THE 1994 UNIVERSITY OF MARYLAND CONFERENCE ON THE

THETA CORRESPONDENCE, DUAL

PAIRS AND AUTOMORPHIC FORMS

LECTURE I:

THE WEIL REPRESENTATION

AND DUAL PAIRS

Brooks Roberts

Let F be a finite field or a local field. We assume that the characteristic of F is not two. Let | | be the valuation on F such that if μ is a Haar measure on F, then μ(xM) = |x|μ(M) for x ∈ F and measurable sets M of μ. Note that if F is a finite field then |x| = 1 for x ∈ F and |0| = 0. Let ℂ^{1} be the complex numbers of absolute value 1. Let W be a finite dimensional vector space over F. Let ⟨ , ⟩ : W × W → F be a bilinear form. Assume that ⟨ , ⟩ is nondegenerate, i.e., for all w ∈ W such that w ≠ 0 there exists w′ ∈ W such that ⟨w, w′⟩ ≠ 0. Assume also that ⟨ , ⟩ is symplectic, i.e., satisfies ⟨w, w′⟩ = −⟨w′, w⟩ for w, w′ ∈ W. Let Sp(W) be the set of linear isomorphisms from W to W such that ⟨g(w), g(w′)⟩ = ⟨w, w′⟩ for w, w′ ∈ W. Then Sp(W) is a group under composition of functions. Note that the elements of Sp(W) act on the left, and that composition is from right to left. Let ψ be a nontrivial additive character of F. This notation will be fixed throughout these lectures.

1. The Weil representation. Our first task is to construct a certain projective representation of Sp(W). This construction involves the representation theory of another group. The Heisenberg group H associated to W is the topological group with underlying set W × F, equipt with the product topology, and multiplication defined by

(w, t)(w′, t′) = (w + w′, t + t′ + \frac{1}{2}⟨w, w′⟩).

If (w, t), (w′, t′) ∈ H then

(w, t)^{−1} = (−w, −t),
(w, t)(w′, t′)(w, t)^{−1} = (w′, t′ + ⟨w, w′⟩),
[(w, t), (w′t′)] = (0, ⟨w, w′⟩).

It follows that the center of H is 0 ⊕ F. There is an obvious identification of the center of H with F that we will often implicitly use. It follows directly from the definition of the

Typeset by AMS-TEX

multiplication for H that the product of a Haar measure on W with a Haar measure on F is a left and right Haar measure on H; in particular, H is unimodular.

The unitary representation theory of H is given by the Stone-Von Neumann Theorem:

Theorem (Stone-Von Neumann). Up to equivalence of unitary representations, there is a unique irreducible unitary representation (ρ, ℋ) of H, called the metaplectic representation, with central character ψ.

¿From this theorem one obtains a projective representation of Sp(W) on ℋ. Define a left action of Sp(W) on H by g · (w, t) = (g(w), t). Define a right action of Sp(W) on the set of representations of H in the following way: if g is in Sp(W) and δ is a representation of H, let δ^{g} be the representation of H with the same space as δ and with action defined by δ^{g}(h) = δ(g · h). Now if g is in Sp(W) then ρ^{g} has the same central character as ρ. By the Stone-Von Neumann theorem, there exists a unitary isomorphism from ρ to ρ^{g}. By Schur’s lemma, this isomorphism is determined up to an element of ℂ^{1}. We denote by ϖ(g) the image of this isomorphism under the projection p from 𝒰(ℋ) to 𝒰(ℋ)/ℂ^{1}. Another application of Schur’s lemma shows that the map

ϖ : Sp(W) → 𝒰(ℋ)/ℂ^{1}

that takes g to ϖ(g) is a homomorphism. We call ϖ the projective Weil representation. Let g ∈ Sp(W) and let T ∈ 𝒰(ℋ). We say that T is a lifting of ϖ(g) if and only if p(T) = ϖ(g), that is, if and only if for all h in H,

T ∘ ρ(h) = ρ(g · h) ∘ T.

