Meromorphic Continuation of Eisenstein Series

Joseph Bernstein School of Mathematical Sciences, Sackler Faculty of Exact Sciences Tel Aviv University, Ramat-Aviv, 69978 Israel

1 Lecture I

Case of G = SL(2, ℝ).

1. Consider the group G = SL(2, ℝ). Set 𝔤 = Lie(G) – Lie algebra of G. U(𝔤) – universal enveloping algebra z(G):= center of U(𝔤).

z(G) is the algebra of bi-invariant differential operators on G.

In case of G = SL(2, ℝ) the algebra z(𝔤) is the polynomial algebra ℂ[C], where c is the Casimir operator (some operator of order 2).

Fix a maximal compact subgroup K = SO(2) in G. Sometimes it is convenient to consider the quotient space G/K. This is a classical upper half plane

H = {2 = x + iy ∈ ℂ | y > 0}

action of G on H is given by fractional linear transformations, and K is the stabilizer of the point x_{0} = i.

2. Fix a discrete subgroup Γ ⊂ G (we mostly are interested in case Γ = SL(2, ℤ)). Consider the space y = Γ\G. This space has canonical right action of G. Hence we get a representation.

(∏, G, F (y)) where F (y) is some space of functions on y.

Basic definition

An automorphic form is a function f ∈ F (y) which satisfies

(i) f is K-finite

(ii) f is z(G)-finite

(iii) f has moderate growth ( = f is tempered). We will discuss the precise definition later.

3. Let A(y) be the space of automorphic forms. By definition it is a union of subspaces of the following sort.

Fix an ideal I ⊂ z(G) of finite codimension. Fix a finite collection σ of irreducible representations of k. Set A_{σ}^{I} = {f ∈ F (y) | If = 0, f ∈ F (y)_{σ}}, where F ((y)_{σ} is the space of a function which under the action of k gives only irreducible representations in σ.

One of the results we would like to explain is

Theorem. for any I, σ the space A_{σ}^{I} is finite dimensional.

Remark. Why we need temperedness condition. For example, consider Γ = SL(2, ℤ) and consider a harmonic Γ-invariant function f on H. Then cf = 0 since on the space F (H) C acts as y^{2}(∂_{x}^{2} + ∂_{y}^{2}). There are too many of such f if we do not compose the temperedness condition. For example for every λ the function f_{λ} = exp(λ, j), where j is j-invariant, is harmonic on Γ\H.

4. Let f be a z(G) – finite function. Then by a standard argument we can decompose f = Σ to where the sum is taken over characters θ of the algebra z(G) and f_{θ} is annihilated by some power of ker(θ : z(G) → ℂ). Thus we can canonically decompose the space A(y) into a direct sum A^{θ}(y).

Similarly, we have decomposition A(y) = ⨁_{ρ∈I_{rr}k} A_{ρ}(y).

5. Consider Γ = SL(2, ℤ). Fix a standard parabolic subgroup P = \{(\begin{matrix}a & b \\ 0 & d \end{matrix})\} ⊂ G. Set U_{p} =unipotent radical of P, U_{p} = \{(\begin{matrix}1 & u \\ 0 & 1 \end{matrix})\}. Set L_{p} = p/U_{p}, L_{p} ≈ ℝ^{∗} (L_{p} is called Levi group). Set Γ_{p} = Γ ∩ p, Γ_{U_{p}} = Γ ∩ U_{p}, Γ_{L_{p}} =Image of Γ_{p} in L_{p}.

We can study automorphic forms on the space y comparing them with automorphic forms on the space y_{p} = Γ_{p}U_{p}\G. G acts on y_{p} on the right and preserves a measure μ. Moreover, if we fix a measure on y we get canonical measure on y_{p}. Thus G and hence z(𝔤) acts on the space F (y_{p}).

Definition. A function f on y_{p} is called automorphic if it is k-finite, z(𝔤)- finite and of moderate growth. A(y_{p}) the space of automorphic functions on y_{p}. Space F (y_{p}) has additional symmetries. Namely consider the basic affine space Aff_{p} = U_{p}\G. Then it has right action of G, but also left action of L_{p}. Note that the action of G on F (Aff_{p}) preserves L^{2}-structure given by a G-invariant measure on μ on Aff_{p}, but the action of L_{p} multiplies μ by a character. So we would always modify the action of L_{p} on the space F (y_{p}) in such a way that it preserves L^{2}-structure, action of L_{p} defines the action of the algebra z(L_{p}).

