Lecture V

In this lecture I will present the proof of meromorphic continuation of Eisenstein series.

1. Let G be a group over ℚ, Γ – an arithmetic subgroup. We denote by G the group G(ℝ) of ℝ-points of G. Our main object of study is the automorphic space y.

Let P be a class of parabolic subgroups defined over ℚ. For such P we constructed the space y_{P} in lecture 4.

We denote by 𝔞_{P} the Lie algebra of the ℚ-split component A_{P} of the center of L_{P}. A_{P} and hence 𝔞 act on the space y_{P} and we consider L^{2}-twisted actions of A_{P} and 𝔞_{P} on F (y_{P}).

2. Let f be an automorphic form on y_{P}. Then it is 𝔞_{P} -finite and hence can be decomposed f = Σf_{x} over some characters x ∈ S_{P} := (𝔞_{P}^{∗})_{ℂ}. If this sum is over a finite subset R ⊂ S_{P} we say that supp_{S_{P}} (f) ⊂ R.

3. We would like to characterize every automorphic form f on y by support of all the cuspidal components of its constant terms.

More precisely, for any parabolic Q consider the form f_{Q} = C_{QG}(f) ∈ F (y_{Q}), take its cuspidal component f_{Q,cusp} and decompose it as a sum over characters x ∈ S_{Q}.

The collection of all pairs (Q, x ∈ S_{Q}) we call cuspidal exponents of f, and the corresponding functions f_{Q,cusp,x} ∈ F_{cusp}(y_{Q})^{x} the cuspidal coefficients of f.

A similar definition works for f ∈ A(F_{p}), only here cuspidal exponents are taken only for Q ⊂ P. Lemma. If all cuspidal coefficients of an automorphic form f vanish then f = 0.

Proof by induction.

3. If P ⊂ P′ then clearly cuspidal coefficients of the function C_{P,P′}(f) are just a part of cuspidal coefficients of f ∈ A(y_{P′}) – namely coefficients corresponding to parabolic subgroups Q ⊂ P.

Geometric lemma allows us also to restrict cuspidal coefficients of Eisenstein operators whenever they converge.

Suppose we are given an automorphic form φ ∈ A(y_{P}). Denote by B the set of cuspidal exponents of φ and φ_{B}, the collection of corresponding cuspidal coefficients.

Consider now the form f = E_{GP} (φ) and assume that the series converges absolutely. Proposition. Cuspidal exponents of f are of the form (Q, w(B_{P′})), where Q is a parabolic subgroup, w : Q → P′ ⊂ P an association of parabolic subgroups and B_{P′} are cuspidal are cuspidal exponents of the form φ corresponding to P′.

Moreover cuspidal coefficients of f are sums of R_{w} (cusp. coefficients of φ).

This proposition follows from the geometric lemma, since in the formula C_{QG}(f) = C_{QG} ∘ E_{GP} (φ) = ∑_{w} E_{QQ′} ∘ R_{w} ∘ C_{P′P} the terms on the right with Q ≠ Q′ will not give any contribution to the cuspidal part.

4. Fix φ ∈ A(y_{P}), consider the family of automorphic forms u_{s} = φ · a^{s} parametrized by points s ∈ S_{P} and set E(φ, s) := E(u_{s}).

As was shown in Borel’s lecture this series is absolutely converging for Re s ≫ 0 in the C-Weyl chamber in 𝔞_{P}^{∗}. Theorem. E(φ, s) has meromorphic continuation.

Let us write the system of equations which are satisfied by E(φ, s) in the domain of convergence.

(i) φ ∈ A_{σ} for some finite subset σ ∈ Ivv/c. Hence we have an equation

e_{σ}E(φ, s) = E(φ, s) .

(ii) There exists an ideal I ⊂ z(L_{P}) of finite codimension such that Iφ = 0. This implies that I_{s}u_{s} = 0, where I_{s} is the shift of the ideal I by s ∈ s_{P}. Hence we have an equation

Ĩ_{s}E(φ, s) = 0

where Ĩ_{s} = I_{s} ∩ z(G).

Let us denote the system of equations (i) and (ii) by Ξ′(s). Theorem. Ξ′(s) is a system of equations in the space F^{temp}(y) of locally finite type.

We will discuss the proof of this theorem in the appendix.

Now we would like to write the third equation which fixes our function f uniquely (in some domain of the variety of parameters S_{P}).

5. Let Q be a parabolic subgroup. Denote by W_{Q→P} the subset of W_{Q}\W/W_{P} which consists of elements w which give association of Q with a parabolic subgroup P′ ⊂ P.

We know from proposition 3 above that Q cuspidal exponents of E(φ, s) are contained in the union of w(B + s), w ∈ W_{Q→P}.

If Q ⊂ P we have an element w = e ∈ W_{Q→P}. The corresponding exponents (B + s) of E(φ, s) we will call principle exponents: all other exponents we call additional exponents. It is clear that for generic s ∈ S these sets are disjoint.

Now for every w ∈ W_{Q→P} consider ideal J = J_{w} ⊂ O(S_{P′}) which consist of functions (polynomial) on S_{P′} which kill c_{P′P} (φ)_{cusp}. Clearly the ideal J + s kills c_{P′P} (u_{s})_{cusp}, and w(J + s) ∈ O(S_{Q}) kills R_{w}c_{P′P} (u_{s})_{cusp}. Now let us write equations (iii):

Now let us write equations (iii):

(iii)′ ∏_{w∈W_{Q→P}} w(J + s)c_{QG}(f) = 0

(iii)′′ ∏_{w∈W_{Q→P}^{add}} w(J + s)c_{QG}(f) = ∏_{w∈W_{Q→P}^{add}} (wJ + s)c_{QP} (u_{s}). Here W_{Q→P}^{add} is the set W_{Q→P} \{e}. These are linear equations for a function f (which is a solution of the system Ξ′(s) and hence an automorphic form).

