THE 1994 UNIVERSITY OF MARYLAND CONFERENCE ON THE

THETA CORRESPONDENCE, DUAL

PAIRS AND AUTOMORPHIC FORMS

LECTURE II:

THE THETA CORRESPONDENCE

AND WITT TOWERS

Brooks Roberts

In this lecture we assume that F is a nonarchimedean local field not of characteristic two.

1. The representations. Since Mp(W) and the inverse images in Mp(W) of subgroups of Sp(W) are not groups of td-type as in [C] or [BZ], we need to define the kind of representations we will consider. Suppose that A is a closed subgroup of Sp(W). Let G be the inverse image of A in Mp(W). In this lecture, we will assume that all representations π of G are such that π|_{ℂ^{1}} is the identity, i.e., π(1, z) acts by multiplication by z for z ∈ ℂ^{1}. The Weil representation satisfies this property. Note also that this definition does not depend on the realization of the metaplectic group. More precisely, if π is a representation of G that has the above property, and if we use an equivalent cocycle to define the metaplectic group, then the transport of π to the metaplectic group defined by the equivalent cocycle also will satisfy the property.

To define further properties that representations of G might have, we need to use a result from the last lecture. In the last lecture it was stated that one can choose the cocycle defining Mp(W) to take values in \{±1\}. Let { \atop Sp(W)} be the extension of Sp(W) by \{±1\} defined by this cocycle. Let Â be the inverse image of A in { \atop Sp(W)}. Then Â is a group of td type. There is a commutative diagram

1 ⟶ ℂ^{1} ⟶ G ⟶ A ⟶ 1
↑^{inc}  ↑  ↑^{id}
1 ⟶ {±1} ⟶ Â ⟶ A ⟶ 1.

A vector in the space of π will be called smooth if it is smooth regarded as a vector in the space of π|Â. We will say that π is smooth if every vector in the space of π is smooth. Similarly, we say that π is admissible if π|Â is admissible. The set of all equivalence classes of irreducible smooth admissible representations of the type that we consider will

Typeset by AMS-TEX

be denoted by Irr(G). Again, these definitions do not depend on the realization of the metaplectic group. Finally, it is easy to extend these definitions to the case when A is part of a degenerate dual reductive pair.

¿From now on, when we refer to the Weil representation we will mean the restriction of ω to the subspace of smooth vectors 𝒮. If 𝒮 is the Schrödinger model as in the previous lecture, then 𝒮 is the space of all locally constant compactly supported functions on Y.

2. The theta correspondence. We define the theta correspondence. Let (A, A′) be an irreducible reductive dual pair in the extended sense discussed at the end of of the last lecture. Let G and G′ be the inverse images of A and A′ in Mp(W), respectively. Let δ be a smooth representation of G × G′. We will say that π ∈ Irr(G) and σ ∈ Irr(G′) correspond if

Hom_{G×G′}(δ, π ⊗_{ℂ} σ) ≠ 0.

Let ℛ(G) be the set of π ∈ Irr(G) such that

Hom_{G}(δ, π) ≠ 0.

Define ℛ(G′) similarly. The correspondence between Irr(G) and Irr(G′) is the set

{(π, σ) ∈ ℛ(G) × ℛ(G′) : Hom_{G×G′}(ω, π ⊗_{ℂ} σ) ≠ 0}.

With respect to this correspondence, there are three important properties that δ might have. First, we will say that Howe duality holds for δ if the correspondence is the graph of a bijection between ℛ(G) and ℛ(G′). Second, we will say that multiplicity preservation holds for δ if for all π ∈ Irr(G) and all σ ∈ Irr(G′),

dim_{ℂ} Hom_{G×G′}(δ, π ⊗_{ℂ} σ) ≤ 1.

To define the third property, let π ∈ Irr(G). Let

δ(π) = δ/ ⋂_{f∈Hom_{G}(δ,π)} ker(f).

Via δ, G × G′ acts on δ(π). Note that δ(π) ≠ 0 if and only if π ∈ ℛ(G). A complete proof of the following result can be found in [MVW]. Analogous results apply for elements of Irr(G′).

