1 Lecture II

Reminder. G = SL(2, ℝ), Γ = SL(2, ℤ). We would like to study representation (Π, G, F (y)), where y = Γ\G.

We fixed a parabolic subgroup P = \{(\begin{matrix}a & b \\ 0 & a^{−1} \end{matrix})\} and consider a space Z = Γ_{p}\G. The natural projection pr : Z → y identifies F (y) with a Ginvariant subspace of F (Z).

1. For every λ ∈ ℝ and T > 0 we introduced a seminorm ρ_{ω_{λ},T} on the space F (Z). When T ≪ 1 this defines a norm on the subspace F (y) and up to equivalence this norm does not depend on T.

Let us analyse properties of operators Π(α) for α ∈ H with respect to this norm. Clearly operator Π(α) extends to the space F (Z), but since the Siegel domain Z_{T} is not G-invariant the operator Π(α) in F (Z) is not continuous with respect to the seminorm ρ_{ωλ,T}.

We will patch this in the following way. Choose a compact ball B ⊂ G which contains supp(α); we assume that B is symmetric, i.e. B^{−1} = B. Choose T′ < T such that Z_{T} B ⊂ Z_{T′}. Then clearly the operator Π(α) is continuous as an operator

(F (2), ρ_{ω,T′}) → (F (Z), ρ_{ω,T}) .

Since on the subspace of interest F (y) the norms ρ_{ω,T} and ω_{ω,T′} are equivalent this is enough for our purposes.

2. The space Z is fibered by compact orbits of the group U acting on the left. This defines on the space F (Z) the cuspidal structure, which is a canonical decomposition

(∗)  F (Z) = F_{cusp}(Z) + F_{const}(Z) .

The projector π_{p} : F (Z) → F_{const}(Z) = F (y_{p}) is given by π_{p}(f)(Z) = ∫_{u∈Γ_{Up}\U_{p}}(uZ)du, which is essentially a constant term of f.

The decomposition (∗) is orthogonal with respect to every seminorm ρ_{ω,T}. The first basic observation is that if a cuspidal function of f ∈ F_{cusp}(X) is of uniform moderate growth then it is rapidly decreasing. Let us formulate the precise estimate which shows this. Fix a basis {ξ_{i}} of the Lie algebra 𝔤. Basic estimate. For a cuspidal function for Z we have an estimate

(∗)  ρ_{ω_{λ+1}T} (f)^{2} ≤ C ∘ ∑_{i} ρ_{ω_{λ},T} (ξ_{i}f)^{2} .

3. Since the functions ξ_{i}f are also cuspidal we can repeat this trick many times provided that f is smooth enough.

We can formalize it in the following way.

For every k ∈ ℕ define a seminorm ρ_{ω,T}^{k} on F (Z) by ρ_{ω,T}^{k} (f)^{2} = Σρ_{ω,T} (df)^{2}, where the sum is over all monomials d in ξ_{i} of degree ≤ k.

This seminorm measures the growth of f together with its derivatives up to order k. Corollary of basic estimate. Let f ∈ F (Z) be a cuspidal function. Then

ρ_{ω_{λ+k,T}} (f) ≤ C(k) · ρ_{ω,T} (f) .

In other words for cuspidal functions we can trade off smoothness for decay. Corollary. Suppose ρ_{ω,T}^{k} (f) < ∞ for every k and f is cuspidal. Then ρ_{ω_{λ},T} (f) < ∞ for every λ.

4. By definition the cuspidal part of the space F (y) coincides with F (y)∩ F_{cusp}(Z). We do not know whether this space is not zero. But we have seen that any smooth function in this space is rapidly decreasing. Proposition. Let A_{cusp}(y) ⊂ A(y) be the subspace of cuspidal automorphic forms. Then

(i) Every cusp form f ∈ A_{cusp}(y) is rapidly decreasing.

(ii) There exists a canonical decomposition A(y) = A_{cusp}(y) ⊕ A_{cusp}^{⊥}, where

A_{cusp}^{⊥} = {f ∈ A(y) , f is orthogonal to A_{cusp}(y)} .

Proof.

(i) Let f ∈ A_{cusp}(y). Then by Karish-Chandra’s result we can find α ∈ H s.t. Π(α)f = f. It is easy to see that ρ_{ω,T}^{k} (Π(α)f) ≤ ρ_{ω,T} (f) · ∥α∥_{k}, where ∥α∥_{k} is a C^{k}-norm of α. Thus the function f is of u.m.g. and hence rapidly decreasing.

(ii) Let us construct a projector

pr_{cusp} : A(y) → A_{cusp}(y) .

Write f_{0} ∈ A(y) as Π(α)f_{0} and consider the linear functional on the space L_{cusp}^{2}(y) given by

f ↦ ⟨f_{0}, Π(α^{+})f⟩ .

Formally we can rewrite this as

⟨f_{0}, Π(α^{+})f⟩ = ⟨Π(α)f_{0}, f⟩ = ⟨f_{0}, f⟩ .

But in fact the expression ⟨f_{0}, Π(α^{+})f⟩ is defined for all f ∈ L_{cusp}^{2}(y) and is continuous in f. This follows from the fact that

ρ_{ω_{k}}(Π(α^{+})f) ≤ ρ_{ω_{0}}^{k}(Π(α^{+})f) ≤ C · ρ_{ω_{0}}(f) = C · ∥f∥_{L^{2}} .

Thus we can find a vector h ∈ L_{cusp}^{2}(y)s.t.⟨h, f⟩ ≡ ⟨f_{0}, Π(α^{+})f⟩.

