LIFTING AUTOMORPHIC FORMS FROM G_{2} TO PGSp(6) USING THE MINIMAL REPRESENTATION

OF E_{7}

GORDAN SAVIN

1. Introduction

This talk concerns the construction of certain automorphic forms. It contains joint work with B. Gross.

Let us begin with some questions.

Question: Give a cusp form on GL_{2}.

Answer: Ramanujan’s Δ function.

Question: Give a cusp form on GL_{3}.

Answer: The Gelbart-Jacquet lift of Δ from GL_{2} Question: Give a cusp form on GL_{n}. Best Approaches: Jacquet-Langlands, base change, etc.

The moral of the story is that, although they are known to exist, it is not easy to construct cusp forms explicitly. However, once we have a cusp form on one group, it is sometimes possible to construct cusp forms on other groups if we have some explicit version of Langlands’ functoriality.

2. A canonical form on G_{2}

First we construct the global form of G_{2} that we will use. Let 𝕆 be the octonion algebra of Cayley numbers. Coxeter has shown that 𝕆 may be defined over ℤ. Hence, we can get an algebra over any ring R by setting 𝕆(R) = 𝕆 ⊗_{ℤ}R. Set G = Aut 𝕆. This is an algebraic group defined over ℤ. As is well known,

• G(ℝ) is the compact real form of G_{2}, and

• for finite p, G(ℚ_{p}) is split G_{2}. (Indeed, this is the only form of G_{2}

over a p-adic field.) Note that, since G is compact at ℝ, G(ℚ)\G(𝔸) is compact.

B. Gross has proved the following facts about G.

Date: Notes by Karl E. Rumelhart.

2 Lifting Automorphic Forms from G_{2} to PGSp(6)

1. G(ℤ) ≅ G_{2}(2). This notation refers to the Chevalley group over 𝔽_{2}. (Note that it is obvious that G(ℤ) is finite because it is a discrete subgroup of the compact group G(ℝ).)

2. Let K_{p} = G(ℤ_{p}) be the hyperspecial maximal compact subgroup of G(ℚ_{p}), and let K_{p}(1) be the first congruence subgroup. We have K_{p}/K_{p}(1) ≅ G_{2}(p). Define

K = ∏_{p} K_{p}, and K′ = K_{2}(1) ∏_{p>2} K_{p}.

Then it is true that

G(𝔸) = G(ℚ) × G(ℝ) × K′.

3. Let W be the irreducible representation of G(ℝ) with highest weight ρ. Note that, as G_{2} has six positive roots, the dimension of W is 2^{6} = 64. Then the following is true

W |_{G(ℤ)≅G_{2}(2)} = st.

Here st is the Steinberg reresentation of the finite Chevalley group G_{2}(2).

Now we construct an automorphic form on G. It should be viewed as an analogue of Ramanujan’s Δ. (Later we will lift is to PGSp_{6} using the minimal representation of E_{7}.)

Proposition 1. There exists a unique automorphic representation π = ⨂_{v} π_{v} ⊆ L^{2}(G(ℚ)\G(𝔸)) which satisfies

1. π_{∞} = W,

2. π_{2} = St, the p-adic Steinberg representation, and

3. for p > 2, π_{p} is spherical.

Proof. Any such π has K′ invariants. Furthermore, using Gross’ decomposition of G(𝔸) and the Peter-Weyl theorem,

L^{2}(G(ℚ)\G(𝔸))^{K′} = L^{2}(G(ℚ)\G(𝔸)/K′)
= L^{2}(G(ℝ))
= ˆ⨁_{E∈Ǧ(ℝ)}E ⊗ E.

This is a representation of K/K′ × G(ℝ) ≅ G_{2}(2) × G(ℝ), and the part corresponding to π must be W ⊗ W. Restricting from G(ℝ) to G(ℤ) gives a representation of G_{2}(2) ≅ G(ℤ) which is st. Since the Steinberg representation of G_{2}(ℚ_{2}) is the only irreducible representation with minimal K-type st, the proposition follows.

3. Lifting to PGSp_{6}

Set G′ = PGSp_{6}. We have the following inclusion of dual groups.

Ĝ = G_{2}(ℂ) ↪^{i} Ĝ′ = Spin_{7}(ℂ).

By the Langlands philosophy, there should be a corresponding lift of automorphic forms. Of course, understanding this is a difficult problem and, even when we have good local control, it is not clear how to realize the lift automorphically. Our idea is use dual pairs.

