THE 1994 UNIVERSITY OF MARYLAND CONFERENCE ON THE

THETA CORRESPONDENCE, DUAL

PAIRS AND AUTOMORPHIC FORMS

LECTURE III:

NONARCHIMEDEAN EXAMPLES

Brooks Roberts

In this lecture we will consider some nonarchimedian examples. We will retain the notation from the last lecture. Suppose also that D = F and τ is the identity. Then (V, ( , )) is a nondegenerate symmetric bilinear space of dimension m, and (V′, ( , )′) is a nondegenerate symplectic bilinear space. As is customary, we will write O(V) for U(V) and Sp(n, F) for U(V′), where n = m′ = Witt(V′); note that V′ depends only on n.

Since m is even, it is possible to introduce splittings and work with O(V) and Sp(n, F) instead of the inverse images G and G′. In the last lecture we already remarked that since V′_{0} = 0, we have G ≅ O(V) × ℂ^{1}. One can also show that since m is even, G′ is split as an extension of Sp(n, F) by ℂ^{1}. The proof of this result requires an explicit understanding of the Rao cocycle, and will be omitted. For the remainder of this lecture we fix a splitting Sp(n, F) ↪ G′ of G′, and let ω be the mixed model from the last lecture, i.e., the Schrödinger model with respect to the complete polarization W = (V ⊗_{F} X′) ⊕ (V ⊗_{F} Y′). Let ω_{m,n} be the composition of

O(V) × Sp(n, F) ↪ G × G′ ↪ Mp(W)

with ω. Here O(V) is the map that takes g to (g, 1). It follows that ω_{m,n}(g, 1)φ = (g⊗1)·φ for g ∈ O(V); again, O(V) acts by left translation composed with inversion on 𝒮(V ⊗_{F} Y′). We will consider the theta correspondence defined by ω_{m,n}.

1. Connection with previous work. The work from the last lecture applies to the theta correspondence defined by ω_{m,n}. If π ∈ Irr(O(V)) and σ ∈ Irr(Sp(n, F)) then one can define representations π′ ∈ Irr(G) and σ′ ∈ Irr(G′) such that π′|_{O(V)} = π and σ′|_{Sp(n,F)} = σ. The analogues of all the results from the last lecture can be deduced from the last lecture using this observation.

2. Two dimensional spaces. We will consider the case

dim_{F} V = 2.

Typeset by AMS-TEX

For the remainder of this lecture, suppose that dim_{F} V = 2. Let us try to describe O(V). Consider the Witt decomposition of V. There are two possibilities: either Witt(V) = 0 or 1. Suppose Witt(V) = 1. In this case, it is easy to describe O(V). The isotropic flag considered in the last lecture is just X_{1}. The elements of the parabolic subgroup associated to X_{1} have the form

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

with respect to the decomposition V = X_{1} ⊕ Y_{1}; here a ∈ F^{×}. Thus, the parabolic group has trivial unipotent radical. We will shortly see that this parabolic subgroup is SO(V); along with the map that sends x_{1} ⊕ y_{1} to y_{1} ⊕ x_{1}, this parabolic subgroup generates O(V). Suppose next Witt(V) = 0. In this case a description of O(V) is not so immediate. The isotropic flag considered in the last lecture is empty, and O(V) has no proper parabolic subgroup. To describe O(V) when Witt(V) = 0 requires a deeper understanding of V, especially in the case when Witt(V) = 0.

Up to isomorphism there are 2|F^{×}/F^{×}| − 1 nondegenerate two dimensional symmetric bilinear spaces. Recall that, in general, two nondegenerate symmetric bilinear spaces of the same dimension are isomorphic if and only if they have the same determinant and Hasse invariant. Thus, up to isomorphism, there are potentially 2|F^{×}/F^{×2}| nondegenerate two dimensional symmetric bilinear spaces. However, there is no nondegenerate two dimensional symmetric bilinear space with determinant −1 and Hasse invariant −1.

