k)}, 0, …, 0,

\overbrace{−1 + \frac{p − q}{2} , …, −1 + \frac{p − q}{2}}^{\frac{1−ϵ}{2} (q−2ℓ)}, −b_{ℓ} + \frac{p − q}{2} , …, −b_{1} + \frac{p − q}{2} )

where k + \frac{1−δ}{2} (p − 2k) + ℓ + \frac{1−ϵ}{2}(q − 2ℓ) ≤ n. II.(G, G′) = (U(p, q), U(r, s)), (K, K′) = (U(p) × U(q), U(r) × U(s)).

σ = (a_{1}, …, a_{k}, 0, …, 0, b_{1}, …, b_{ℓ}; c_{1}, …, c_{m}, 0, …, 0, d_{1}, …, d_{n})+

\frac{1}{2}(r − s, …, r − s; s − r, …, s − r) →

σ′ = (a_{1}, …, a_{k}, 0, …, 0, d_{1}, …, d_{n}; c_{1}, …, c_{m}, 0, …, 0, b_{1}, …, b_{ℓ})+

\frac{1}{2}(p − q, …, p − q; q − p, …, q − p)

where the obvious inequalities hold:

k + ℓ ≤ p, m + n ≤ q, k + n ≤ r, m + ℓ ≤ s

III. (G, G′) = (Sp(p, q), O^{∗}(2n)), (K, K′) = (Sp(p) × Sp(q), U(n)). IV. (G, G′) = (GL(m, ℝ), GL(n, ℝ)), (K, K′) = (O(m), O(n). V. (G, G′) = (GL(m, ℂ), GL(n, ℂ)), (K, K′) = (U(m), U(n)). VI. (G, G′) = (O(m, ℂ), Sp(2n, ℂ)), (K, K′) = (O(m), Sp(n)). VII. (G, G′) = (GL(m, ℍ), GL(n, ℍ)), (K, K′) = (Sp(m), Sp(m)).

As an example we sketch one of the calculations involved in the proof of this Proposition.

Example 6.8 Let (G, G′) = (O(p, q), Sp(2n, ℝ)). Then

(K, M′) = (O(p) × O(q), Sp(2n, ℝ) × Sp(2n, ℝ))

and (M, K′) = (U(p, q), U(n)). Finally

(K_{M}, K_{M′}) = (U(p) × U(q), U(n) × U(n)).

The embeddings Sp(2n, ℝ) ↪ Sp(2n, ℝ) × Sp(2n, ℝ) and U(n) ↪ U(n) × U(n) are diagonal, and the restriction of an outer tensor product π ˇ⊗π′ is the ordinary tensor product.

Let σ = (a_{1}, …, a_{k}, 0, …, 0; δ) ˇ⊗(b_{1}, …, b_{ℓ}, 0, …, 0; ϵ). Then provided

(6.9)(a)  k + \frac{1 − δ}{2} (p − 2k) ≤ n
ℓ + \frac{1 − ϵ}{2}(q − 2ℓ) ≤ n

we have

τ′ = (a_{1} + \frac{p}{2}, …, a_{k} + \frac{p}{2}, \overbrace{\frac{p}{2} + 1, …, \frac{p}{2} + 1}^{\frac{1−δ}{2} (p−2k)}, \frac{p}{2}, …, \frac{p}{2}) ˇ⊗

(6.9)(b)  (−\frac{q}{2}, …, −\frac{q}{2}, \overbrace{−\frac{q}{2} − 1, …, −\frac{q}{2} − 1}^{\frac{1−ϵ}{2} (q−2ℓ)}, −b_{ℓ} − \frac{q}{2}, …, −b_{1} − \frac{q}{2})

Then

τ = (a_{1} + \frac{n}{2}, …, a_{k} + \frac{n}{2}, \overbrace{1 + \frac{n}{2}, …, 1 + \frac{n}{2}}^{\frac{1−δ}{2} (p−2k)}, \frac{n}{2}, …, \frac{n}{2}) ˇ⊗

