Eisenstein series on affine Kac-Moody groups over function fields

H. Garland constructed Eisenstein series on affine Kac-Moody groups over the field of real numbers. He established the almost everywhere convergence of these series, obtained a formula for their constant terms, and proved a functional equation for the constant terms. In a subsequent paper, the convergence of the Eisenstein series was obtained. In this paper, we define Eisenstein series on affine Kac-Moody groups over global function fields using an adelic approach. In the course of proving the convergence of these Eisenstein series, we also calculate a formula for the constant terms and prove their convergence and functional equations.


Introduction
Classical Eisenstein series are central objects in the study of automorphic forms. The classical Eisenstein series were generalized to the case of reductive groups and studied by R. Langlands [13,14]. As in the classical case, he found these Eisenstein series have certain analytic properties as well as Fourier series expansions where L-functions appear in the constant terms. Because of this relationship, these L-functions inherit important analytic properties from the Eisenstein series. This approach to studying automorphic L-functions is known as the Langlands-Shahidi method. (See [7] for a survey.) The Eisenstein series over function fields were studied by G. Harder in [8].
In [4], H. Garland defines and studies Eisenstein series on affine Kac-Moody group over R. He proved the almost everywhere convergence of the series while placing specific emphasis on calculating the constant term, finding its region of convergence, and proving functional equations of the constant term. More precisely, letĜ R be an affine Kac-Moody group over R. For each character χ of the positive partÂ of the torus, he defines a function Φ χ :Â → C × and, as in the classical case, uses the Iwasawa decompositionĜ R =KÂÛ to extend Φ χ to a function onĜ R . Garland extends this group by the automorphism e −rD ∈ Aut(V λ R ), where r > 0 and D is the degree operator of the Kac-Moody Lie algebra associated withĜ R . Setting Φ χ (ge −rD ) = Φ χ (g), he defines an Eisenstein series E χ on the spaceĜ R e −rD ⊆ Aut(V λ R ) by whereΓ is a discrete subgroup ofĜ R .
2000 Mathematics Subject Classification. Primary 22E67; Secondary 11F70. 1 With suitable conditions on the character χ, Garland proves the almost everywhere convergence of E χ and calculates the constant term of this series, E # χ , representing it as a sum over the affine Weyl groupŴ : w∈Ŵ (ae −rD ) w(χ+ρ)−ρc (χ, w).
Here the functionc(χ, w) is a ratio of completed Riemann zeta functions, ξ(s) := π −s/2 Γ( s 2 )ζ(s). After establishing when this infinite sum converges, he proves functional equations for the constant term E # χ . In [5], Garland proves the Eisenstein series E χ converge absolutely. For an arbitrary field F , we can construct an affine Kac-Moody groupĜ F ( [2]). Let V be the set of places of F . In this paper, we consider the fields F ν for ν ∈ V, completions of a global function field F . We work adelically and define an Eisenstein series E χ onĜ A , a restricted direct product of the groupsĜ Fν . Then we calculate the constant term of the Eisenstein series E χ and, as in Garland's work, find that we can express the constant term as an infinite sum over the affine Weyl group. Moreover, this expression contains c(χ, w)-functions composed of ratios of ζ F , the zeta function for the function field F . This calculation leads to a proof of convergence of the Eisenstein series.
Theorem 0.1. Let χ ∈ĥ * such that Re(χ(h α i )) < −2 for i = 1, . . . , l + 1, and let m = (m ν ) ν∈V be a tuple such that m ν ∈ Z ≥0 and 0 < ν m ν < ∞. Then the Eisenstein series is convergent for all (h, u) ∈Ĥ A ×Û A /(Û A ∩Γ F ). (See (3.12) for the definition of η mD .) The zeta function ζ F , which appears in the c(χ, w)-functions of the constant term, satisfies a functional equation. Using this, we prove that the constant term of the Eisenstein series satisfies a family of functional equations indexed by elements in the affine Weyl group. This paper has seven sections. In Section 1, we will provide a basic construction of an affine Kac-Moody Lie algebra and its corresponding groups, and then proceed to Section 2 where we prove an Iwasawa decomposition for these groups. Section 3 uses the Iwasawa decomposition to define an Eisenstein series. We also describe the characters we use for this definition and our analogue of the automorphism e −rD that appears in Garland's definition above. In Section 4 we calculate the constant term of our series. The content of Section 5 establishes the region of convergence for this infinite sum, which we use in Section 6 to prove the convergence of the Eisenstein series E χ . Finally, in Section 7 we make use of the functional equation for ζ F to prove functional equations for the constant term of our Eisenstein series.
We thank M. Patnaik for helpful comments. We also thank Keith Conrad, who made many useful comments. Finally, we thank the referee whose suggestions improved this paper very much.