We say that a function ω : Sp(W) → 𝒰(ℋ) is a lifting of ϖ if and only if for all g in Sp(W), ω(g) is a lifting of ϖ(g), and ω(1) = 1. For each lifting ω of ϖ there is an associated cocycle c : Sp(W) × Sp(W) → ℂ^{1} defined by the condition

c(g, g′)ω(gg′) = ω(g)ω(g′).

Here we regard Sp(W) as acting trivially on ℂ^{1}. This cocycle in turn defines an extension Mp(W) of Sp(W) by ℂ^{1}:

1 → ℂ^{1} → Mp(W) → Sp(W) → 1.

The group law for Mp(W) is given by

(g, z)(g′, z′) = (gg′, c(g, g′)zz′)

for g, g′ ∈ Sp(W) and z, z′ ∈ ℂ^{1}. One obtains a unitary representation of Mp(W) on ℋ by defining ω(g, z) = zω(g). The cocycles corresponding to different liftings of ϖ are all equivalent, and hence determine equivalent extensions of Sp(W) by ℂ^{1} and an element of H^{2}(Sp(W), ℂ^{1}). We will abuse notation, and call any of the extensions determined by these cocycles the metaplectic group. Similarly, we call any of the representations the Weil representation of the metaplectic group. Before moving on, suppose that W = 0. From above, Sp(W) = 1, H = 0×F, and (ρ, ℋ) = (ψ, ℂ). It follows that Mp(W) = 1×ℂ^{1} ≅ ℂ^{1} and ω acts by ω(1, z) = z for z ∈ ℂ^{1}.

2. Unitary induction. We will use unitary induction to construct and analyze models of ρ. We recall the definition of unitary induction.

Suppose that G is separable locally compact group and that H is a closed subgroup of G. Let α be a character of H. We will define

Ind_{H}^{G} α,

the unitary representation of G induced from α. Assume that there is a continuous homomorphism χ from G to the positive real numbers such that

χ(h) = \frac{△_{H}(h)}{△_{G}(h)}

for h ∈ H. Here △_{H} and △_{G} are the modular functions of G and H, respectively. Then there exists a regular Borel measure dg on H\G such that

f ↦ ∫_{H\G} f(g) dg

is a positive nonzero functional on C_{0}(H\G) and

∫_{H\G} f(gg′) dg = χ(g′) ∫_{H\G} f(g) dg

for f ∈ C_{0}(H\G) and g′ ∈ G. Now let the space of Ind_{H}^{G} α be the space of ℂ valued measurable functions on G such that

f(hg) = α(h)f(g)

for h ∈ H and g ∈ G and

∫_{H\G} |f(g)|^{2} dg < ∞.

Then Ind_{H}^{G} α is a Hilbert space with inner product

(f, f′) = ∫_{H\G} f(g)\overline{f′(g)} dg.

Define an action of G on Ind_{H}^{G} α by

(g · f)(g′) = χ(g′)^{− \frac{1}{2}} f(gg′).

With this action, Ind_{H}^{G} α is a unitary representation of G.

3. Models for ρ. Let us now explain what groups G and H and character α we will use to get models of ρ. For G we of course take H. There are many choices for H and α, but the choices we will consider are all of the same general type. Suppose A is a closed additive subgroup of W. Then A × F is a closed normal subgroup of H. For H we will take A × F. For α, we will take characters ψ_{A} that extend ψ. Since H is unimodular and A×F is normal in H, A×F is unimodular; see [HR], p. 206. One can also see this directly as in the proof that H is unimodular. We note that Ind_{A×F}^{H} ψ_{A} has central character ψ, and is thus a candidate for ρ. The next theorem gives a condition for the irreducibility of Ind_{A×F}^{H} ψ_{A}.

Theorem. If (A × F)/ ker(ψ) is maximal abelian in H/ ker(ψ) then Ind_{A×F}^{H} ψ_{A} is irreducible, and Ind_{A×F}^{H} ψ_{A} ≅ ρ.