Lemma. (Kavish-Chandra). z(L_{p}) ⊃ z(G) and z(L_{p}) is an f-generated z(G) module.

We will identify y_{p} = Γ_{L_{p}}\Aff_{p}. Thus on the space F (y_{p}) we have an additional action of z(L_{p}). Clearly that z(G)-finite functions are also z(L_{p})- finite, but z(L_{p}) has more characters than z(G). Thus for A(y_{p}) we have finer decomposition A(y_{p}) = ΣA^{x}, f-characters of z(L_{p}).

Explicit description Aff_{p} = ℝ^{2}\0. Action of L_{p} on F (Aff_{p}) is given by homotheties (twisted). Generator of L_{p} acts as an operator δ = −(x_{1}∂_{1} + x_{2}∂_{2} + 1). We have δ^{2} = C + 1.

A^{s}(y_{p}) consists of homogeneous k-finite functions of degree −s − 1 y_{p} = {ℝ^{2}\0}/{±1}.

6. Operators E and C.

Spaces F (y) and F (y_{p}) are connected via canonical operators

E : F (y_{p}) ⟶ F (y) (Eisenstein operator)

C : F (y) ⟶ F (y_{p}) (constant term operator) .

In order to define them notice that the choice of a measure on y canonically defines a measure on y_{p}, since the measure on U_{p} can be normalized by the condition μ(Γ_{U_{p}}\U_{p}) = 1.

Set Z_{p} = Γ_{p}\G and consider a double fibration y_{p} ← Z → y. This double fibration defines operators E and C. Explicitly

E(φ) = ∑_{γ∈Γ/Γ_{p}} γ(φ)

C(f) = ∫_{u∈U_{p}/Γ_{Up}} uf .

These operators are partially defined since the series for E does not always converge. But where they are defined they are adjoint.

The subspace of F (y) annihilated by the constant term operator C is called the space of cuspidal functions and denoted F_{cusp}(y). Note that the operator C maps A(y) into A(y_{p}). Its kernel A_{cusp}(y) is called the space of cuspidal automorphic functions (or cusp forms). One of the important results we will prove is the following Lemma. Any cusp form f ∈ A_{cusp}(y) is rapidly decreasing. Corollary. There exists a canonical decomposition A(y)A_{cusp}(y)⊕A_{Eis}(y), where A_{Eis} = A_{cusp}^{⊥}.

6’. Operators E and C could be also described geometrically.

Let us identify U\G with the space Aff_{p} = ℝ^{2}\0. It has a canonical G-invariant measure which is the standard volume on ℝ^{2}. In our case y_{p} = Aff_{p}/{±}.

The space y we can interpret as the space of all immodular lattices Λ ⊂ ℝ^{2}.

With these interpretations we have

E(φ)(Λ) = ∑_{u∈Λ/±1,u−primitive} φ(u)

C(f)(u) = ∫_{x∈ℝ/ℤ} f(Λ(u, v_{x})dx, where

we fixed a vector v ∈ ℝ^{2} such that the volume (u, v) equals 1, set for every x ∈ ℝ v_{x} = v + xu ∈ ℝ^{2} and denote by Λ(u, v_{x}) the lattice generated by u and v_{x}.

7. Eisenstein operator E allows to produce automorphic forms on y from automorphic forms on much simpler space y_{p}.

Namely, start with an automorphic form φ ∈ A(y_{p}) and consider a family of automorphic forms on y_{p} given by u_{s} = φ, a^{s}, where a : y_{p} → ℝ^{+∗} is defined by a(y) = ∥y∥^{−2}.

We will prove the theorem.

Theorem. (i) For Res ≫ 0 the series E(φ, s) = E(u_{s}) is absolutely convergent and is an automorphic form on y. (ii) The function s ↦ E(φ, s) ∈ F (y) is holomorphic in s for Res ≫ 0. (iii) The function E(s) has meromorphic continuation to the whole complex plane of parameter s.

8. Convolution operators. Let H = c_{c}^{∞}(G) be considered as an algebra with respect to convolution. Any representation (π, G, V) defines a representation of this algebra π(α) = ∫ α(g)π(g)dμ in the space V. This is a smoothing operator.