(6.9)(c)  (−\frac{n}{2}, …, −\frac{n}{2}, \overbrace{−1 − \frac{n}{2}, −1 − \frac{n}{2}}^{\frac{1−ϵ}{2} (q−2ℓ)}, −b_{ℓ} − \frac{n}{2}, …, −b_{1} − \frac{n}{2})

Finally σ′ exists if and only if

(6.9)(d)  k + \frac{1 − δ}{2} (p − 2k) + ℓ + \frac{1 − ϵ}{2}(q − 2ℓ) ≤ n

in which case

σ′ = (a_{1} + \frac{p − q}{2} , …, a_{2} + \frac{p − q}{2} , \overbrace{1 + \frac{p − q}{2} , …, 1 + \frac{p − q}{2}}^{\frac{1−δ}{2} (p−2k)}, \frac{p − q}{2} , …, \frac{p − q}{2} ,

(6.9)(e) \overbrace{−1 + \frac{p − q}{2} , …, −1 + \frac{p − q}{2}}^{\frac{1−ϵ}{2} (q−2ℓ)}, −b_{ℓ} + \frac{p − q}{2} , …, −b_{1} + \frac{p − q}{2} )

Therefore H ≃ ∑ σ ˇ⊗σ′ where the sum runs over σ, σ′ satisfying (6.9)(d).

§7. Infinitesimal character.

§8. Induction principle.

The discussion in this section is closely related to that of section II.6 of Roberts’ lectures. The essential point is the computation of the Jacquet module of the oscillator representation with respect to the nilpotent radical of a parabolic subgroup of a dual pair. This was originally discussed in [36] in somewhat different terms, and first expressed this way in [25] for orthogonal/symplectic pairs over p–adic fields. This was generalized in [30] to all type I pairs. The case of the real dual pairs (O(2p, 2q), Sp(2n, ℝ)) is treated in [29], general real symplectic pairs in [4], and complex groups in [5].

We begin with a rough discussion of the principle. For the moment we ignore the covering groups, and write G · G′ = G × G′, etc. Consider a dual pair (G, G′) in some Sp(2n, ℝ) and parabolic subgroups P = MN and P′ = M′N′ of G and G′ respectively. We assume, as is often the case, that (M, M′) is itself a dual pair in some Sp(2m, ℝ). Let ω be an oscillator representation for S̃p(2n, ℝ) restricted to G × G′, and let ω_{M} be an oscillator representation for M × M′. The Jacquet module ω_{N×N′} of ω is a representation of M × M′, and a calculation shows it has ω_{M}ζ as a quotient for some character ζ of M × M′. Suppose there is a non–zero M × M′ map

ϕ : ω_{M} → σ ˇ⊗σ′

for some representation σ ⊗ σ′ of M × M′. That is θ(ψ, M, M′)(σ) = σ′. It follows from the computation of the Jacquet module that there is a non–zero G × G′–map

Φ : ω → Ind_{P}^{G}(σξ) ˇ⊗Ind_{P′}^{G′}(σ′ξ′)

for some characters ξ, ξ′. That is θ(ψ, G, G′)(π) = π′ for some irreducible constituents π and π′ of Ind_{P}^{G}(σξ) and Ind_{P′}^{G′}(σ′ξ′) respectively.

This produces a large part of the correspondence for (G, G′). The problem, in general, is to determine the constituents of the induced modules in the image of Φ.

The same discussion holds in the p–adic case as well. In the real case, one has the extra very powerful information on K–types, and this is the primary tool for attacking the question. For example if π and a K–type σ of lowest degree in π are known, this determines a K′–type θ(ψ, H)(σ) in π′, and this may be enough to describe π′ explicitly.

Another important difference from the p–adic case is that the Jacquet–functor is exact over a p–adic field but not over ℝ, so the precise information about the filtration of the Jacquet module of [25] [30], is not available over ℝ. At the same time there is no notion of “supercuspidal” representation over ℝ, so the whole notion of “first occurrence” (cf. Roberts’ talks) takes on a somewhat different form.