We can construct models of these symmetric bilinear spaces in the following way. Let d ∈ F^{×}/F^{×2} and ϵ ∈ {±1}, and assume that (d, ϵ) ≠ (−1, −1). Define a two dimensional F algebra by

E = E(d, ϵ) = \{ \begin{matrix}F × F if d = −1 \\ F (\sqrt{−d}) if d ≠ −1. \end{matrix}

Let Gal(E/F) = {1, τ}. Let V = V (d, ϵ) be E regarded as an F vector space. If ϵ = 1, then define a nondegenerate symmetric bilinear form on V by

(x, y) = \frac{T_{F}^{E}(xτ(y))}{2}.

If d ≠ −1, then fix a representative δ for the nontrivial coset of F^{×}/ N_{F}^{E}(E^{×}). If ϵ ≠ 1, so that d ≠ −1, then define a nondegenerate symmetric bilinear form on V by

(x, y) = δ \frac{T_{F}^{E}(xτ(y))}{2}.

The nondegenerate symmetric bilinear space (V, ( , )) has determinant d and Hasse invariant ϵ.

3. The admissible dual. We will now determine the admissible dual of O(V). To do so, we recall the following well known result. For a proof see [L].

Theorem. Let E^{1} be the set of u ∈ E^{×} such that N_{F}^{E}(u) = 1. Then

SO(V) ≅ E^{1}, O(V) ≅ E^{1} ⋊ Gal(E/F).

Here E^{1} acts on V by multiplication, and E^{1} ⋊ Gal(E/F) acts by (u, ϵ)(x) = uϵ(x).

Using the theorem we can deduce some simple properties of SO(V) and O(V). By the theorem, it follows that SO(V) is abelian and the center of O(V) is {±1}. Also, O(V) is compact if and only if Witt(V) = 0, i.e., V is anisotropic.

We can also compute Irr(O(V)). By the theorem, SO(V) is abelian. It follows that the elements of Irr(SO(V)) are continuous quasi-characters of SO(V). ¿From elements of Irr(SO(V)) we can construct elements of Irr(O(V)) by induction. However, not all elements of Irr(O(V)) will give induce irreducibly. Let α ∈ Irr(SO(V)). We will say that α is generic if τ ·α ≠ α, i.e., α ≠ α^{−1}. We will say that α is singular if τ ·α = α, i.e., α = α^{−1}. If α is generic, then ind_{SO(V)}^{O(V)} α is irreducible; in this case, call this representation π(α).

Suppose α is singular. Then ind_{SO(V)}^{O(V)} α is reducible and, in fact, the direct sum of two quasi-characters of O(V). Define the quasi-character π(α) of O(V) by π(α)(g, ϵ) = α(g). Then ind_{SO(V)}^{O(V)} α is the direct sum of π(α) and π(α)⊗_{ℂ} det. We have the following theorem:

Theorem. We have

Irr(O(V)) = {π(α) : α generic} ∪ {π(α) : α singular} ∪ {π(α) ⊗_{ℂ} det : α singular}.

Finally, we consider the classification of Irr(O(V)) in terms of induction from parabolic subgroups. Suppose first that Witt(V) = 1. We saw above that SO(V) is a proper parabolic subgroup of O(V). It follows that every element of Irr(O(V)) is a constituent of an induced representation from a proper parabolic subgroup. No element of Irr(O(V)) is supercuspidal. Suppose that Witt(V) = 0. We saw above that O(V) has no proper parabolic subgroups. Since O(V) is compact, every element of Irr(O(V)) is supercuspidal.

4. Nonvanishing. We will now completely determine what representations of O(V) occur in the theta correspondence with Sp(n, F).

Theorem. Let ℛ_{n}(O(V)) be the elements of Irr(O(V)) that occur in the theta correspondence between Irr(O(V)) and Irr(Sp(n, F)). Then

ℛ_{0}(O(V)) = {1},
ℛ_{1}(O(V)) = Irr(O(V)) − {det},
ℛ_{n}(O(V)) = Irr(O(V)),

for n ≥ 2.

Proof. It is clear that ℛ_{0}(O(V)) = {1}. Let π ∈ Irr(O(V)) − {det}. We need to show that π ∈ ℛ_{1}(O(V)), i.e.,

Hom_{O(V)}(ω_{1}, π) ≠ 0.