There is a condition which will help us find closed additive subgroups A of W such that (A × F)/ ker(ψ) is maximal abelian in H/ ker(ψ). Let A be a closed additive subgroup of W. Define

A^{⊥} = {w ∈ W : ⟨w, a⟩ ∈ ker(ψ) for all a ∈ A}.

Then A^{⊥} is a closed additive subgroup of W. If A is a subspace of W then one can show that A^{⊥} is, as the notation indicates, the space of w ∈ W such that ⟨w, a⟩ = 0 for all a ∈ A.

Lemma. Let A ⊂ W be a closed additive subgroup. Then (A × F)/ ker(ψ) is a maximal abelian subgroup of H/ ker(ψ) if and only if A^{⊥} = A.

4. The Schrödinger model. In this section we define and analyze the Schödinger model of the projective Weil representation. The main result is the construction of an explicit lifting ω of ϖ.

To obtain the Schrödinger model for the projective Weil representation, we choose A = X, where X is a maximal isotropic subspace of W. That is, X is a subspace of W such that ⟨x, x′⟩ = 0 for all x, x′ ∈ X and X is maximal with respect to this property. Such a subspace is called a Lagrangian , and we denote the set of all Lagrangians by ℒ(W). For the remainder of this section, fix a Lagrangian X. For ψ_{X}, we take the character defined by

ψ_{X}(x, t) = ψ(t).

Since X is isotropic, ψ_{X} is a character that extends ψ.

To verify X^{⊥} = X and for further work, we need to recall some facts about Lagrangians. The dimension of X over F is half of the dimension of W over F. There exists another Lagrangian Y such that W = X ⊕ Y. Any decomposition of W into the direct sum of two Lagrangians is called a complete polarization of W. For the remainder of this section fix such a Lagrangian Y. There exists an ordered basis {x_{1}, …, x_{n}} for X and an ordered basis {y_{1}, …, y_{n}} for Y such that the matrix of ⟨ , ⟩ with respect to the ordered basis for W formed by first taking the ordered basis for X and then the ordered basis for Y is

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

Such a basis is called a symplectic basis. For the remainder of this lecture, we fix such a symplectic basis. From the explicit form of ⟨ , ⟩ it is easy to see that ⟨ , ⟩ gives a nondegenerate pairing ⟨ , ⟩ : X × Y → F. This pairing gives identifications X^{∗} ≅ Y and Y^{∗} ≅ X. Finally, X^{⊥} = X.

Now we define the Schrödinger model. Note that H = (X × F) ⋊ (Y × 0). It follows that the elements of Ind_{X×F}^{H} ψ_{X} are determined on Y × 0. Define

Ind_{X×F}^{H} ψ_{X} → L^{2}(Y)

by φ ↦ φ|_{Y ×0}, where we identify Y × 0 with Y. We can select a Haar measure on Y so that this map is an isometry. By transporting the action of H from the first space to the second space via this isometry, we obtain a unitary representation of H on L^{2}(Y) with central character ψ. This representation is given by the formula:

ρ(x + y, t)φ(y′) = ψ(t + ⟨y′, x⟩ + \frac{1}{2}⟨y, x⟩)φ(y + y′)

for x ∈ X, y ∈ Y t ∈ F and φ ∈ L^{2}(Y). We can also transport the projective Weil representation ϖ to a projective representation Sp(W) → 𝒰(L^{2}(Y))/ℂ^{1} that we will also call ϖ. This projective representation is called the Schrödinger model.

We can describe the elements of Sp(W) in the following way. Let g ∈ Gl(W). We can write

g = ( \begin{matrix}a & b \\ c & d \end{matrix} )

where a ∈ Hom_{F} (X, X), b ∈ Hom_{F} (Y, X), c ∈ Hom_{F} (X, Y) and d ∈ Hom_{F} (Y, Y). By this we mean that g(x + y) = (a(x) + b(y)) + (c(x) + d(y)) for x ∈ X and y ∈ Y. Now g ∈ Sp(W) if and only if ⟨gw, gw′⟩ = ⟨w, w′⟩ for all w, w′ ∈ W. Using this condition, one can find exact conditions on a, b, c and d that are satisfied if and only if g ∈ Sp(W). Also, if one uses the symplectic basis from above, then the matrix for g ∈ Sp(W) will be the two by two matrix whose entries are the matrices for a, b, c and d. Note that if g is as above, then

g^{−1} = ( \begin{matrix}d^{∗} & −b^{∗} \\ −c^{∗} & a^{∗} \end{matrix} ) .