We turn now to a more careful description. We consider only type I dual pairs, so we are given a division algebra D of dimension d = 1, 2 or 4 over ℝ with involution (cf. §4). We are also given V (resp. W) a vector space over D equipped with a non–degenerate Hermitian (resp. skew–Hermitian) form and G = U(V) is the isometry group of this form (resp. G′ = U(W).) Then (G, G′) is an irreducible reductive dual pair in Sp(𝕎) ≃ Sp(2dn, ℝ), where 𝕎 = V ⊗_{D} W considered as a real vector space.

We consider an orthogonal direct sum

(8.1)(a)  V = V_{+} ⊕ V^{0} ⊕ V_{−}

Here the form restricted to V^{0} is non–degenerate, V_{+} and V_{−} are isotropic, and the form defines a perfect pairing on V_{+} × V_{−}. The stabilizer P = MN of V_{−} in G is a parabolic subgroup of G, with M ≃ GL(D, V_{+}) × U(V^{0}).

Similarly we consider

(8.1)(b)  W = W_{+} ⊕ W^{0} ⊕ W_{−}

with P′ = M′N′, and M′ ≃ GL(D, W_{+}) × U(W^{0}).

Fix ψ and let ω be the corresponding (smooth) oscillator representation of S̃p(2dn, ℝ). We consider this as a representation of G̃ · G̃′ by restriction. Let F be the corresponding Fock model, considered as a (𝔤 ⊕ 𝔤′, K̃ · K̃′)–module.

Now (M, M′) is a dual pair in Sp(𝕎_{M}) where 𝕎_{M} = (V_{+} ⊗_{D} W_{+}) ⊕ (V_{−} ⊗_{D} W_{−}) ⊕ 9V^{0} ⊗ W^{0}), considered as a real vector space. Let ω_{M} be the oscillator representation of this metaplectic group, with corresponding Fock model F_{M}. A technical headache is that the inverse images of M and M′ in S̃p(𝕎_{M}) may not be isomorphic to M̃ and M̃′, so denote these covers \overline{M}, \overline{M}′. We consider ω_{M} as a representation of \overline{M} ·\overline{M}′, and F_{M} as a (𝔪⊕𝔪′, \overline{K}_{M} \overline{K}_{M′})–module (with the obvious notation for maximal compact subgroups of M and M′ and their covers).

Let P̃ ≃ M̃N and P̃′ ≃ M̃′N′ be the inverse images of P and P′ in Sp(𝕎).

Theorem 8.2.

(1) There is a non–zero surjective P̃ · P̃′ equivariant map

Φ : ω ↠ ω_{M}χ

where χ is some character of \overline{M} · \overline{M}′. Here \overline{P} · \overline{P}′ acts on the right hand side by the given action of \overline{M} · \overline{M}′, with N · N′ acting trivially.

(2) There is a non–zero surjective (𝔪 ⊕ 𝔪′, K̃_{M} · K̃_{M′})–map
Φ : F ↠ F_{M}χ.

Here F may be replaced by the Lie algebra homology group H_{0}(𝔫 ⊕ 𝔫′, F).

(3) Suppose σ (resp. σ′) is an irreducible (𝔪, \overline{K}_{M}) (resp. (𝔪′, \overline{K}_{M′})) module, and σ corresponds to σ′ in the duality correspondence for the dual pair (M, M′). Then there is a non–zero map

Φ : ω → IndG̃ ; P̃(σζ) ˇ⊗IndG̃′ ; P̃′(σ′ζ′)

where ζ, ζ′ are certain characters of M̃ and M̃′.

(4) In the setting of (3), some irreducible constituents of

IndG̃ ; P̃(σζ)

and

IndG̃′ ; P̃′(σ′ζ′)

correspond via the duality correspondence for (G, G′).