Lemma. There exists a smooth G′ representation Θ(π) such that

δ(π) ≅ π ⊗_{ℂ} Θ(π)

as representations of G × G′. Moreover, Θ(π) is unique up to isomorphism.

We say that strong Howe duality holds for δ if for every π ∈ ℛ(G) the representation Θ(π) has unique nonzero irreducible quotient θ(π), and for every σ ∈ ℛ(G′) the representation Θ(σ) has a unique nonzero irreducible quotient θ(σ). Note that if strong Howe duality holds for δ and π ∈ ℛ(G) and σ ∈ ℛ(G′) then θ(π) ∈ ℛ(G′) and θ(σ) ∈ ℛ(G).

The following lemma will used in the proof of the next proposition.

Lemma. Let π ∈ Irr(G). Then

⋂_{T ∈Hom_{G}(δ,π)} ker(T) = ⋂_{U ℂ vector space} ⋂_{T ∈Hom_{G}(δ,π⊗_{ℂ}U)} ker(T).

Proof. This follows from the previous lemma. □

The three properties are related by the following proposition:

Proposition. Strong Howe duality holds for δ if and only if Howe duality and multiplicity preservation hold for δ. If strong Howe duality holds for δ, then the map θ : ℛ(G) → ℛ(G′) is the bijection given by Howe duality, and has as inverse the map θ : ℛ(G′) → ℛ(G).

Proof. Suppose that Strong Howe duality holds for δ.

We first show that Howe duality holds for δ. We need to show that (1) every π ∈ ℛ(G) and σ ∈ ℛ(G′) occurs in the correspondence, and (2) for all π ∈ Irr(G) and σ_{1}, σ_{2} ∈ Irr(G′),

Hom_{G×G′}(δ, π ⊗_{ℂ} σ_{1}) ≠ 0, Hom_{G×G′}(δ, π ⊗_{ℂ} σ_{2}) ≠ 0 ⟹ σ_{1} ≅ σ_{2};

for all σ ∈ Irr(G′) and π_{1}, π_{2} ∈ Irr(G),

Hom_{G×G′}(δ, π_{1} ⊗_{ℂ} σ) ≠ 0, Hom_{G×G′}(δ, π_{2} ⊗_{ℂ} σ) ≠ 0 ⟹ π_{1} ≅ π_{2}.

The statement (1) follows since there are nonzero G × G′ maps

δ → δ(π) → π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} θ(π)

and

δ → δ(σ) → Θ(σ) ⊗_{ℂ} σ → θ(σ) ⊗_{ℂ} σ.

To prove (2) we will show that if π ∈ Irr(G) and σ ∈ Irr(G′) correspond then θ(π) ≅ σ and θ(σ) ≅ π. Now since π and σ correspond, there is a nonzero G × G′ map:

δ → π ⊗_{ℂ} σ.

This map factors through the map δ → δ(π), so that there is a nonzero G × G′ map

δ(π) → π ⊗_{ℂ} σ.

Hence, there is a nonzero G′ map

Θ(π) → σ.

By strong Howe duality, θ(π) ≅ σ. Similarly, θ(σ) ≅ π.

Next, we prove that multiplicity preservation holds for δ. Suppose that π ∈ Irr(G) and σ ∈ Irr(G′), and

T_{1}, T_{2} : δ → π ⊗_{ℂ} σ

are G×G′ maps. We must show that one of these maps is a multiple of the other. Without loss of generality we may assume that one of these maps is nonzero, and so by the last paragraph, σ = θ(π). Now by the lemma, these maps are zero on

⋂_{T ∈Hom_{G}(δ,π)} ker(T).

Take quotients and denote the resulting maps again by T_{1} and T_{2}:

T_{1}, T_{2} : π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} θ(π).

These maps must factor through the quotient map

π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} θ(π),

via, say, \overline{T_{1}} and \overline{T_{2}}, respectively. By Schur’s lemma, one of \overline{T_{1}} and \overline{T_{2}} is a multiple of the other, and multiplicity preservation follows.

Now suppose that Howe duality and multiplicity preservation hold. Let π ∈ ℛ(G). By Howe duality, there exists σ ∈ Irr(G′) such that π and σ correspond. It follows that there is a nonzero G × G′ map

δ(π) ≅ π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} σ.