To prove this we will essentially argue as in the proof of the existence of the stable range as in the last lecture. Note, though, that that proof breaks down since the stabilizer in O(V) of every point of V is nontrivial. However, under the action of SO(V) the stabilizers of the points of V − {0} are trivial.

Suppose first that π is one dimensional. Then π = π(α) or π = π(α) ⊗_{ℂ} det for some singular α ∈ Irr(SO(V)). Define an element x of V in the following way. If π(τ) = 1 then let x = 1. Suppose π(τ) = −1. Then α ≠ 1; let g ∈ SO(V) be such that α(g) ≠ 1. Let x ∈ V = E be such that x/τ(x) = g; by Hilbert’s theorem 90, x exists. Now define L : ω_{1} → π by

L(φ) = ∫_{SO(V)} α(h)^{−1}ω_{1}(h, 1)φ(x) dh + π(τ) ∫_{SO(V)} α(h)^{−1}ω_{1}(hτ, 1)φ(x) dh.

Since φ ∈ 𝒮(V), it follows that the integrals converge. A computation shows that L ∈ Hom_{O(V)}(ω_{1}, π). It remains to show that L ≠ 0. Using the identification SO(V) ≅ E^{1} and α = α^{−1}, we have

L(φ) = ∫_{E^{1}} α(u)φ(ux) du + π(τ) ∫_{E^{1}} α(u)φ(uτ(x)) du
= ∫_{E^{1}} α(u)φ(ux) du + π(τ) ∫_{E^{1}} α(u)φ(u(τ(x)/x)x) du
= (1 + π(τ)α(x/τ(x))) ∫_{E^{1}} α(u)φ(ux) du.

It now suffices to show that the last integral is nonzero for some φ ∈ 𝒮(V). We leave this as a simple exercise.

Suppose now that π = π(α) for some generic α ∈ Irr(SO(V)). We can take the space of π(α) to be ℂ ⊕ ℂ with the action

π(α)(g)(z_{1} ⊕ z_{2}) =α(g)z_{1} ⊕ α(g)^{−1}z_{2}
vπ(α)(τ)(z_{1} ⊕ z_{2}) =z_{2} ⊕ z_{1}

for g ∈ SO(V) and z_{1}, z_{2} ∈ ℂ. Define L : ω → π by

L(φ) = ∫_{SO(V)} α(g)^{−1}ω_{1}(g, 1)φ(1) dg ⊕ ∫_{SO(V)} α(g)^{−1}ω(gτ, 1)φ(1) dg.

Then L is a well defined element of Hom_{O(V)}(ω_{1}, π), and L ≠ 0.

Next, we need to show that det ∉ ℛ(O(V)). It suffices to show that any linear functional L on 𝒮(V) invariant under SO(V) is invariant under τ.

Suppose first that V is anisotropic so that SO(V) is compact. Let φ ∈ 𝒮(V). Integrating the equation

L(ω_{1}(g, 1)φ) = L(φ)

for g ∈ SO(V), we find that L(φ) = L(φ′), where φ′ ∈ 𝒮(V) is given by

φ′(x) = ∫_{SO(V)} ω_{1}(g, 1)φ(x) dg.

Here the measure give SO(V) measure 1. Since φ′ is invariant under SO(V), it is invariant under τ. It follows that L is invariant under O(V).

Suppose next that V is isotropic. We will argue as in section 4 of [P], using the corollary of section 1.4 of [B], which appears as Theorem 4.4 in [P]. Define V → F by x ↦ (x, x). Clearly, O(V) preserves the fibers of this map. Let F be a fiber of the map lying over x. We need to show that any distribution on F invariant under SO(V) is invariant under O(V). If x = 0 this follows from Lemma 4.6 of [P]. If x ≠ 0 then F is isomorphic to SO(V) as an SO(V) set, and it suffices to show that any distribution on SO(V) invariant under translation by SO(V) is invariant under inversion. We leave the verification of this as exercise.

Finally, the remaining statement follows from the theorem on the stable range from the last lecture. □

5. An example and a conjecture of Kudla-Prasad. A fundamental problem in theory of the theta correspondence is to determine when a representation is in the correspondence. In particular, one would like to know the smallest integer n(π) such that π ∈ ℛ_{n(π)}(O(V)), i.e., the first time π occurs in the correspondence. Note that by the existence of the stable range, n(π) exists, and

n(π) ≤ dim_{F} V.