Here, as usual, if t is a linear map between vector spaces over F, then t^{∗} is the dual map. We abuse notation by implicitly making the identifications X^{∗} ≅ Y and Y^{∗} ≅ X mentioned above. Thus, if a, b, c and d are as above, then a^{∗} ∈ Hom_{F} (Y, Y), b^{∗} ∈ Hom_{F} (Y, X), c^{∗} ∈ Hom_{F} (X, Y), d^{∗} ∈ Hom_{F} (X, X), and a^{∗}, b^{∗}, c^{∗} and d^{∗} are the unique elements of these spaces such that

⟨a(x), y⟩ = ⟨x, a^{∗}(y)⟩, ⟨x, d(y)⟩ = ⟨d^{∗}(x), y⟩

for x ∈ X, y ∈ Y, and

⟨b(y), y′⟩ = ⟨b^{∗}(y′), y⟩, ⟨x, c(x′)⟩ = ⟨x′, c^{∗}(x)⟩

for x, x′ ∈ X and y, y′ ∈ Y. In the symplectic basis, the matrices for a^{∗}, b^{∗}, c^{∗} and d^{∗} are the transposes of the matrices of a, b, c and d.

We will need a more explicit description. Let P be the parabolic subgroup P of Sp(W) consisting of all p ∈ Sp(W) such that p(X) = X. This subgroup is called the Siegel parabolic. Let p ∈ Sp(W). A computation shows that p ∈ P if and only if

p = ( \begin{matrix}a & b \\ 0 & a^{∗−1} \end{matrix} )

for some a ∈ Gl(X) and b ∈ Hom_{F} (Y, X) such that ab^{∗} = ba^{∗}. To obtain the remaining elements of Sp(W), consider the action of Sp(W) on ℒ(W). This action is transitive, and thus, ℒ(W) is isomorphic to the set Sp(W)/P. We can also consider the action of P on ℒ(W), or equivalently, the action of P on Sp(W)/P. This action has n + 1 orbits, and these orbits are described in the following way. Fix an integer k between 0 and n. Then the k^{th} orbit is the set of Langrangians whose intersections with X have dimension k; see [Rao], p. 346. Regarding this action as an action of P on Sp(W)/P, we obtain a decomposition of Sp(W) into a disjoint union of n + 1 double cosets of P. As a representative for the k^{th} double coset we can take the element τ_{k} whose matrix in the symplectic basis is

( \begin{matrix}0 & 0 & 1_{k} & 0 \\ 0 & 1_{n−k} & 0 & 0 \\ −1_{k} & 0 & 0 & 0 \\ 0 & 0 & 0 & 1_{n−k} \end{matrix} ) .

Thus,

Sp(W) = P τ_{0}P ∪ P τ_{1}P ∪ P τ_{2}P ∪ ⋯ ∪ P τ_{n}P.

We now find a lifting for ϖ|_{P}.

Proposition. Define

ω : P → 𝒰(L^{2}(Y))

by

ω ( \begin{matrix}a & b \\ 0 & a^{∗−1} \end{matrix} ) φ(y) = | det(a)|^{1/2}ψ(\frac{1}{2}⟨ba^{∗}y, y⟩)φ(a^{∗}y).

Then for each p ∈ P, ω(p) is a lifting of ϖ(p). Moreover, ω is a homomorphism.

Proof. This can be proven by a direct computation. Instead, we give a conceptual proof. Form the semidirect product of H with P with respect to the action of P on H. Extend ψ_{X} to a character of the subgroup (X × F) ⋊ P of H ⋊ P by defining

ψ_{P} ((x, t), p) = ψ(t).

