On the Notion of an Automorphic Representation *

The irreducible representations of a reductive group over a local field can be obtained from the square-integrable representations of Levi factors of parabolic subgroups by induction and formation of subquotients [2], [4]. Over a global field F the same process leads from the cuspidal representations, which are analogues of square-integrable representations, to all automorphic representations.

Suppose P is a parabolic subgroup of G with Levi factor M and σ = ⊗σ_{v} a cuspidal representation of M(A). Then Ind σ = ⊗_{v}Ind σ_{v} is a representation of G(A) which may not be irreducible, and may not even have a finite composition series. As usual an irreducible subquotient of this representation is said to be a constituent of it.

For almost all v, Ind σ_{v} has exactly one constituent π_{v}^{∘} containing the trivial representation of G(O_{v}). If Ind σ_{v} acts on X_{v} then π_{v}^{∘} can be obtained by taking the smallest G(F_{v})-invariant subspace V_{v} of X_{v} containing nonzero vectors fixed by G(O_{v}) together with the largest G(F_{v})- invariant subspace U_{v} of V_{v} containing no such vectors and then letting G(F_{v}) act on V_{v}/U_{v}.

Lemma 1. The constituents of Ind σ are the representations π = ⊗π_{v} where π_{v} is a constituent of Ind σ_{v} and π_{v} = π_{v}^{∘} for almost all v.

That any such representation is a constituent is clear. Conversely let the constituent π act on V/U with 0 ⊆ U ⊆ V ⊆ X = ⊗X_{v}. Recall that to construct the tensor product one chooses a finite set of places S_{0} and for each v not in S_{0} a vector x_{v}^{∘} which is not zero and is fixed by G(O_{v}). We can find a finite set of places S, containing S_{0}, and a vector x_{S} in X_{S} = ⊗_{v∈S}X_{v} which are such that x = x_{S} ⊗ (⊗_{v∉S}x_{v}^{∘}) lies in V but not in U.

Let V_{S} be the smallest subspace of X_{S} containing x_{S} and invariant under G_{S} = Π_{v∈S}G(F_{v}). There is clearly a surjective map

V_{S} ⊗ (⊗_{v∉S}V_{v}) → V/U,

* Appeared in Proceedings of Symposia in Pure Mathematics Vol. 33 (1979), part 1, pp. 203–207 and if v_{0} ∉ S the kernel contains V_{S} ⊗ U_{v_{0}} ⊗ (⊗_{v∉S∪{v_{0}}}V_{v}). We obtain a surjection V_{S} ⊗ (⊗_{v∈S}V_{v}/U_{v}) → V/U with a kernel of the form U_{S} ⊗ (⊗_{v∈S}V_{v}/U_{v}), U_{S} lying in V_{S}. The representation of G_{S} on V_{S}/U_{S} is irreducible and, since Ind σ_{v} has a finite composition series, of the form ⊗_{v∈S}π_{v}, π_{v} being a constituent of Ind σ_{v}. Thus π = ⊗π_{v} with π_{v} = π_{v}^{∘} when v ∉ S.

The purpose of this note is to establish the following proposition.*

Proposition 2. A representation π of G(A) is an automorphic representation if and only if π is a constituent of Ind σ for some P and some σ.

The proof that every constituent of Ind σ is an automorphic representation will invoke the theory of Eisenstein series, which has been fully developed only when the global field F has characteristic zero [3]. One can expect however that the analogous theory for global fields of positive characteristic will appear shortly; so there is little point in making the restriction to characteristic zero explicit in the proposition. Besides, the proof that every automorphic representation is a constituent of some Ind σ does not involve the theory of Eisenstein series in any serious way.

We begin by remarking some simple lemmas.

Lemma 3. Let Z be the center of G. Then an automorphic form is Z(A)-finite.

This is verified in [1].

Lemma 4. Suppose K is a maximal compact subgroup of G(A) and φ an automorphic form with respect to K. Let P be a parabolic subgroup of G. Choose g ∈ G(A) and let K′ be a maximal compact subgroup of M(A) containing the projection of gKg^{−1} ∩ P (A) on M(A). Then

φ_{P} (m; g) = ∫_{N(F)\N(A)} φ(nmg)dn

is an automorphic form on M(A) with respect to K′.