Hence, there is nonzero G′ map

Θ(π) → σ.

It follows that Θ(π) has a nonzero irreducible quotient. To see that this quotient is unique, suppose that α_{1} and α_{2} are subrepresentations of Θ(π) such that Θ(π)/α_{1} and Θ(π)/α_{2} are nonzero and irreducible. We must show that α_{1} = α_{2}. Now there are nonzero G × G′ compositions

δ → δ(π) → π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} Θ(π)/α_{1}
δ → δ(π) → π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} Θ(π)/α_{2}.

By Howe duality, Θ(π)/α_{1} ≅ Θ(π)/α_{2} ≅ σ. We thus have nonzero G × G′ compositions

δ → δ(π) → π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} Θ(π)/α_{1} → π ⊗_{ℂ} σ
δ → δ(π) → π ⊗_{ℂ} Θ(π) → π ⊗_{ℂ} Θ(π)/α_{2} → π ⊗_{ℂ} σ.

By multiplicity preservation, these maps are multiples of each other. It follows that the compositions

Θ(π) → Θ(π)/α_{1} → σ
Θ(π) → Θ(π)/α_{2} → σ

are multiples of each other. Hence, they have the same kernel. This means that α_{1} = α_{2}.

The last statement of the theorem already has been proven. □

The Weil representation gives a representation of G × G′, and we can consider if this representation satisfies Howe duality, multiplicity preservation or strong Howe duality. There is a map

G × G′ → Mp(W).

If (A, A′) are a degenerate irreducible reductive dual pair, then this map is given by ((g, z), (g′, z′)) ↦ (1, zz′). We will abuse notation, and denote the composition of this map with ω by ω|_{G×G′}. In the degenerate case, ω|_{G×G′} is the trivial representation that sends ((g, z), (g′, z′)) to multiplication by zz′. We call the correspondence defined by this representation the theta correspondence. The following theorem was proven by R. Howe in the case when G and G′ are of type II or of type I and unramified [MVW], and by J.-L. Waldspurger in the case when G and G′ are of type I [W].

Theorem (Howe, Waldspurger). If the residual characteristic of F is odd, then ω|_{G×G′} satisfies strong Howe duality.

An important element of the proof is the use of the lattice model. This model corresponds to the model of the metaplectic representation obtained by letting A in the construction of the last lecture be an 𝔒_{F} module of maximal rank.

3. Some questions. Having defined the theta correspondence, in the remainder of this lecture we will consider two questions.

Question 1. If π ∈ Irr(G), then does π occur in the theta correspondence between Irr(G) and Irr(G′)?

Question 2. Can we construct supercuspidal representations using the theta correspondence?

Both questions have answers in the same context, but for the moment let us only consider the first question. With regard to the first question, we cannot expect every representation of Irr(G) to correspond to a representation of Irr(G′). After all, G′ may vary. For example, if G′ is small compared to G, then we would not expect an injection of Irr(G) into Irr(G′). On the other hand, this objection is removed if G′ is large compared to G. Apparently, to understand when an element of Irr(G) occurs in the theta correspondence we should vary G′. Let us make this precise.

We need some notation. For the remainder of this lecture we will only consider irreducible reductive dual pairs of type I. Let D be a division algebra over F with involution τ. By Witt’s theorem, a finite dimensional nondegenerate Hermitian D-vector space can be written as an orthogonal direct sum of an anisotropic Hermitian D-vector space and a number of Hermitian hyperbolic planes ℍ over D. The Hermitian hyperbolic plane ℍ over D has space ℍ = D ⊕ D, with form defined by

((d_{1}, d_{2}), (d′_{1}, d′_{2})) = d_{1}τ(d′_{2}) + d_{2}τ(d′_{1}).