Note: This essentially follows from the computation of the top term of the filtration of the Jacquet module [25,30] applied twice, once each to P ⊂ G and P′ ⊂ G′. Note: Some covering problems are being swept under the rug. The main point is that σ and ζ are representations of \overline{M}, but that σζ may be identified with a representation of M̃. In the orthogonal–symplectic case this is written out in painful detail in [2], and in somewhat less detail but still painfully in [4]. Similar but not identical covering issues are treated in the p–adic case in [30]. In [29] the same issues are addressed in a situation where the covering groups are all trivial.

sketch of proof. The main point is to choose the proper (mixed) model of the ω. For a complete polarization of 𝕎 we take

X = (V ⊗_{D} W_{+}) ⊕ (V_{+} ⊗_{D} W^{0}) ⊕ X^{0}
Y = (V ⊗_{D} W_{−}) ⊕ (V_{−} ⊗_{D} W^{0}) ⊕ Y^{0}

where X_{0}, Y_{0} is an arbitrary complete polarization of V^{0} ⊗_{D} W^{0}. Then ω is realized on the Schwarz space S(Y). For ω_{M} we take

X_{M} = (V_{−} ⊗_{D} W_{+}) ⊕ X^{0}
Y_{M} = (V_{+} ⊗_{D} W_{−}) ⊕ Y^{0}

so ω_{M} is realized on S(Y_{M}).

An explicit calculation shows that restriction from Y to Y_{M} intertwines the action of P̃, P̃′ up to certain characters, which gives (1).

The corresponding statement for the Fock models follows from Theorem 3.*. In fact it follows that up to terms of lower degree, Φ : F ↠ F_{M} is given by restriction.

Statement (3) is an immediate consequence of Frobenius reciprocity [45,Proposition 6.3.5]. The final statement follows from (3) by composing Φ with the map from the image of Φ in the induced representation to an irreducible quotient of this image.

We note that in this generality there is very little that can be said about the image of Φ and an irreducible quotient of it. Even if the induced representation has a unique irreducible quotient, there is no reason a priori that it should be in the image of Φ.

Open problem:

Prove a version of Theorem 8.2 for cohomological induction.

It appears that some version of Theorem 8.2 should hold with parabolic induction from M replaced by cohomological induction from the Levi factor L of a theta– stable parabolic [33,45]. This is true in many examples, and some calculations indicate it is true in some generality. The homology groups that enter into Frobenius reciprocity in this setting ([45,Proposition 6.3.2]) are not solely in degree zero. As a result in the analogue of Theorem 8.2(2) it is necessary to calculate some higher homology groups.

Such a theorem together with Theorem 8.2 would go very far towards a complete explicit understanding of the duality correspondence.

§9. Examples.

We give a few examples of the local theta–correspondence over ℝ. Our first examples are very simple, and then increase in difficulty and decrease in detail, before petering out again at the end.

Example 1. (O(1), Sp(2n, ℝ)).

The center of Sp(2n, ℝ) is O(1) ≃ ±1, and this forms a dual pair. The inverse image Õ(1) of O(1) in S̃p(2n, ℝ) is isomorphic to ℤ/4ℤ (n odd) or ℤ/2ℤ × ℤ/2ℤ (n even). This group has two genuine characters χ and χ′ which correspond to the two irreducible summands of the oscillator representation ω_{ψ}. The labeling of χ and and χ′ is a matter of convention, or equivalently of a choice of isomorphism Õ(1) ≃ ℤ/4ℤ or ℤ/2ℤ × ℤ/2ℤ. The same two characters also give the two constituents of ω\overline{ψ}.

Example 2. (U(1), U(n)).

Here U(1) is the center of the maximal compact subgroup K = U(n). This dual pair describes the restriction of the oscillator representation to K̃, which is quite important.