This gives us the projection proof A(y) to L_{cus[}^{2}(y). Clearly this cusp is well defined and commutes with actions of k and z(G) (even of U(𝔤)). In particular, pr(A(y)) ⊂ A_{cusp}(y). This is a required projector pr_{cusp} : A(y) → A_{cusp}(y) which leads to a direct sum decomposition.

5. Proof for the basic estimate. Let us assume that our basis ξ_{i} of 𝔤 is orthonormal w.r.t. some K invariant metric on 𝔤. We use this metric to define the left invariant Riemannian metric on G and hence Riemannian metric on Z. Then our inequality (∗) can be written as

∫_{Z_{T}} ω_{λ+1}^{2}f^{2}dμ ≤ C · ∫ ω_{x}^{2} · (grad f)^{2}dμ .

5.(i) Now let us use the lemma from elementary calculus. Lemma. Consider the unit circle S^{1} = ℝ/ℤ with natural parameter a (and hence metric). Then for any smooth function h on S^{1} we have

∫_{S′} |h|^{2}du − | ∫ hdu|^{2} ≤ ∫ |gradh|^{2}du .

This lemma has an immediate corollary. Corollary. Let ν : S^{1} → M be a smooth morphism of S^{1} into some Riemannian manifold M. Suppose that the Lifschitz constant of ν is ≤ ν; in other words at every point x ∈ S^{1} the tangent morphism dν : T_{x}(S^{1}) → T_{ν(x)}(M) has norm ≤ ν. Then for every smooth function from M we have

∫ ν^{∗}|f|^{2}du − | ∫ ν^{∗}fdu|^{2} ≤ v^{2} · ∫ ν^{∗}|grad f|^{2}du .

Indeed l.h.s. is bounded by ∫ |grad(ν^{∗}f)|^{2}du, and pointwise we have an estimate

|grad(ν^{∗}f)^{2} ≤ r^{2} · ν^{∗}(|gradf|^{2}) .

This proves the corollary.

5(ii). now let us come back to the basic inequality.

Let D = A · K. for every point d ∈ D consider a morphism ν_{d} : S^{1} = Γ_{U}\U → Z given by u ↦ ud.

Then we can write the integral ∫_{Z_{T}} ω^{2}|f|^{2}dμ as an integral over D_{T} of ∫ ν_{d}^{∗}(|f|^{2})du with respect to some measure ν on D.

Similarly we can rewrite the integral ∫_{2T} ω^{2}|grad f|^{2}dμ. Thus the basic inequality is reduced to the following estimate for every d

∫ ν_{d}^{∗}|f|^{2}du ≤ (∫ ν_{d}^{∗}|grad f|^{2}du)a(d)^{−2} .

But this follows from Corollary 5(i) and the observation that ν_{d} has Lipschitz constant ≤ a(d)^{−1}.

It is very clear from a geometric picture of the norm structure of the space Z̃ = Z/K.

Q.E.D.

6. Several more corollaries of the basic estimate.

Proposition Let f ∈ A(y) an automorphic form. Consider C(f) ∈ A(y_{p}) and realize both f and C(f) as a functions on Z¿ Then f̃ = f − C(f) is rapidly decreasing, i.e.

ρ_{ω_{λ},T}^{k} (f̃) < ∞ for every λ, k, T .

This was essentially proven in Proposition 4.

Warning. The function f̃ does not usually lie in the space F (y), so it has nothing to do with a cuspidal part of f described in Proposition 4(ii).

7. Proposition. Fix a function α ∈ H and consider an operator Π(α) : F (Z) → F (Z). Choose T′ < T such that the operator Π(α) is continuous from (F (Z), ρ_{ω,T′}) to (F (Z), ρ_{ω,T}). Then on the subspace of cuspidal functions this operator is Hilbert-Schidt.

Proof. I will describe a more sophisticated proof than what is done usually, but in my opinion it is more understandable.

Step 1. Π(α) is a continuous operator from F (Z)_{cusp,ρ_{T′,ω}} to F (Z)_{cusp,ρ_{T,ω}^{k}} for any k.

This is true since all derivatives can be shifted to α.

Step 2. The identity operator F (Z)_{cusp}ρ_{ω_{λ},T′}^{k} → F (Z)_{cusp}, ρ_{ω_{λ+e}}^{k−e} is a continuous operator.

Step 3. Proposition. Identity operator

F (Z), ρ_{ω_{λ+2},T′}^{3} → F (Z), ρ_{ω_{λ},T}

is a Hilbert-Schmidt operator.

In other words, if we sufficiently weaken the bounds for smoothness and decay of functions on Z, then the identity operator is a Hilbert-Schmidt operator.

This is a special case of a general statement about any Lie group. We will prove this proposition in an appendix.

8. Corollary. Fix an ideal I ⊂ z(G) and a subset σ ∈ Irrk as above. Then the space A_{σ}^{I}(y) is finite dimensional and the dimension of this space can be effectively bounded.

Proof. Consider the constant term operator C : A(y) → A(y_{p}). Then the image C(A_{σ}^{I}(y)) lies in the subspace A_{σ}^{I}(y_{p}) which dimension is very easy to bound since y_{p} is a very simple space.

Thus it is enough to bound the dimension of KerC ∩ A_{σ}^{I}(y) = A_{σ,cusp}^{±}(y). By Harish-Chandra lemma there exists α ∈ H such that Π(α) = Id on the space A_{σ}^{I}.

Consider the operator Π(α) on L_{cusp}^{2}(y). Then Proposition 7 easily implies that this is a Hilbert-Schmidt operator and moreover, it allows us to explicitly bound its Hilbert-Schmidt norm, ∥Π(α)∥_{HS}^{2} ≤ D. Since on the subspace A_{σ,cusp}^{I} ⊂ L_{cusp}^{2}(y) this operator is an identity, we get that dim A_{σ,cusp}^{I} ≤ D.