The number of these hyperbolic planes in the decomposition is called the Witt index of the Hermitian D-vector space. Suppose that (V_{0}, ( , )) is a finite dimensional anisotropic Hermitian D-vector space. For each nonnegative integer m, let V = V_{m} be the orthogonal direct sum of V_{0} with m copies of the Hermitian hyperbolic plane over D. Beginning with a anisotropic skew Hermitian D-vector space, similarly define V′ = V′_{m′} for m′ a nonnegative integer. Let W = W_{m,m′} = V ⊗_{D} V′. By the last lecture, A = A_{m} = U(V_{m}) and A′ = A′_{m′} = U(V′_{m′}) are an irreducible reductive dual pair. As above, let G and G′ be the inverse images of A and A′ in Mp(W), respectively. Employing the same abuse of notation as above, let ω_{m,m′} = ω|_{G×G′}.

Returning to the first question, we will fix m and vary m′. This will change the size of G′.

4. Linearity and the mixed model. To investigate the first question we need another model of the Weil representation. This model, called the mixed model, is based on the behavior of the metaplectic representation with respect to direct sums. Suppose that W_{1} and W_{2} are subspaces of W such that W is the orthogonal direct sum of W_{1} and W_{2}. Let H_{1} and H_{2} be the Heisenberg groups corresponding to W_{1} and W_{2}, respectively. Then a computation shows that the map from H_{1} × H_{2} to H that takes ((w_{1}, t_{1}), (w_{2}, t_{2})) to (w_{1} + w_{2}, t_{1} + t_{2}) is an epimorphism with kernel {((0, t), (0, −t)) : t ∈ F }. Suppose that ρ_{1} and ρ_{2} are metaplectic representations of H_{1} and H_{2}, respectively. Then clearly ρ_{1} ⊗_{ℂ} ρ_{2} is trivial on the kernel. We will abuse notation and regard ρ_{1} ⊗_{ℂ} ρ_{2} as a representation of the quotient. Using the isomorphism with H, we can consider ρ_{1} ⊗_{ℂ} ρ_{2} as a representation of H. It is clear that ρ_{1} ⊗_{ℂ} ρ_{2} has central character ψ. Thus, ρ_{1} ⊗_{ℂ} ρ_{2} is a metaplectic representation of H. Let ρ = ρ_{1} ⊗_{ℂ} ρ_{2}.

The behavior of Weil representations with respect to orthogonal direct sums is also simple. Let the notation be as in the last paragraph. The orthogonal decomposition W = W_{1} ⊕ W_{2} determines an inclusion of Sp(W_{1}) × Sp(W_{2}) in Sp(W). Suppose that ϖ_{1} and ϖ_{2} are the projective Weil representations of Sp(W_{1}) and Sp(W_{2}) corresponding to ρ_{1} and ρ_{2}. Let ω_{1} and ω_{2} be liftings of ϖ_{1} and ϖ_{2}, respectively. Let ϖ be the projective Weil representation of Sp(W) corresponding to ρ. Then a computation show that ω_{1} ⊗_{ℂ} ω_{2} is a lifting of ϖ|_{Sp(W_{1})×Sp(W_{2})}. Fix a lifting ω of ϖ such that ω|_{Sp(W_{1})×Sp(W_{2})} = ω_{1} ⊗_{ℂ} ω_{2}. Let Mp(W_{1}), Mp(W_{2}) and Mp(W) be defined with respect to ω_{1}, ω_{2} and ω, respectively. There is a natural map from Mp(W_{1}) × Mp(W_{2}) to Mp(W) that takes ((g_{1}, z_{1}), (g_{2}, z_{2})) to ((g_{1}, g_{2}), z_{1}z_{2}). We have

ω((g_{1}, g_{2}), z_{1}z_{2}) = ω_{1}(g_{1}, z_{1}) ⊗ ω_{2}(g_{2}, z_{2})

for (g_{1}, z_{1}) ∈ Mp(W_{1}) and (g_{2}, z_{2}) ∈ Mp(W_{2}). If c_{1}, c_{2} and c are the cocycles defining Mp(W_{1}), Mp(W_{2}) and Mp(W), a computation shows that for (g_{1}, g_{2}), (g′_{1}, g′_{2}) ∈ Sp(W_{1}) × Sp(W_{2}) ⊂ Sp(W),

c((g_{1}, g_{2}), (g′_{1}, g′_{2})) = c_{1}(g_{1}, g′_{1})c_{2}(g_{2}, g′_{2}).