For general α the operator π(α) does not preserve the subspace of k-finite vectors. However, if we consider α in the subalgebra H^{k} ⊂ H of Ad(k)- invariant functions the operator π(α) preserves each subspace V_{σ}, σ ∈ IrrK.

Proposition (Harish-Chandra). Fix I ⊂ z(G) and σ ⊂ IrrK as above. Then for any neighborhood U of point c ∈ G one can find a function α ∈ H^{K}, supported in U, such that π(α) = Id on the subspace V_{σ}^{I}.

In particular, for all automorphic forms f ∈ A_{σ}^{I} we have Π(α)f = f. In particular this implies that f is smooth of uniform moderate growth.

9. The main analytic difficulty in studying the space F (y) is that the space y is not compact and thus we have to understand the structure of y near ∞.

Let us try to see the structure of y by considering it as a metric space (metric comes from a left invariant metric on G) and “looking at it from afar”, i.e. paying attention only to large scale structures.

On a large scale y is equivalent to the space ỹ = y/k. The space ỹ is easier to understand since ỹ = Γ\H. Replacing ỹ by its standard fundamental domain in H we see that on a large scale it is equivalent to the segment [1, ∞) ⊂ ℝ^{+∗}. Let me describe a method how to use these observations in order to describe the analytic structures on the space F (y).

10. Consider the subgroup Γ_{p} = Γ ∩ P and set Z = Γ_{p}\G. We have the natural projection Z → y and thus we can identify the space of functions F (y) as a subspace of F (Z). We will see that the space Z has relatively simple structure so that all the estimates in the space of functions F (Z) are relatively simple. We will develop these estimates and then apply them to the complicated subspace F (y) ⊂ F (Z).

Geometrically the space Z̃ = Z/K, which equals Γ_{p}\H = Γ_{u}\H is a cylinder (quotient of H by integral translations in x coordinate). However, since the metric on H is given by ds^{2} = \frac{dx^{2}+dy^{2}}{y^{2}} , it is better to think about Z̃ as a horn;

PICTURE FOR Z̃ We have seen that for the description of the space ỹ we need only the narrow part of this horn. This motivates the following definition. Consider natural projection a : Z → ℝ_{∗}^{+} given by a(z) = y – coordinate of the point Z module K, considered as a point of H. Definition. For every T > 0 consider domain Z_{T} = {z ∈ Z | a(z) ≥ T }.

We call these domains Z_{T} Siegel domains, though formally Siegel considered corresponding domains in G, not in Z. Reduction theory. (i) For every T the projection pr : Z_{T} → y has finite fibers of size bounded by C(T). (ii) If T ≪ 1, then pr : Z_{T} → y is onto.

Thus given a function f on y we can completely describe its analytic properties in terms of analytic properties of its lifting f^{∗} ∈ F (Z) restricted to a Siegel domain Z_{T} with some fixed T ≪ 1.

11. For every λ ∈ ℝ consider a weight function ω = ω_{λ} on Z defined by ω_{λ}(z) = a(z)^{λ}. Now for a given weight function ω on Z and a number T defines a seminorm on the space T (Z)

ρ_{ω,T} (f)^{2} = ∫_{Z_{T}} ∥ωf∥^{2}dμ_{Z}

(roughly ρ_{ωT} measures the size of the function ωf on Z_{T}).

Reduction theory immediately implies Proposition Consider the seminorm ρ_{ω,T} on the subspace F (y) ⊂ F (Z).

(i) Suppose T ≪ 1. Then ρ_{ω,T} is a norm on F (y).

(ii) Given T, T′, both ≪ 1, the norms ρ_{ω,T} and ρ_{ω,T′} are equivalent.

Thus using the space Z we have defined on F (y) a system of norms ρ_{ω} defined up to equivalence. We denote L_{ω}^{2}(y) the completion of the space F (y) with respect to the norm ρ_{ω}. This gives us a decreasing family of subspaces L_{ω_{λ}}^{2}(y) in F (y). Definition. Let f be a function on y.

(i) We say that f is of moderate growth (or that f is a tempered function on y) if f ∈ L_{ω_{λ}}^{2}(y) for some weight ω_{λ}(i.e. ρ_{ω_{λ},T} (f) < ∞).

(ii) We say that f is rapidly decreasing at infinity if f ∈ L_{ω}^{2}(y) for all λ (i.e. ρ_{ω_{λ},T} (f) < ∞) for every λ).