For generic s these equations just mean that cuspidal support of f lies in the union of w(B + s), ad that coefficients of principle exponents are the same as coefficients of u_{s}.

In order to prove the theorem about meromorphic continuation of E(φ, s) we would use the continuations principle. It is based on the following Proposition. There exists an open domain v ⊂ S_{P} such that the system of equations Ξ(s) = Ξ′(s)∪ (iii) has a unique solution.

This proposition is based on a key lemma which shows that under some conditions one cuspidal exponent dictates coefficients of another exponent.

Namely, let Q, Q′ be two parabolic subgroups. We say that they are adjacent if there exists a parabolic subgroup R containing Q, Q′ whose semisimple rank is s.s.rankQ + 1 = s.s.rankQ′ + 1 and a nontrivial element w ∈ W_{Q} s.t. wQ ≈ Q′ (associated inside R). In other words Q, Q′ are maximal parabolic subgroups in R which are now trivially associated inside R.

6. Key lemma. Let Q′, Q′′ be two classes of parabolic subgroups in G. We say that they are adjacent if we can realize Q′ and Q′′ as subgroups of G such that

(i) Q′ ∩ Q′′ contains Levi subgroups of Q′ and Q′′.

(ii) Consider parabolic subgroup R generated by Q′ and Q′′. Then Q′ and Q′′ are maximal (non-trivial) parabolic subgroups in R.

Let us consider the space L_{Q′,Q′′} = {ξ ∈ 𝔞_{Q′}|(ξ, 𝔞_{R}^{∗}) = 0}. This is a onedimensional space, and it has a canonical nonzero functional ρ_{L} := ρ_{Q′} − ρ_{R}.

For any weight η ∈ S_{Q′} we can define a number λ(η) = (η|L_{Q′,Q′′})/ρ_{L}. Key lemma. Let f be any automorphic form on G, (Q′, Q′′) – adjacent parabolic subgroups. Let χ be a weight in S_{Q′} Denote by χ̃ the same weight considered as an element of S_{Q′′} when we identify S_{Q′} and S_{Q′′} by restricting characters to the intersection Q′ ∩ Q′′.

Suppose that Reλ(x) ≫ 0. Then the coefficient of cuspidal exponent f_{Q′,cusp,χ} completely determines the coefficient of the cuspidal exponent f_{Q′′,cusp,χ̃}.

Based on the key lemma the proof proceeds as follows.

Choose a vector η inside the Weyl chamber c_{P} ⊂ 𝔞_{P}^{∗}. Fix a very large number Λ ∈ ℝ and consider an open domain U ⊂ S_{P} defined by Λη + B, where B is the unit ball in S_{P} We claim that for s in domain U principle exponents of any form f determine all additional exponents.

Indeed, let Q be a parabolic subgroup and w : Q ∼ P′ ⊂ P be its association with a subgroup P′ ⊂ P.

We can and will assume that w has minimal length ℓ in its double coset W_{Q}\W/W_{P}.

Then we can construct a sequence of adjacent parabolic groups Q = Q_{0} ∼ Q_{1} ∼ Q_{2} ∼ … ∼ Q_{ℓ} = P′ in such a way that the element η will be positive with respect to any pair (Q_{i}, Q_{i+1}).

This implies that for any exponent χ ∈ Λη + B + R its real part will be ≫ 0 for any pair (Q_{i}, Q_{i+1}).

By the key lemma this implies that the χ component of P′ = Q_{ℓ} the determines χ-component of Q_{ℓ−1}, which determines χ-component of Q_{ℓ−2} and so on to χ-component of Q = Q_{0}. Thus the principle component χ determined corresponding component for Q. QED

8. Proof of key lemma.

Step 1. We can replace our form f by f′ = C_{RG}(f) ∈ A(y_{R}).

Step 2. As usual we can compare K-finite forms on y_{R} with K-finite automorphic forms on its Levi group L_{R}. The groups Q′, Q′′ induce two adjacent parabolic subgroups in L_{R}.

Thus we can reduce the proof to the case R = G.

Step 3. We can replace G by its derived group and assume G is semisimple.

Step 4. Using Langlands decomposition we can assume that he form f has cuspidal components only on one class of associate parabolic subgroups, in our case just Q′ and Q′′.

Step 5. Decomposing f with respect to the action of z(G) we can assume that the cuspidal component of f on Q′ and Q′′ appear only at exponents X at Q′ and X̃ at Q′′.

Step 6. L^{2}-argument. Suppose that f_{Q′,cusp,X} = 0. Then the only nontrivial constant term of f is the cuspidal exponent f_{Q′′,cusp,X̃} (since Q′, Q′′ are maximal subgroups). Since exponent X̃ is negative with respect to Q′′ and the cuspidal coefficient f_{Q′′,cusp,X} is rapidly decreasing on the derived group of L′′_{Q} we can conclude that this constant term lies in L^{2}(Z_{T}), where Z_{T} is a Siegel domain.

Since f is well approximated by its constant terms it lies in L^{2}(y).

Now the same argument as for SL(2) shows that if Re x is very large then the eigenvalue of the Casimir operator on f should be real and positive, which contradicts the fact that it is a negative operator.