In general, n(π) will depend on π. However, S.S. Kudla and D. Prasad have made conjecture involving n(π) that should hold for all π ∈ Irr(O(V)) for any nondegenerate even dimensional symmetric bilinear space.

In our example we found that

n(1) =0,
n(π) =1 if π ≠ 1, det,
n(det) =2.

It follows that for all π ∈ Irr(O(V)),

n(π) + n(π ⊗_{ℂ} det) = 2.

S.S. Kudla and D. Prasad have conjectured:

Conjecture (Kudla-Prasad). If V is a nondegenerate even dimensional symmetric bilinear space, then for all π ∈ Irr(O(V)),

n(π) + n(π ⊗_{ℂ} det) = dim_{F} V.

6. Construction of supercuspidals. Using the computation of ℛ_{n}(O(V)) from above and a theorem from the last lecture, we can construct supercuspidal representations of Sp(n, F). In this section we assume that the residual characteristic of F is odd so that the theorem of Howe and Waldspurger applies. Recall from the last lecture that if π ∈ Irr(O(V)) is supercuspidal, then the theta lift of π to Sp(n(π), F) is supercuspidal. We have the following theorem.

Theorem. Suppose that V is anisotropic. Then

θ(1) = 1 ∈ Irr_{sc}(Sp(0, F)),
θ(π) ∈ Irr_{sc}(Sp(1, F)) for π ∈ Irr(O(V)) − {1, det},
θ(det) ∈ Irr_{sc}(Sp(2, F)).

We make some remarks about the supercuspidal representations constructed in the theorem. For the rest of this section assume that V is anisotropic. Apparently, the theorem constructs all the supercuspidal representations of Sp(1, F) = Sl(2, F) as V ranges over all anisotropic nondegenerate two dimensional symmetric bilinear spaces. See Shalika and [GG]. One way to prove this would be to show how each of the representations constructed is induced from a compact subgroup. See [Ca].

The supercuspidal representations θ(det) ∈ Irr_{sc}(Sp(2, F)) are also interesting. The representations θ(det) have been shown to be induced from a compact subgroup. See Asmuth. In the case when d is a unit, the compact subgroup can be taken to be Sp(2, 𝔒_{F}), and the representation is the composition of the natural map

Sp(2, 𝔒_{F}) → Sp(2, 𝔒_{F} /𝔓_{F}).

with a representation of Sp(2, 𝔒_{F} /𝔓_{F}) called θ_{10}, of dimension q(1 − q)^{2}/2. This representation was constructed in [Sr]. Evidently, θ_{10} is cuspidal, does not possess a Whittaker model and is unipotent.

This example plays a key role in the construction of the counterexample to the generalized Ramanujan conjecture from [HPS]. This conjecture says that the local components of cuspidal automorphic representations should be tempered. See [Sa]. This conjecture is true for cuspidal automorphic representations of Gl(2, 𝔸_{ℚ}) that come from classical holomorphic cusp forms. See [G]. A representation of Gl(2, F) is tempered if it has a unitary central character and its matrix coefficients lie in space L^{2+ϵ}(Gl(2, F)/F^{×}) for all ϵ > 0. The tempered representations of Gl(2, F) are then the discrete series representations, along with the irreducible principal series representations π(μ_{1}, μ_{2}) with μ_{1} and μ_{2} unitary; the last representations form the continuous series representations of Gl(2, F). The remaining unitary representations of Gl(2, F) are the complementary series representations. To see how to obtain a counterexample for Sp(2, F), suppose that F is a number field. If π = ⊗_{v}π_{v} is a cuspidal automorphic representations of O(V (𝔸_{F})), then the global theta lift θ(π) of π to Sp(2, 𝔸_{F}) is nonzero. Suppose that π_{v} is det for one v. Then θ(π) is cuspidal. It follows that θ(π) = ⊗_{v}θ(π_{v}). It turns out that θ(π_{v}) is not tempered for all π_{v} ∈ Irr(O(V (F_{v}))) − {det}, and thus we have a counterexample to the generalized Ramanujan conjecture.

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