Fix ψ. With choices as in §5, example II (preceding Proposition 5.6) the correspondence from U(1) to U(n) is (k + \frac{n}{2}) → (k + \frac{1}{2}, \frac{1}{2}, …, \frac{1}{2}), k = 0, 1, 2, . . .. The K̃–types with k even (resp. odd) constitute the irreducible summand ω_{ψ}^{+} (resp. ω_{ψ}^{−}). The lowest K̃–types of these two summands (both in the sense of Vogan, and of lowest degree) are (\frac{1}{2}, …, \frac{1}{2}) and (\frac{3}{2}, \frac{1}{2}, …, \frac{1}{2}) respectively.

Note that the K̃–types lie along a “line”, i.e. their highest weights are obtained from the highest weight of the lowest K̃–type by adding multiples of a single vector. This is a condition of [46]. In fact the four irreducible summands of the oscillator representations are the only non–trivial unitary representations of S̃p(2n, ℝ) (n ≥ 1) with K̃–types along a line. The oscillator representation is particularly “small”, and the duality correspondence is due in part to this. It is interesting to note that it is necessary to pass to the two fold cover of Sp(2n, ℝ) to find these especially small representations. This is analogous to the spin representations of the two-fold cover Spin(n) of SO(n).

Once ψ is fixed, the dual oscillator representation ω_{ψ}^{∗} = ω\overline{ψ} has K̃–spectrum (\frac{1}{2}, …, \frac{1}{2}, \frac{1}{2} − k) (k = 0, 1, …).

Example 3. (O(n), SL(2, ℝ)). The duality correspondence in this case is essentially the classical theory of spherical harmonics [19,20]. With the appropriate coordinates the Fock model is on ℂ[z_{1}, …, z_{n}]. One of the operators coming from the (complexified) Lie algebra of SL(2, ℝ) is the Laplacian Δ = ∑_{i} \frac{d^{2}}{dz_{i}^{2}} , and the harmonics in the sense of section 5 are the kernel of the Laplacian. Then r^{2} = ∑_{i} z_{i}^{2} and ∑ z_{i} \frac{d}{dz_{i}} + \frac{n}{2} together with Δ span the image of 𝔰𝔩(2, ℂ). Theorem 5.3 (= Theorem 9 of [19]) reduces to the classical statements of spherical harmonics in this case.

Example 4. (GL(1, ℝ), GL(1, ℝ))

This example is an illustration of the principle that no dual pair is too simple to be taken lightly. The p–adic case is treated in [30,Chapter 3, §III.7, pg. 65], and has some non–trivial analytic content. The corresponding statement over ℝ for the smooth vectors is similar. Here we discuss only the Fock model.

Write the dual pair as (G_{1}, G_{2}); of course the images of G_{1} and G_{2} in SL(2, ℝ) coincide. We embed G_{1} ↪ SL(2, ℝ) as x → ι_{1}(x) = diag(x, x^{−1}). Write S̃L(2, ℝ) a as {(g, ±1) | g ∈ SL(2, ℝ)} with the usual cocycle [38]. Let G̃L(1, ℝ) be the two– fold cover of GL(1, ℝ) defined by the cocycle c(x, y) = (x, y)_{ℝ}. Here (x, y)_{ℝ} is the Hilbert symbol [40] and equals 1 unless x < 0, y < 0. This is isomorphic to the det^{\frac{1}{2}} cover, or alternatively to ℝ^{∗} ∪ iℝ^{∗} ⊂ ℂ^{∗}. Then ι̃_{1} : (x, ϵ) → (ι(x), ϵ) gives an isomorphism of G̃L(1, ℝ) with the inverse image of GL(1) in S̃L(2, ℝ). Let 𝔤_{1} ≃ ℂ be the complexified Lie algebra of G_{1}, and let K_{1} = ±1 be its maximal compact subgroup, so K̃_{1} ≃ ℤ/4ℤ.

Fix ψ and the Fock model F = ℂ[z]. We seek to show that every genuine irreducible (𝔤_{1}, K̃_{1})–module is a quotient of F. The action of K̃_{1} breaks up F into its two irreducible summands F^{+} = ℂ[z^{2}] and F^{−} = zℂ[z^{2}] for S̃L(2, ℝ) (cf. example 1). So it is enough to show any character of 𝔤_{1} occurs as a quotient of either summand.