We can use the last two paragraphs to construct a model of the Weil representation of Sp(V ⊗_{D} V′) in which the action of G is fairly explicit. The idea is make the action of G as linear as possible, using the fact that in the Schrödinger model the action of Levi component of the Siegel parabolic is linear. Now by definition, V′ = V′_{0} ⊥ m′ℍ′, where ℍ′ is the skew Hermitian hyperbolic plane ℍ′ = D ⊕ D. The space m′ℍ′ has two natural subspaces:

X′ = X′_{m′} = (D ⊕ 0) ⊥ (D ⊕ 0) ⊥ ⋯ ⊥ (D ⊕ 0)
Y′ = Y′_{m′} = (0 ⊕ D) ⊥ (0 ⊕ D) ⊥ ⋯ ⊥ (0 ⊕ D).

The subspaces are totally isotropic. We have a direct sum decomposition

V′ = X′ ⊕ V′_{0} ⊕ Y′.

We will call this decomposition the Witt decomposition. The pairing ( , )′ : X′×Y′ → D is nondegenerate, and V_{0} is the orthogonal complement to X′⊕Y′. Similar comments apply to V. Let W_{1} = V ⊗_{D} (X′ ⊕ Y′) and W_{2} = V ⊗_{D} V′_{0}. There is a commutative diagram:

A ⟶ Sp(W_{1}) × Sp(W_{2})
id↓  ↓
A ⟶  Sp(W).

Note that (V ⊗_{D} X′) ⊕ (V ⊗_{D} Y′) is a complete polarization of V ⊗_{D} (X′ ⊕ Y′). Let ω_{1} be the Schrödinger model with respect to V ⊗_{D} Y′. Clearly, A acts naturally on 𝒮(V ⊗_{D} Y′) by left translation composed with inversion.

Proposition (Mixed model). Let the notation be as above. If g ∈ A, z ∈ ℂ^{1}, φ_{1} ∈ 𝒮(V ⊗_{D} Y′) and φ_{2} ∈ 𝒮_{2}, then

ω(g ⊗ 1, z)(φ_{1} ⊗ φ_{2}) = (g ⊗ 1) · φ_{1} ⊗ ω_{2}(g ⊗ 1, z)φ_{2}.

Proof. The element (g ⊗1, z) is the image of ((g ⊗1, 1), (g ⊗1, z)) from Mp(W_{1})×Mp(W_{2}). Hence,

ω(g ⊗ 1, z)φ = ω_{1}(g ⊗ 1, 1)φ_{1} ⊗ ω_{2}(g ⊗ 1, z)φ_{2}.

With respect to the above complete polarization of W_{1}, g ⊗ 1 has the form

( \begin{matrix}g ⊗ 1 & 0 \\ 0 & g ⊗ 1 \end{matrix} ) .

By the action of the Siegel parabolic in the Schrödinger model the result now follows; note that in the notation of the last lecture, (g ⊗ 1)^{∗} = g^{−1} ⊗ 1. □

The mixed model also allows us to resolve a question that the careful reader may have raised. Namely, it is not clear, at least initially, that the inverse image G of A in Sp(W) depends only on m, and not on m′, as an extension of A by ℂ^{1}. This is, in fact, the case. Suppose that m′′ is another nonnegative integer. Let the cocycles defining Mp(W_{m,m′}) and Mp(W_{m,m′′}) be as above, and use the same lifting of ϖ_{2} in both models. Then the map (g⊗1, z) ↦ (g⊗1, z) is an equivalence between the inverse images of A in Mp(W_{m,m′}) and Mp(W_{m,m′′}) regarded as extensions of A by ℂ^{1}. The reader should carefully consider the degenerate cases. In particular, this remark asserts that if V′_{0} = 0, then G ≅ A × ℂ^{1} for all m′!

We now have sequences:

⋮
Irr(G_{m+1})
Irr(G_{m})
Irr(G_{m−1})
⋮
Irr(G_{0})

⋮
Irr(G′_{m′+1})
Irr(G′_{m′})
Irr(G′_{m′−1})
⋮
Irr(G′_{0})

We will refer to these sequences as the Witt towers.