It is clear that the growth conditions are hereditary and that φ_{P} (·; g) is smooth and K′- finite. That it transforms under admissible representations of the local Hecke algebras of M is a consequence of theorems in [2] and [4].

* The definition of an automorphic representation is given in the paper [1] by A. Borel and H. Jacquet to which this paper was a supplement.

We say that φ is orthogonal to cusp forms if ∫_{ΩG_{der}(A)} φ(g)ψ(g) dg = 0 whenever ψ is a cusp form and Ω is a compact set in G(A). If P is a parabolic subgroup we write ψ⊥P if φ_{P} (·; g) is orthogonal to cusp forms on M(A) for all g. We recall a simple lemma [3].

Lemma 5. If φ⊥P for all P then φ is zero.

We now set about proving that any automorphic representation π is a constituent of some Ind σ. We may realize π on V/U, where U, V are subspaces of the space 𝒜 of automorphic forms and V is generated by a single automorphic form φ. Totally order the conjugacy classes of parabolic subgroups in such a way that {P } < {P′} implies rank P ≤ rank P′ and rank P < rank P′ implies {P } < {P′}. Given φ let {P_{φ}} be the minimum {P } for which {P } < {P′} implies φ⊥P′. Amongst all the φ serving to generate π choose one for which {P } = {P_{φ}} is minimal. If ψ ∈ 𝒜 let ψ_{P} (g) = ψ_{P} (1, g) and consider the map ψ → ψ_{P} on V. If U and V had the same image we could realize π as a constituent of the kernel of the map. But this is incompatible with our choice of φ and hence if U_{P} and V_{P} are the images of U and V we can realize π in the quotient V_{P} /U_{P}.

Let 𝒜_{P}^{∘} be the space of smooth functions ψ on N(A)P (F)\G(A) satisfying the following conditions.

(a) ψ is K-finite.

(b) For each g the function m → ψ(m, g) = ψ(mg) is automorphic and cuspidal. Then V_{P} ⊆ 𝒜_{P}^{∘}. Since there is no point in dragging the subscript P about, we change

notation, letting π be realized on V/U with U ⊆ V ⊆ 𝒜_{P}^{∘}. We suppose that V is generated by a single function φ.

Lemma 6. Let A be the center of M. We may so choose φ and V that there is a character χ of A(A) satisfying φ(ag) = χ(a)φ(g) for all g ∈ G(A) and all a ∈ A(A).

Since P (A)\G(A)/K is finite, Lemma 3 implies that any φ in 𝒜_{P}^{∘} is A(A)-finite. Choose V and the φ generating it to be such that the dimension of the span Y of {l(a)φ|a ∈ A(A)} is minimal. Here l(a) is left translation by a. If this dimension is one the lemma is valid. Otherwise there is an a ∈ A(A) and α ∈ C such that 0 < dim(l(a) − α)Y < dimY. There are two possibilities. Either (l(a) − α)U = (l(a) − α)V or (l(a) − α)U ≠ (l(a) − α)V. In the second case we may replace φ by (l(a) − α)φ, contradicting our choice. In the first we can realize π as a subquotient of the kernel of l(a) − α in V.

What we do then is to choose a lattice B in A(A) such that BA(F) is closed and BA(F)\A(A) is compact. Amongst all those φ and V for which Y has the minimal possible dimension we choose one φ for which the subgroup of B defined as {b ∈ B|l(b)φ = βφ, β ∈ C} has maximal rank. What we conclude from the previous paragraph is that this rank must be that of B. Since φ is invariant under A(F) and BA(F)\A(A) is compact, we conclude that Y must now have dimension one. The lemma follows.

Choosing such a φ and V we let ν be that positive character of M(A) which satisfies

ν(a) = |χ(a)|, a ∈ A(A),

and introduce the Hilbert space L_{2}^{∘} = L_{2}^{∘}(M(F)\M(A), χ) of all measurable functions ψ on M(Q)\M(A) satisfying the following conditions.

(i) For all m ∈ M(A) and all a ∈ A(A), ψ(ag) = χ(a)ψ(g). (ii) ∫_{A(A)M(Q)\M(A)} ν^{−2}(m)|ψ(m)|^{2}dm < ∞.