Let H be the group of adjoint type over ℚ with the following properties: H^{0}(ℝ) = E_{7,3} (this is the group that acts on the exceptional tube domain), and H(ℚ_{p}) is split. H has been studied by Satake. Its restricted root system over ℚ is of type C_{3}. The long root spaces are 1-dimensional and the short root spaces are 8-dimensional. In fact, the short root spaces can be identified with 𝕆. Morally, one can think of H as PGSp_{6}(𝕆). It is now easy to see how G×G′ embeds as a dual pair in H. Since ℚ ↪ 𝕆(ℚ), G′ = PGSp_{6}(ℚ) ↪ PGSp_{6}(𝕆) = H. Moreover, since ℚ = 𝕆^{G}, it is clear that the centralizer of G is isomorphic to G′.

Let Π_{v} be the minimal representation of H_{v}. It is a result of Kim that

⨂_{v} Π_{v} ⊆ L^{2}(H(ℚ)\H(𝔸)).

This representation should be viewed as an analogue of the Weil representation. In particular, it can be used to provide explicit liftings.

We turn to results on correspondences. There are two types of local results (real and p-adic) and global results.

Real. Since G(ℝ) is compact, we can write

Π_{∞} = ⨁_{E∈Ǧ(ℝ)}E ⊗ Θ(E).

It can be shown that Θ(E) is always holomorphic discrete series and that the correspondence E ↦ Θ(E) is a Langlands’ lift.

p-adic. For π a representation of G(ℚ_{p}), define Θ(π) = {π′|π ⊗ π′ is a quotient of Π_{p}}. Also, we will write π_{S} for the spherical representation of G with Satake parameter S. Similarly, π_{S′} is the spherical representation of G′ with Satake parameter S′. Then, with Magaard, we have obtained the following results.

1. If π_{S} is tempered, then Θ(π_{S}) = {π_{i(S)}}. (Recall that i: Ĝ ↪ Ĝ′.)

2. If π_{S′} is a quotient of Π_{p}, then S′ = i(S) for some S.

3. Θ(St) = {St′}. Here St, St′ are the Steinberg representations of G, G′ respectively. Note that this result is predicted by the local Langlands correspondence since the inclusion i takes regular nilpotent elements to regular nilpotent elements.

Global. (These results are motivated by work of Ginzburg, Rallis and Soudry.) For π any form on G, define Θ(π) by the usual theta integral. There are two types of results about Θ(π): cuspidality and non-vanishing.

Cuspidality. Θ(π) is cuspidal if at least one p-adic component is generic, that is, has a Whittaker model. Note that St is generic.

Non-vanishing. Let P = MN be the Siegel parabolic of G′. We have M ≅ GL_{3} and N ≅ 3 × 3 symmetric matrices. The action of M on N is given by gA = (det g)^{−1}gAg^{T}. Our idea for proving nonvanishing of the theta lift is to study Fourier coefficeints along N. To show that a lift is non-zero, it is enough to show that one of its Fourier coefficients is non-zero. We will do this by relating it to a certain period functional on G. To define the period we will need the following result on M-orbits in N.

Proposition 2. M orbits of rank three elements in N are parametrized by quaternion algebras over ℚ.

Proof. A matrix A in N defines a quadratic form by q(x) = x^{T} Ax. Thus, with A we can associate the even Clifford algebra D = Cliff^{+}(q). By the general formula, dim D = 2^{3−1} = 4.

Fix a non-trivial additive character Ψ: 𝔸/ℚ → ℂ^{∗}. For each A ∈ N(ℚ) set Ψ_{A}(X) = Ψ(trace(AX)). Now, for σ′ any automorphic representation on G′, we can associate the “Ath Fourier coefficient”, λ′_{A}: σ′ → ℂ given by

λ′_{A}(f′) = ∫_{N(ℚ)\N(𝔸)} f′(X)ψ_{A}(X)dX.

On the other hand, by the proposition there is a quaternion algebra, D, corresponding to A. Assume that D is definite at ∞ so that D ↪ 𝕆. Let C be the centralizer of D in G. C is isomorphic to the norm one elements in D. Now if π is any automorphic representation of G, we can define a period functional λ_{D}: π → ℂ by

λ_{D}(f) = ∫_{C(ℚ)\C(𝔸)} f(x)dx.

We have the following result.

Theorem 1. Let π be an automorphic representation of G. Then the functional λ_{D} ≠ 0 on π if and only if the functional λ′_{A} ≠ 0 on Θ(π).

Using the theorem, we can prove that the canonical form on G constructed above has a non-zero lift. Indeed, just take A to be the 3 × 3 identity matrix. The corresponding quaternion algebra is Hamilton’s quaternions. It is easy to see that in this case the period functional is non-zero. Thus, the lift has a non-zero Fourier coefficient. In particular, the lift does not vanish.

We summarize our results in the following corollary.

Corollary 1. There exists a cusp form π′ on G′ such that 1. π′_{∞} is holomorphic discrete series with infinitesimal character 2ρ′.

2. π′_{2} = St′.

3. For p > 2, π′_{p} is spherical with the Satake parameter coming from G_{2}.