5. Nonvanishing. We now give answers to the first question. We will show that if a representation occurs in the theta correspondence, then it continues to occur as one proceeds up the Witt tower. We will also indicate how far one must go up the tower for all representations to occur in theta correspondence.

Theorem (Persistence). Let π ∈ Irr(G_{m}). If

Hom_{G_{m}}(ω_{m,m′}, π) ≠ 0

then for all m′′ ≥ m′,

Hom_{G_{m}}(ω_{m,m′′}, π) ≠ 0.

Proof. By hypothesis, there is a surjective G_{m} map

𝒮(V ⊗_{D} Y′_{m′}) ⊗_{ℂ} 𝒮_{2} → π.

Since m′′ ≥ m′, restriction is a surjective G_{m} map

𝒮(V ⊗_{D} Y′_{m′′}) → 𝒮(V ⊗_{D} Y′_{m′}).

The result now follows by composition. □

We will use induced representations in the proof of the next theorem. Suppose that B is a closed subgroup of A. Let H and B̂ be the inverse images of B in G and Â, respectively. Suppose that β is a smooth representation of H. Let Ind_{H}^{G} β be the set of all functions φ from G to the space of β such that φ(hg) = β(h)φ(g) for h ∈ H and g ∈ G and φ|Â ∈ IndÂ ; B̂ β. The last condition means that there exists a compact open subgroup K of Â such that φ(gk) = φ(g) for g ∈ Â and k ∈ K. Let c-Ind_{H}^{G} β be the G subspace of Ind_{H}^{G} β of functions φ such that φ|Â ∈ c-IndÂ ; B̂, i.e., there is a compact subset K ⊂ Â such that the support of φ|Â lies in B̂K. Note that Ind_{H}^{G} β and c-Ind_{H}^{G} β are representations of the type defined in the first section.

Lemma. Let π ∈ Irr(G_{m}). Suppose that β is a smooth representation of H = 1 × ℂ^{1}, i.e., a complex vector space. Then

Hom_{G_{m}}(c-Ind_{H}^{G_{m}} β, π) ≠ 0.

Proof. We may assume that the cocycle defining Mp(W) takes values in {±1}. Then the inclusion of Â in G_{m} is just the identity. Let v ∈ π, v ≠ 0. The stabilizer of v in Â contains a compact open subgroup which can be taken to be of the form K × 1. Fix u ∈ β, u ≠ 0. Let φ_{0} be the function from G_{m} to U defined by φ_{0}(g, z) = zu if (g, z) ∈ K × ℂ^{1} and 0 otherwise. Then φ_{0} ∈ c-Ind_{H}^{G_{m}} β. Let f be a linear functional on β that takes u to 1. Fix a left Haar measure on Â. Define L : c-Ind_{H}^{G_{m}} β → π by

L(φ) = ∫Â f(φ(x^{−1}))π(x)v dx.

Then L is well defined, and L((g, 1)φ) = π(g, 1)L(φ) for g ∈ A. It follows that L ∈ Hom_{G_{m}}(c-Ind_{H}^{G} β, π). Finally, L(φ_{0}) ≠ 0. □

Theorem (Stable range). Let π ∈ Irr(G_{m}). If m′ ≥ dim_{D} V_{m}, then Hom_{G_{m}}(ω_{m,m′}, π) ≠ 0.

Proof. We first consider the action of A on V ⊗_{D} Y′. It is convenient to use the F isomorphism

V ⊗_{D} Y′ ≅ Hom_{D}(X′, V).

Here X′ is regarded as a right D module via the right action xd = τ(d)x. This isomorphism can be deduced from the canonical isomorphism of V ⊗_{D} Y′ with Hom_{D}(Hom_{D}(Y′, D), V) and the isomorphism of right D modules Hom_{D}(Y′, D) ≅ X′ given by the nondegenerate pairing ( , )′ : X′ × Y′ → D. Via this isomorphism the right action of A on V ⊗_{D} Y′ gives a right action of A on Hom_{D}(X′, V). This action is given by T g = g^{−1} ∘ T. Let T ∈ Hom_{D}(X′, V). and consider the A orbit 𝒪 of T. The set 𝒪 is closed, and if W is the orthogonal complement to the image of T, the stabilizer of T can be identified with U(W). Thus, as an A set, 𝒪 ≅ U(W)\A.