We will use the unitary representation of H ⋊ P induced from ψ_{P}. As indicated above, this induced representation depends on some choices, which we now make. By p. 210-211 of [HR], we have

△_{(X×F)⋊P} (h, p) =△_{X×F} (h)△_{P} (p)δ′(p)
△_{H⋊P} (h, p) =△_{H}(h)△_{P} (p)δ(p).

Here δ′(p) and δ(p) are such that

∫_{X×F} f(p · h) dh = δ′(p) ∫_{X×F} f(h) dh

for p ∈ P and f ∈ C_{0}(X × F), and

∫_{H} f(p · h) dh = δ(p) ∫_{H} f(h) dh

for p ∈ P and f ∈ C_{0}(H). Using the explicit descriptions of the Haar measures on X × F and H, we find that δ = 1 and δ′(p) = | det(a)|^{−1}, if p is as above. Hence,

\frac{△_{(X×F)⋊P} (h, p)}{△_{H⋊P} (h, p)} = | det(a)|^{−1}.

We now define the unitary representation of H ⋊ P induced from ψ_{P} with respect to the continuous homomorphism χ from H ⋊ P defined by χ(h, p) = | det(a)|^{−1}.

Now define

Ind_{(X×F)⋊P}^{H⋊P} ψ_{P} → Ind_{X×F}^{H} ψ_{X}

by f ↦ f|_{H}. We can choose the measure in the definition of the first induced representation so that this map is an isomorphism of unitary representations of H. Composing, we obtain an isomorphism of unitary representations of H:

Ind_{(X×F)⋊P}^{H⋊P} ψ_{P} ⟶^{∼} L^{2}(Y).

Let p ∈ P. Using this isomorphism, we can transfer the action of p from the first space to the second; a computation shows that the action of p is ω(p). If follows that ω is a homomorphism. Finally, since

(1, p)(h, 1) = (p · h, 1)(1, p)

in H ⋊ P, we see that ω(p) lifts ϖ(p). □

To find liftings for the ϖ(τ_{k}) we make some observations. We have already noted that X ≅ Y^{∗}, via the pairing ⟨ , ⟩. Using this pairing and the nontrivial additive character ψ, we can also obtain a pairing X × Y → ℂ^{1} by mapping (x, y) to ψ(⟨x, y⟩). This pairing gives isomorphisms X ≅ Ŷ and Y ≅ X̂, where X̂ and Ŷ are the groups of characters of X and Y, respectively. The symplectic basis also gives an identification ι of X and Y. We obtain a composition of isometries:

L^{2}(Y) → L^{2}(X) → L^{2}(Ŷ).

Let F ∈ 𝒰(L^{2}(Y)). By composing with the above composition we can regard F as an isometry from L^{2}(Y) to L^{2}(Ŷ). A computation shows that F is a lifting of ϖ(τ_{n}) if and only if for χ, χ′ ∈ Ŷ, y ∈ Y and φ ∈ L^{2}(Y),

F (χφ)(χ′) = F (φ)(χχ′),
F (φ( + y))(χ) = χ(y)^{−1}F (φ)(χ).

There is an obvious choice for an isometry that satisfies these conditions. Let F be the Fourier transform. More precisely, fix a Haar measure on Y, and let F be the unique isometry from L^{2}(Y) to L^{2}(Ŷ) such that

F (φ)(χ) = ∫_{Y} φ(y)χ(y) dy

for φ ∈ 𝒮(Y) and χ ∈ Ŷ. Then F satisfies the above conditions. Thus, regarded as an element of 𝒰(L^{2}(Y)), F is a lifting of ϖ(τ_{n}). We now let ω(τ_{n}) be the Fourier transform F with the Haar measure chosen so that F is self dual, that is, such that F^{2}(φ) = φ(− ) for φ ∈ L^{2}(Y). The explicit formula for ω(τ_{n}) is

ω(τ_{n})φ(y) = ∫_{Y} φ(y′)ψ(⟨ι^{−1}(y), y′⟩) dy′

for φ ∈ 𝒮(Y) and y ∈ Y. Liftings for the ϖ(τ_{k}) for k less than n can be similarly obtained. These will be partial Fourier transforms.