L_{2}^{∘} is a direct sum of irreducible invariant subspaces and if ψ ∈ V then m → ψ(m, g) lies in L_{2}^{∘} for all g ∈ G(A). Choose some irreducible component H of L_{2}^{∘} on which the projection of ψ(·, g) is not zero for some g ∈ G(A).

For each ψ in V define ψ′(·, g) to be the projection of ψ(·, g) on H. It is easily seen that, for all m_{1} ∈ M(A), ψ′(mm_{1}, g) = ψ′(m, m_{1}g). Thus we may define ψ′(g) by ψ′(g) = ψ′(1, g). If V′ = {ψ′|ψ ∈ V }, then we realize π as a quotient of V′. However if δ^{2} is the modular function for M(A) on N(A) and σ the representation of M(A) on H then V′ is contained in the space of Ind δ^{−1}σ.

To prove the converse, and thereby complete the proof of the proposition, we exploit the analytic continuation of the Eisenstein series associated to cusp forms. Suppose π is a representation of the global Hecke algebra ℋ, defined with respect to some maximal compact subgroup K of G(A). Choose an irreducible representation k of K which is contained in π. If E_{κ} is the idempotent defined by κ let ℋ_{κ} = E_{κ}ℋE_{k} and let π_{κ} be the irreducible representation of ℋ_{κ} on the κ-isotypical subspace of π. To show that π is an automorphic representation, it is sufficient to show that π_{κ} is a constituent of the representation of ℋ_{κ} on the space of automorphic forms of type κ. To lighten the burden on the notation, we henceforth denote π_{κ} by π and ℋ_{κ} by ℋ.

Suppose P and the cuspidal representation σ of M(A) are given. Let L be the lattice of rational characters of M defined over F and let L_{C} = L ⊗ C. Each element μ of L_{C} defines a character ξ_{μ} of M(A). Let I_{μ} be the κ-isotypical subspace of Ind ξ_{μ}σ and let I = I_{0}. We want to show that if π is a constituent of the representation of I then π is a constituent of the representation of ℋ on the space of automorphic forms of type κ.

If {g_{i}} is a set of coset representatives for P (A)\G(A)/K then we may identify I_{μ} with I by means of the map φ → φ_{μ} with

φ_{μ}(nmg_{i}k) = ξ_{μ}(m)φ(nmg_{i}k).

In other words we have a trivialization of the bundle {I_{μ}} over L_{C}, and we may speak of a holomorphic section or of a section vanishing at μ = 0 to a certain order. These notions do not depend on the choice of the g_{i}, although the trivialization does.

There is a neighborhood U of μ = 0 and a finite set of hyperplanes passing through U such that for μ in the complement of these hyperplanes in U the Eisenstein series E(φ) is defined for φ in I_{μ}. To make things simpler we may even multiply E by a product of linear functions and assume that it is defined on all of U. Since it is only the modified function that we shall use, we may denote it by E, although it is no longer the true Eisenstein series. It takes values in the space of automorphic forms and thus E(φ) is a function g → E(g, φ) on G(A). It satisfies

E(ρ_{μ}(h)φ) = r(h)E(φ)

if h ∈ ℋ and ρ_{μ} is Ind ξ_{μ}σ. In addition, if φ_{μ} is a holomorphic section of {I_{μ}} in a neighborhood of 0 then E(g, φ_{μ}) is a holomorphic in μ for each g, and the derivatives of E(φ_{μ}), taken pointwise, continue to be in 𝒜.

Let I_{r} be the space of germs of degree r at μ = 0 of holomorphic sections of I. Then φ_{μ} → ρ_{μ}(h)φ_{μ} defines an action of ℋ on I_{r}. If s ≤ r the natural map I_{r} → I_{s} is an ℋhomomorphism. Denote its kernel by I_{r}^{s}. Certainly I_{0} = I. Choosing a basis for the linear forms on L_{C} we may consider power series with values in the κ-isotypical subspace of 𝒜, Σ_{|α|≤r}μ^{α}ψ_{α}. ℋ acts by right translation on this space and the representation so obtained is, of course, a direct sum of that on the κ-isotypical automorphic forms. Moreover φ_{μ} → E(φ_{μ}) defines an ℋ-homomorphism λ from I_{r} to this space. To complete the proof of the proposition all one needs is the Jordan-Hölder theorem and the following lemma.