Consider the restriction map

𝒮(V ⊗_{D} Y′) → 𝒮(𝒪) ≅ 𝒮(U(W)\A).

Since 𝒪 is closed, this map is a surjective A map. Tensoring with ω_{2}, by the proposition from the last section, we obtain a surjective G map

ω_{m,m′} → 𝒮(U(W)\A) ⊗_{ℂ} ω_{2}.

Suppose now m′ ≥ dim_{D} V, i.e., dim_{D} X′ ≥ dim_{D} V. Then we can select T so that T is surjective, and consequently W = 0 and U(W) = 1. Let H = 1 × ℂ^{1}. Consider the map

𝒮(A) ⊗_{ℂ} ω_{2} → c-Ind_{H}^{G} ω_{2}|_{H}

that sends φ_{1} ⊗ φ_{2} to φ, where φ(g, z) = φ_{1}(g)ω_{2}(g, z)φ_{2}. This map is an isomorphism of G representations. An application of the lemma now completes the proof. □

6. Parabolic subgroups. Before showing how the theta correspondence can be used to construct supercuspidal representations we need to make some remarks about parabolic subgroups.

Consider the maximal isotropic flag

X_{1} ⊂ X_{2} ⊂ ⋯ ⊂ X_{m}

in V = V_{m}. Fix 1 ≤ i ≤ m, and consider the parabolic subgroup of A consisting of g ∈ A such that g(X_{i}) = X_{i}. With respect to the decomposition

V = X_{i} ⊕ V_{m−i} ⊕ Y_{i}

the elements of the parabolic subgroup have the form

( \begin{matrix}a & ∗ & ∗ \\ 0 & g & ∗ \\ 0 & 0 & a^{∗}^{−1} \end{matrix} )

where a ∈ Gl_{D}(X_{i}) and g ∈ U(V_{m−i}). The parabolic subgroup has a Levi decomposition with Levi factor the group of elements of the form

( \begin{matrix}a & 0 & 0 \\ 0 & g & 0 \\ 0 & 0 & a^{∗}^{−1} \end{matrix} )

with a ∈ Gl_{D}(X_{i}) and g ∈ U(V_{m−i}), and unipotent radical the group of elements of the form

( \begin{matrix}1 & ∗ & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & 1 \end{matrix} ) .

Let P_{i} and M_{i} be the inverse images of the parabolic subgroup and the above Levi factor in G, respectively. It is a fact, that we will not prove, that the inverse image of the above unipotent radical is split as an extension of the unipotent radical by ℂ^{1}. Identify the unipotent radical with its image N_{i} under a splitting. We will use analogous notation for G′_{m′}.

7. Supercuspidality. We will now show how the theta correspondence can be used to construct supercuspidal representations of G′_{m′} for some m′ given a supercuspidal representation of G_{m}. By the above theorem, we know that if π ∈ Irr(G) then π will eventually occur in the theta correspondence with Irr(G′_{m′}). We will prove that if m′ is the smallest integer such that π occurs in the theta correspondence with Irr(G′_{m′}) and if π supercuspidal, then θ(π) is supercuspidal.

This theorem depends on the computation of the Jacquet modules of ω_{m,m′} with respect to the N′_{i}. To state the result, we need some more notation. For each nonnegative integer k, let σ_{k} be the representation of Gl_{D}(X_{k}) × Gl_{D}(X_{k}) on 𝒮(Gl_{D}(X_{k})) given by

σ_{k}(g_{1}, g_{2})φ(g) = φ(g_{1}^{−1}gg_{2}).

If k and i are nonnegative integers with i ≥ k, let Q′_{ik} be the inverse image in M′_{i} of the group of elements

( \begin{matrix}\begin{matrix}a′ & ∗ \\ 0 & a′′ \end{matrix} & 0 & 0 \\ 0 & g′ & 0 \\ 0 & 0 & \begin{matrix}a′^{∗−1} & 0 \\ ∗ & a′′^{∗−1} \end{matrix} \end{matrix} )

where a′ ∈ Gl_{D}(X_{i−k}), a′′ ∈ Gl_{D}(X_{k}) and g′ ∈ U(V′_{m′−i}). A proof of the following theorem can be found in Kudla or MVW.