The action of the Lie algebra 𝔤_{1} of GL(1, ℝ) is by the operator X = z^{2} − \frac{d^{2}}{dz^{2}} (cf. §3). Therefore F^{±} are each free modules for this action, and every character of 𝔤_{1} occurs as a quotient of F^{±} in a unique way. To be explicit we consider F^{+}. Fix λ ∈ ℂ ≃ 𝔤_{1}^{∗}, with λ(X) = λ, and let N_{λ} = 𝔘(𝔤) · (z^{2} − λ). By induction this is a codimension one subspace of F^{+}. The image \overline{1} of 1 in the quotient F^{+}/N_{λ} is non–zero, and X · \overline{1} = \overline{z^{2}} = λ\overline{1}. The case of F^{−} is similar.

The embedding of the second copy G_{2} of GL(1) is ι_{2} : x = diag(x^{−1}, x). The natural choice of ι̃_{2} is (x, ϵ) → (ι_{2}(x), ϵ)). With this convention ι̃_{2}^{−1} ∘ ι̃_{1} : G̃_{1} → G̃_{2} takes (x, ϵ) to (x^{−1}, ϵ) = (x, ϵsgn(x))^{−1}.

With these choices, the duality correspondence is

χ → χ^{−1}sgn

for any genuine (𝔤_{1}, K̃_{1}) character χ. The sgn term comes from the twist by sgn(x) in ι_{2}^{−1} ∘ ι_{1}.

Of course ι̃_{2} can be modified to eliminate the twist by sgn. Also the oscillator representation itself restricted to this dual pair can be normalized by tensoring with a genuine character of G̃L(2, ℝ) so that the correspondence takes χ → χ^{−1} as χ runs over characters of GL(1, ℝ). This is what is normally done. Example 5. (GL(m, ℝ), GL(m, ℝ)).

This generalizes Example 4. Let G be the det^{\frac{m}{2}} cover of GL(mℝ). Then (G_{1}, G_{2}) is a dual pair with G_{1} ≃ G ≃ G_{2}, and with the natural choice of these isomorphisms the duality correspondence takes π to π^{∗}⊗sgn, as π runs over all irreducible genuine representations of G. As in Example 4 the twist by sgn can be eliminated, and G replaced by GL(m, ℝ).

The proof is by induction on m, starting with m = 1 by Example 4, and the induction principle of §8. We normalize the correspondence to eliminate the covering groups. We write an irreducible representation π of GL(m) as the unique irreducible quotient of

Ind_{MN}^{G_{1}}(σ ⊗ 11)

where M ≃ GL(m−1)×GL(1). By induction and Theorem 8.2 there is a non–zero map from ω to the tensor product of the induced module for G_{1} and a similar induced modules for G_{2} with σ replaced by σ^{∗}. If these induced modules are irreducible the result is immediate. The general case follows from a deformation of parameters argument as in [5], the main point being that we have enough control over K–types to determine at least an irreducible quotient of the image of this map. The K–type information is crucial because the induced module for G_{2} will have unique irreducible submodule, and not a quotient. This information is not available in the p–adic case, and these elementary techniques are not enough to determine the correspondence in this case (cf. the preceding reference [30] ).

Example 6. (GL(m, ℝ), GL(n, ℝ)) Based on 5 the general case of GL(m, ℝ) is straightforward [29]. We normalize the correspondence and eliminate the covering groups. Suppose m ≤ n. Then every representation π of GL(m, ℝ) occurs in the correspondence. Let MN ≃ GL(m) × GL(n − m) × N be the usual maximal parabolic subgroup of GL(n). Then θ(π) is the is the unique irreducible constituent of

Ind_{MN}^{GL(n)}(π ⊗ 11 ⊗ 11)

containing a certain K–type of multiplicity one.