Using the above decomposition of Sp(W), the liftings constructed so far can be combined to obtain a lifting of ϖ. For a proof of this theorem, see [Rao].

Theorem. Define

ω : Sp(W) → 𝒰(L^{2}(Y))

by ω(pτ_{k}p′) = ω(p)ω(τ_{k})ω(p′) for p, p′ ∈ P and k between 0 and n. Then ω is well defined, and lifts ϖ.

The function ω is called the standard lifting of the Schrödinger model of ϖ. The corresponding model of the Weil representation ω is called the Schrödinger model of ω.

5. The Rao cocycle. We retain the notation from the previous section. In this section we consider the cocycle c_{R} associated to the standard lifting ω of ϖ. This cocycle is called the Rao cocycle of Sp(W). Apparently, the Rao cocycle depends on the choice of polarization and symplectic basis, as does ω. The next proposition follows from the last section.

Proposition. We have (1) c_{R}(p, p′) = 1 for p, p′ ∈ P;

(2) c_{R}(τ_{k}, τ_{k′}) = 1 for 0 ≤ k, k′ ≤ n;

(3) c_{R}(p_{1}gp_{2}, p_{2}^{−1}g′p_{3}) = c_{R}(g, g′) for p_{1}, p_{2}, p_{3} ∈ P and g, g′ ∈ Sp(W).

Proof. The statement (1) follows from the fact that ω|_{P} is a homomorphism. To prove (2), one lets W be the group generated by the τ_{k}. This is a finite group, and ω|_{W} is a homomorphism. We leave the proof of (3) as an easy exercise. □

¿From the third formula one can derive a general formula for the Rao cocycle. In [Rao], p. 347, it is proven that if g and g′ are in Sp(W) then there exist p_{1}, p_{2} and p_{3} in P, τ and τ′ in W, and b ∈ Hom_{F} (Y, X) such that b^{∗} = b and g = p_{1}τup_{2}^{−1} and g′ = p_{2}τ′p_{3}, where

u = ( \begin{matrix}1 & b \\ 0 & 1 \end{matrix} ) .

By the third formula, c_{R}(g, g′) = c_{R}(τu, τ′). Here b is related to a certain quadratic form, the Leray invariant, which is associated to the triple of Lagrangians Y, g′^{−1}Y, gY. It remains to compute c_{R}(τu, τ′). This can be achieved with the aid of the concept of the Weil index.

6. Further results. There are two other important results in [Rao]. The first result shows that the cocycle defining the metaplectic group can be normalized to take values in {±1}.

Theorem. The cocycle defining Mp(W) is equivalent to a cocycle with values in {±1}.

In fact, one can explicitly specify a coboundry to multiply the Rao cocycle by to obtain a cocycle with values in {±1}. See [Rao], section 5. The resulting cocycle is called the normalized Rao cocycle.

The next theorem describes exactly when the cocycle defining Mp(W) is trivial, i.e., is a coboundary.

Theorem. If W ≠ 0, then the cocycle defining Mp(W) is trivial if and only if F is finite or ℂ.

7. A note on notation in [Rao]. We make some comments about how to move between the notation in [Rao] and our notation. First, in [Rao] the elements of Sp(W) are written as acting on the right, and the composition of elements of Sp(W) is from left to right. This means that the group law for Sp(W) in [Rao] is different from the group law that we have been using. Of course, the groups are isomorphic: the isomorphism from the set Sp(W) with group law as above to the set Sp(W) with group law as in [Rao] maps g to g^{−1}. If one takes V^{∗} = X, V = Y and e_{1} = −y_{1}, …, e_{n} = −y_{n}, e_{1}^{∗} = x_{1}, …, e_{n}^{∗} = x_{n}. in [Rao], then one finds that ω(g) = r(g^{−1}) and c_{R}(g, g′) = c(g^{−1}, g′^{−1}) for g and g′ in Sp(W). Here r is called the standard Weil representation and c is called the multiplier of r in [Rao]. In particular, if the element g of Sp(W) is written, as above,