Theorem. Let 1 ≤ i ≤ m′. Let r = min(i, m). Then the representation (ω_{m,m′})_{N′_{i}} of G_{m} × M′_{i} has a filtration

0 = F_{r+1} ⊂ F_{r} ⊂ F_{r−1} ⊂ ⋯ ⊂ F_{1} ⊂ F_{0} = (ω_{m,m′})_{N′_{j}}

such that

F_{k}/F_{k+1} ≅ Ind_{P_{k}×Q′_{ik}}^{G_{m}×M′_{i}} β_{k}.

Here β_{k} has space the space of σ_{k} ⊗_{ℂ} ω_{m−k,m′−i}, and acts by

β_{k}((( \begin{matrix}a & ∗ & ∗ \\ 0 & 1 & ∗ \\ 0 & 0 & a^{∗−1} \end{matrix} ) , z),(( \begin{matrix}\begin{matrix}a′ & ∗ \\ 0 & a′′ \end{matrix} & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \begin{matrix}a′^{∗−1} & 0 \\ ∗ & a′′^{∗−1} \end{matrix} \end{matrix} ) , z′))

= zz′ξ_{k}(det(a))ξ′_{k}(det(a′))ξ′_{k}′(det(a′′))σ_{k}(a, a′′) ⊗ 1

for a, a′′ ∈ Gl_{D}(X_{k}), a′ ∈ Gl_{D}(X_{i−k}), z, z′ ∈ ℂ^{1} and

β_{k}((( \begin{matrix}1 & 0 & 0 \\ 0 & g & 0 \\ 0 & 0 & 1 \end{matrix} ) , z), (( \begin{matrix}1 & 0 & 0 \\ 0 & g′ & 0 \\ 0 & 0 & 1 \end{matrix} ) , z′)) = 1 ⊗ ω_{m−k,m′−i}((g, z), (g′, z′))

for g ∈ U(V_{m−k}), g′ ∈ U(V′_{m′−i}) and z, z′ ∈ ℂ^{1}. Also, ξ_{k}, ξ′_{k} and ξ′′_{k} are quasi-characters of F^{×}.

Theorem. Let π ∈ Irr(G). Suppose that

Hom_{G_{m}}(ω_{m,j}, π) = 0

for 0 ≤ j < m′, but that there exists τ ∈ Irr(G′_{m′}) such that

Hom_{G_{m},G′_{m′}} (ω_{m,m′}, π ⊗_{ℂ} τ) ≠ 0.

If π is supercuspidal then τ is supercuspidal.

Proof. (Sketch) Let 1 ≤ i ≤ m. We must show that τ_{N′_{i}} = 0. Suppose that for some 1 ≤ i ≤ m, τ_{N′_{i}} ≠ 0. Since

Hom_{G_{m}×G′_{m′}} (ω_{m,m′}, π ⊗_{ℂ} τ) ≠ 0

we have

Hom_{G_{m}×M′_{i}}((ω_{m,m′})_{N′_{i}}, π ⊗_{ℂ} τ_{N′_{i}}) ≠ 0.

We claim that

Hom_{G_{m}×M′_{i}}(F_{1}, π ⊗_{ℂ} τ_{N′_{i}}) = 0.

We prove this by induction. Suppose that 1 < k ≤ r + 1 and

Hom_{G_{m}×M′_{i}}(F_{k+1}, π ⊗_{ℂ} τ_{N′_{i}}) = 0.

By the induction hypothesis, it suffices to show that

Hom_{G_{m}×M′_{i}}(F_{k}/F_{k+1}, π ⊗_{ℂ} τ_{N′_{i}}) = 0.

This follows from the characterization of F_{k}/F_{k+1} as an induced representation and the fact that π is supercuspidal.

It follows now that

Hom_{G_{m}×M′_{i}}(F_{0}/F_{1}, π ⊗_{ℂ} τ_{N′_{i}}) ≠ 0.

Hence,

Hom_{G_{m}}(ω_{m,m′−i}, π) ≠ 0.

This contradicts the hypothesis. □

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