A similar result holds for (GL(m, ℂ), GL(n, ℂ)) [5], and presumably also for (GL(m, ℍ), GL(n, ℍ)).

Example 7. (O(m, ℂ), Sp(2n, ℂ))

We parameterize irreducible representations of complex groups by pairs (λ, ν) as in [31]. We fix an orthogonal group G_{1} = O(2m + τ, ℂ) (τ = 0, 1) and consider the family of dual pairs (G_{1}, G_{2}(n)) with G_{2}(n) = Sp(2n, ℂ). Given an irreducible representation π_{1} of G_{1}, there exists a non–negative integer n(π_{1}) such that π occurs in the duality correspondence for (G_{1}, G_{2}(n)) if and only if n ≥ n(π_{1}). In the p–adic case this is covered in the lectures of Brooks Roberts, see also [5,29].

For unexplained notation see [5].

Theorem ([5,Theorem 2.8].

Let π_{1} = L(μ_{1}, ν_{1}) be an irreducible representation of G_{1}.

Define the integer k = k[μ_{1}] by writing μ_{1} = (a_{1}, …, a_{k}, 0, …, 0; ϵ) with a_{1} ≥ a_{2} ≥ ⋯ ≥ a_{k} > 0. Write ν_{1} = (b_{1}, …, b_{m}), and define the integer 0 ≤ q = q[μ_{1}, ν_{1}] ≤ m − k to be the largest integer such that 2q − 2 + τ, 2q − 4 + τ, …, τ all occur (in any order) in {±b_{k+1}, ±b_{k+2}, …, ±b_{m}}. After possibly conjugating by the stabilizer of μ_{1} in W, we may write

μ_{1} = (\overbrace{a_{1}, …, a_{k}}^{k}, \overbrace{0, 0, …, 0}^{m−q−k} \overbrace{0, 0, …, 0}^{q}; ϵ)
ν_{1} = (\overbrace{b_{1}, …, b_{k}}^{k}, \overbrace{b_{k+1}, …, b_{m−q}}^{m−q−k}, \overbrace{2q − 2 + τ, 2q − 4 + τ, …, τ}^{q}).

Let μ′_{1} = (a_{1}, …, a_{k}), ν′_{1} = (b_{1}, …, b_{k}), and ν′′_{1} = (b_{k+1}, …, b_{m−q}).

Then n(π_{1}) = m − ϵq + \frac{1−ϵ}{2}τ, and for n ≥ n(π_{1}), θ(π_{1}) = L(μ_{2}, ν_{2}), where

μ_{2} = (μ′_{1}, \overbrace{1, …, 1}^{\frac{1−ϵ}{2} (2q+τ)}, 0, 0, …, 0)
ν_{2} = (ν′_{1}, \overbrace{2q − 1 + τ, 2q − 3 + τ, …, 2ϵq + 1 + ϵτ}^{\frac{1−ϵ}{2} (2q+τ)}, ν′′_{1},
2n − 2m − τ, 2n − 2m − 2 − τ, …, −ϵ(2q + τ) + 2).

Example 8. (O(p, q), Sp(2n, ℝ)) with p + q = 2n, 2n + 1, 2n + 2.

In these examples the groups are the same “size”, and are of particular interest from the point of view of L–functions (cf. the lectures of Steve Kudla). This is the opposite extreme of the stable range [12]. The case (O(2, 2), Sp(4, ℝ)) is in [35] (O(4, 0) and O(0, 4) are in [22], see §5.). The cases (O(p, q), Sp(2n, ℝ)) with p + q = 2n, 2n + 2 and p, q even are in [29], p, q odd are only missing because of covering group technicalities. Finally (O(p, q), Sp(2nℝ)) is in [4], this is similar to [29] except that the covering groups are unavoidable.