Lemma 7. For r sufficiently large the kernel of λ is contained in I_{r}^{0}.

Since we are dealing with Eisenstein series associated to a fixed P and σ we may replace E by the sum of its constant terms for the parabolic associated to P, modifying λ accordingly. All of these constant terms vanish identically if and only if E itself does. If Q_{1}, …, Q_{m} is a set of representatives for the classes of parabolics associated to P let E_{i}(φ) be the constant term along Q_{i}. We may suppose that M is a Levi factor of each Q_{i}. Define ν(m) for m ∈ M(A) by ξ_{μ}(m) = e^{⟨μ,ν(m)⟩}. Thus ν(m) lies in the dual of L_{R}. If φ ∈ I_{μ}, the function of E_{i}(φ) has the form

E_{i}(nmg_{j}k, φ) = ∑_{α=1}^{a} ∑_{β=1}^{b} p_{α}(ν(m))ξ_{ν_{β}(μ)}(m)ψ_{αβ}(m, k).

Here ψ_{αβ} lies in a finite-dimensional space independent of μ and g_{j}; ν_{β}, 1 ≤ β ≤ b is a holomorphic function of μ; and {p_{α}} is a basis for the polynomials of some degree. This representation may not be unique. The next lemma implies that we may shrink the open set U and then find a finite set h_{1}, …, h_{n} in G(A) such that E(φ_{μ}) is 0 for μ ∈ U, φ_{μ} ∈ I_{μ} if and only if the numbers E_{i}(h_{j}, φ_{μ}), 1 ≤ i ≤ m, 1 ≤ j ≤ n are all 0.

Lemma 8. Let U be a neighborhood of 0 in C^{l}, ν_{1}, …, ν_{k} holomorphic functions on U and p_{1}, …, p_{a} a basis for the polynomials of some given degree. Then there is a neighborhood V of 0 contained in U and a finite set y_{1}, …, y_{b} in C^{l} such that if μ ∈ V then

(∗)  Σp_{i}(y)e^{νj(μ)·y} = 0

for all y if and only if it is 0 for y = y_{1}, …, y_{b}.

To prove this lemma one has only to observe that the analytic subset of U defined by the equation (∗), y ∈ C^{l}, is defined in a neighborhood of 0 by a finite number of them.

We may therefore regard E as a function on U with values in the space of linear transformations from the space I, which is finite-dimensional, to the space C^{mn}. One knows from the theory of Eisenstein series that E_{μ} is injective for μ in an open subset of U. Then to complete the proof of the proposition, we need only verify the following lemma.

Lemma 9. Suppose E is a holomorphic function in U, a neighborhood of 0 on C^{l}, with values in Hom(I, J), where I and J are finite-dimensional spaces, and suppose that E_{μ} is injective on an open subset of U. Then there is an integer r such that if φ_{μ} is analytic near μ = 0 and the Taylor series of E_{μ}φ_{μ} vanishes to order r then φ_{0} = 0.

Projecting to a quotient of J, we may assume that dim I = dim J and even that I = J. Let the first nonzero term of the power-series expansion of det E_{μ} have degree s. Then we will show that r can be taken to be s + 1. It is enough to verify this for l = 1, for we can restrict to a line on which the leading term of E_{μ} still has degree s. But then multiplying E fore and aft by nonsingular matrices we may suppose it is diagonal with entries z^{α}, 0 ≤ α ≤ s. Then the assertion is obvious.

In conclusion I would like to thank P. Deligne, who drew my attention to a blunder in the first version of the paper.

References

1. A. Borel and H. Jacquet, Automorphic forms and automorphic representations, these Proceedings, part 1, pp. 189–202.

2. P. Cartier, Representations of 𝔭-adic groups: A survey, these Proceedings, part 1, pp. 111– 155.

3. R.P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Math., vol. 544, Springer, New York, 1976.

4. N. Wallach, Representations of reductive Lie groups, these Proceedings, part 1, pp. 71–86.