g = ( \begin{matrix}a & b \\ c & d \end{matrix} )

where a ∈ Hom_{F} (X, X), b ∈ Hom_{F} (Y, X), c ∈ Hom_{F} (X, Y) and d ∈ Hom_{F} (Y, Y), then in the notation of [Rao],

g^{−1} = ( \begin{matrix}a^{∗} & −b^{∗} \\ −c^{∗} & d^{∗} \end{matrix} ) .

Thus,

ω ( \begin{matrix}a & b \\ c & d \end{matrix} ) = r ( \begin{matrix}a^{∗} & −b^{∗} \\ −c^{∗} & d^{∗} \end{matrix} ) .

If g ∈ Sp(W) then the matrix of g in the basis {x_{1}, …, y_{n}} is the matrix of g^{−1} in the basis {e_{1}, …, e_{n}^{∗}}. Define x_{R} : Sp(W) → k^{×}/k^{×2} by x_{R}(p_{1}τ_{k}p_{2}) = det((p_{1}p_{2})|_{X}) for p_{1} and p_{2} in P and k between 0 and n. A computation shows that x_{R}(G) = x(g^{−1}) for g ∈ Sp(W) where x is as in [Rao].

Second, the group H of [Rao], called the Heisenberg group, is different from but related to H. The group H of [Rao] has as underlying set W × ℂ^{1} and multiplication defined by

(w, z)(w′, z′) = (w + w′, zz′ψ(⟨y, x′⟩))

where w = y + x and w′ = y′ + x′, with x, x′ ∈ X and y, y′ ∈ Y. The paper [Rao] also defines a right action of Sp(W), with the law of composition of [Rao], on H. There is a homomorphism i from H to H that maps (w, t) to (w, ψ(t + (1/2)⟨y, x⟩)), where w = y + x with x ∈ X and y ∈ Y. Under this map, i(g · h) = i(h) · g for h ∈ H and g ∈ Sp(W).

8. Reductive dual pairs. If A and A′ are groups and δ is a representation of A×A′, then δ defines a correspondence between representations of A and A′. That is, δ defines a subset of the cartesian product of the set of representations of A with the set of representations of A′: if π is a representation of A and σ is a representation of A′, declare that π and σ correspond if and only if there exists a nonzero A × A′ map from δ to the tensor product of π and σ.

Though the correspondence defined by an arbitrary δ need not be interesting, there are many instances when the correspondence has a nontrivial structure. For example, if G is a compact group, take A and A′ to be G, and let δ be the representation on L^{2}(G) defined by right and left translation. Then δ defines a correspondence between the set of finite dimensional irreducible unitary representations of G and itself, and the correspondence is the graph of the bijection that takes a representation to its contragredient. Classical invariant theory furnishes other examples.

We will use the Weil representation to obtain some other examples. Our strategy will be to consider pairs of subgroups of Sp(W) that commute and are as large as possible. The inverse images of such a pair in Mp(W) will also commute, by the following proposition. By restricting the Weil representation to the product of the inverse images we obtain a representation as above. For a proof of the next proposition, see [MVW].

Proposition. Let g and h be in Sp(W), and suppose g̃ and h̃ are elements of Mp(W) that project onto g and h, respectively. If g and h commute, then g̃ and h̃ commute.

To be more precise, suppose that (A, A′) is a pair of subgroups of Sp(W). We say that (A, A′) is a dual pair if A is the centralizer in Sp(W) of A′ and A′ is the centralizer in Sp(W) of A. If (A, A′) is dual pair, then (A, A′) is a reductive dual pair if the actions of A and A′ on W are absolutely reductive. A reductive dual pair (A, A′) is reducible if there exist nonzero orthogonal F subspaces W and W′ of W such that W = W ⊕W′ and (A, A′) preserve W and W′. In this case, A ≅ A|_{W} × A|_{W′}, A′ ≅ A′|_{W} × A′|_{W′}, and (A|_{W} , A′|_{W}), and (A|_{W′}, A′|_{W′}) are reductive dual pairs of Sp(W) and Sp(W′), respectively. If no such subspaces exist, then (A, A′) is an irreducible reductive dual pair. The irreducible reductive dual pairs fall naturally into two types. For the proofs of the next two results see [H].