We first consider the case p, q even. In this case the covering of Sp(2n, ℝ) splits and the correspondence can be written in terms of the linear groups. Roughly speaking the correspondence in these cases is “functorial”, and a number of nice properties hold which fail in general. In particular the minimal K–type in the sense of Vogan is always of minimal degree in this situation.

The duality correspondence is described explicitly in terms of Langlands parameters, and these match up in a natural way. This can be expressed in terms of a homomorphism between the L–groups. The disconnectedness of O(p, q) can be avoided in this range; at most one of π and π ⊗ sgn occur in the correspondence.

If p + q = 2n + 2 every representation of Sp(2n, ℝ) occurs in the correspondence with some O(p, q) (perhaps more than one). If p + q = 2n every representation of Sp(2n, ℝ) occurs with at most one O(p, q) (but some may fail to occur.) This suggests that p + q = 2n + 1 should be particularly nice, and this is the case. Fix δ = ±1 and consider the dual pairs (O(p, q), Sp(2n, ℝ)) with (−1)^{q} = δ and p + q = 2n + 1. The covering of Sp(2n, ℝ) is the metaplectic group, and the representations which occur are all genuine. We twist by a genuine character of the cover of O(p, q) to pass to representations of the linear group.

Fix ψ. Then every genuine representation of S̃p(2n, ℝ) occurs with precisely one O(p, q), and of every pair of representations π and π ⊗ sgn of O(p, q) (these are not isomorphic) precisely one occurs. By restricting to SO(p, q), this establishes a bijection between the set

S̃p(2n, ℝ)_{genuine}

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).

The notion of functoriality is not well–defined for the non–linear group S̃p(2n, ℝ). Nevertheless this correspondence is “functorial” in some sense. It is naturally described in terms of Langlands parameters, and (Vogan) lowest K–types are always of lowest degree. The orbit correspondence [12] is a bijection between the regular semisimple coadjoint orbits for Sp(2n, ℝ) and the union of those for O(p, q). At least philosophically this underlies the correspondence. Again similar but not quite so clean results hold in the case p + q = 2n, 2n + 2.

Example 10. Results of [29]

The dual pairs (O(p, q), Sp(2n, ℝ)) with p, q even are discussed extensively by Moeglin in [29]; the results on p + q = 2n, 2n + 2 of the preceding example are a very special case.

Roughly speaking, Moeglin first considers discrete series representations which correspond to discrete series in a dual pair with (G_{1}, G_{2}) the same size. Theorem 8.2 then produces certain constituents of some induced modules which correspond. The idea is to use the K–type information of §6 to determine these constituents, including their Langlands parameters. This program works provided the (Vogan) lowest K–types and K–types of lowest degree coincide for the representations in question.

The result is an explicit description of the correspondence in cases in which it is functorial. The precise conditions under which this holds are in terms of the lowest K–types; they are quite technical in general, and Example 8 is the cleanest special case. The worst case from this point of view is (O(p, q), Sp(2n, ℝ)) with n very small compared to p, q and p − q large. In this regard we mention that the case (O(p, q), SL(2, ℝ)) is thoroughly described in [13], which interprets results of [43] and [37] in these terms, and also discussed in [20].

Example 10. Representations with cohomology.

Unitary representations with (𝔤, K)–cohomology have been classified [47]. These have regular integral infinitesimal character, i.e. the same infinitesimal character as that of a finite–dimensional representation.

Let (G_{1}, G_{2}) be a dual pair, and let π be an irreducible representation of G_{1} with non–zero (𝔤, K)–cohomology (possibly with coefficients). Also assume the infinitesimal character of π′ = θ(π) is regular and integral. The explicit description of the correspondence in this case is due to Jian-Shu Li [28]. It turns out that π′ is a discrete series representation. The special case when (G_{1}, G_{2}) are in the stable range [12] is in [3], and in some greater generality in [29].

It is conjectured that the representations with cohomology exhaust the unitary representations with regular integral infinitesimal character. If this holds, then Li’s result would imply that the theta–correspondence preserves unitarity in the case of regular integral infinitesimal character.

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