Proposition. Let (A, A′) be an irreducible reductive dual pair. Then either AA′ acts irreducibly on W, or there exists a complete polarization W = X ⊕ Y such that AA′ preserves and acts irreducibly on both X and Y.

We say that an irreducible reductive dual pair (A, A′) is of type I if the first possibility of the proposition holds; if the second possibility holds we say that the pair (A, A′) is of type II.

The irreducible dual pairs of each type can be precisely characterized. To do so, we will need some definitions. Suppose that D is a division algebra over F, not necessarily commutative, equipt with an involution τ; a map τ from D to D is called an involution if τ is additive, the composition of τ with itself is the identity, and τ(dd′) = τ(d′)τ(d) for d and d′ in D. Let V be a left vector space over D. A map ( , ) : V × V → D is called sesquilinear if ( , ) is additive in both variables, and (dv, d′v′) = d(v, v′)τ(d′) for d, d′ ∈ D and v, v′ ∈ V. We say that ( , ) is Hermitian if (v, v′) = τ(v′, v) for v, v′ ∈ V. We say that ( , ) is skew Hermitian if (v, v′) = −τ(v′, v) for v, v′ ∈ V. If V′ is a right D vector space, a map ( , )′ : V′ × V′ → D is called sesquilinear if ( , )′ is additive in both variables, and (wd, w′d′) = τ(d)(w, w′)d′ for d, d′ ∈ D and w, w′ ∈ V′. We say that ( , )′ is Hermitian or skew Hermitian if exactly the same conditions hold as in the case of left D vector spaces.

Theorem. Let (A, A′) be an irreducible dual reductive pair.

(1) The pair (A, A′) is of type I if and only if there exists a division algebra D over F with involution ι, an Hermitian right D vector space (V, ( , )), an skew Hermitian left D vector space (V′, ( , )′), an isomorphism W ≅ V ⊗_{D} V′ as vector spaces over F such that under this identification, ⟨ , ⟩ corresponds to T_{D/F} ∘(( , ) ⊗ τ ∘ ( , )′) and A and A′ correspond to the isometry groups U(V) and U(V′) of ( , ) and ( , )′, respectively.

(2) The pair (A, A′) are of type II if and only if there exists a complete polarization W = X ⊕ Y, a division algebra D over F , a right D vector space U and a left D vector space U′ and an F isomorphism X ≅ U ⊗_{D} U′ such that AA′ preserves X and Y and under this identification, A and A′ are identified with Gl_{D}(U) and Gl_{D}(U′), respectively.

In the next lecture it will also be important to consider what might be called degenerate irreducible reductive dual pairs. Suppose that D is a division algebra over F with involution τ, (V, ( , )) is an Hermitian right D-vector space, and (V′, ( , )′) is an skew Hermitian left D vector space. Form the F vector space W = V ⊗_{D} V′, and equip it with the nondegenerate symplectic bilinear form T_{D/F} ∘(( , ) ⊗ τ ∘ ( , )′). Then U(V) and U(V′) can be regarded as subgroups of Sp(W) if and only if V ≠ 0 and V′ ≠ 0 or V = 0 and V′ = 0. In the next lecture we also will need to consider the case when exactly one of V and V′ is 0. Though in this case U(V) and U(V′) can no longer be regarded as subgroups of Sp(W) = 1, we will still say that (U(V), U(V′)) is an irreducible reductive dual pair. In this case, we say that U(V) × ℂ^{1} and U(V′) × ℂ^{1} are the inverse images of U(V) and U(V′) in Mp(W), respectively. Similar comments apply to the case of type II irreducible reductive dual pairs.

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