Affine Kac-Moody Lie Algebras and Groups
In this section, we describe the affine Kac-Moody groups that we use to define our Eisenstein series. We will first fix notations for an affine Kac-Moody Lie algebraĝ e . Then, following Chevalley's construction, we will use automorphisms of a representation space V λ ofĝ e to define the associated affine Kac-Moody groupĜ λ .
1.1. Affine Kac-Moody Lie Algebras. Let g be a simple, real Lie algebra. Then we define and endowĝ e with the standard bracket operation. (See [5] or [11].) In the expression above, c is a central element and D is the degree operator that acts as t d dt on R[t, t −1 ] ⊗ g and annihilates c. We set where h is the Cartan subalgebra of g. We also set Let ∆ be the classical root system of g, and denote the simple roots by {α 1 , . . . , α l } and the highest root by α 0 . Then the affine roots∆ ofĝ e contain l + 1 simple roots {α 1 , . . . α l+1 }. By setting δ = α 0 + α l+1 , we can describe the set of affine roots∆ associated toĝ as (1.4)∆ = {α + nδ | α ∈ ∆, n ∈ Z} ∪ {nδ | n ∈ Z =0 }.
We have where W is the classical Weyl group, and T is a group of translations that we can index by H ∈ h Z ( [10], [11]). As with g, the algebraĝ has a Chevalley basis which we can construct using the Chevalley basis for g ( [2]). First, fix a Chevalley basis for g, Now we define some important elements ofĝ. For each a = α + nδ ∈∆ W , we let ξ a = t n ⊗ E α and ξ i (n) = t n ⊗ h α i for i = 1, . . . , l.
Also, we set h i = h α i for i = 1, . . . , l, and h l+1 = −h α 0 + 2 (α 0 ,α 0 ) c. Using these elements, we fix a Chevalley basis for the algebraĝ: Finally, we denote the Z-span ofΨ byĝ Z . Thenĝ Z is closed under the bracket operation [ , ] forĝ e . Usingĝ Z , we can make sense of an affine Kac-Moody Lie algebra over an arbitrary field F by setting (1.8)ĝ e F = (F ⊗ ZĝZ ) ⊕ F D. Let D denote the set of λ ∈ (ĥ e ) * such that for i = 1, . . . , l + 1 we have λ(h α i ) ∈ Z ≥0 and λ(h α i ) = 0 for some i. This is the set of dominant, integral, normal weights ofĝ e . In [2] we see that for each λ ∈ D, we have an irreducibleĝ e -module, V λ , with a highest weight vector v λ .
for any a ∈∆ W and m ∈ Z ≥0 . We fix this Z-module V λ Z and call it the Chevalley form of V λ . The representation space V λ and V λ Z decompose into a direct sum of weight spaces, V λ µ and V λ µ,Z = V λ µ ∩ V λ Z , respectively. As a highest weight module, we know that any weight µ of V λ is of the form In the next subsection, we will use the elements ofĝ e F to describe some special automorphisms of the vector space V λ F . These automorphisms generate the affine Kac-Moody group over the arbitrary field F .

Construction of Affine Kac-Moody Groups.
Let F be an arbitrary field. For a ∈∆ W and s ∈ F , we define the automorphism χ a (s) of V λ F as s n ξ n a n! .
By (1.9) we know that this definition works for fields of arbitrary characteristic. For each v ∈ V λ F and a ∈∆ W , there exists an n 0 such that for all n ≥ n 0 we have ξ n a n! · v = 0.
Hence for each v ∈ V λ F and a ∈∆ W , the sum in (1.11) acts as a finite sum ( [2]). We let F ((X)) be the field of Laurent series in the variable X with coefficients from F . Then σ ∈ F ((X)) has an expression as σ = i≥i 0 For each v ∈ V λ F there exists an i k such that for all i ≥ i k , χ α+iδ (s) · v = v for any s ∈ F and so for each v the product in (1.12) acts as a finite product ( [2]).
As a result of these observations, each χ α (σ) is an automorphism of the representation space V λ F . Finally, we make the following definition.

Remark 1.15.
(1) Since we are considering the automorphisms of V λ F , our group depends on the choice of λ. We fix a λ ∈ D and drop the λ from our notation.
(2) One may note that in the construction ofĜ F we only used elements ofĝ and ignored the degree operator D. In section 3, we will extend our groupĜ F by a particular automorphism η mD related to D, thereby establishing a more complete relationship between g e F =ĝ F ⊕ F D and our group. Garland extends his group in a similar way by the automorphism e −rD for r > 0 ( [4], [5]).
In the next section, we will begin working with this group when F is a field with a non-Archimedean absolute value. Our first objective is to develop an Iwasawa decomposition forĜ F in this case, from which we will be able to begin defining our Eisenstein series.

Iwasawa Decomposition for Affine Kac-Moody Groups
In this section, we prove an Iwasawa decomposition forĜ F , where F is a local field with a non-Archimedean absolute value. In particular, we will apply this result to the groupsĜ Fν =Ĝ ν , where F ν is a completion of a global function field F .
We now consider the particular case ofĜ Fν :=Ĝ ν , where F ν is an arbitrary field with a non-Archimedean discrete valuation ν. For x ∈ F ν let |x| ν denote the absolute value that corresponds to the valuation ν. We define O ν = x ∈ F ν |x| ν ≤ 1 and P ν = x ∈ F ν |x| ν < 1 , noting that P ν is the unique maximal ideal of the ring O ν .
For any a ∈∆ W and s ∈ F × ν , we set Likewise, for α ∈ ∆ and nonzero σ ∈ F ν ((X)), we set Using these elements, we can define the subgroups ofĜ ν that appear in the Iwasawa decomposition of the affine Kac-Moody group. We let F ν [[X]] ⊂ F ν ((X)) denote the ring of power series over F ν . We then set and defineB ν to be the group generated byÛ ν andĤ ν , which can be realized as a semi-direct productĤ ν ⋉Û ν . Finally, setK where O ν is defined above.
We continue by choosing a specific set of coset representatives for each Y i . To this end, we recall the following lemma from [2], §16: In particular, if we take w = w i for some i, each element x ∈B ν w iBν has an expression as x = χ α i (s)w i y with y ∈B ν . Now since we choose our Y i 's to be representatives of the coset spaceB ν w iB /B ν , we can choose our Y i 's to consist of elements of the form For an arbitrary field F , and any a ∈∆ W , we have a homomorphism ϕ a : SL 2 (F ) →Ĝ F that is defined by the conditions: For the field F ν , we know that the group SL 2 (F ν ) contains the subgroup B consisting of upper triangular matrices, and the subgroup K = SL 2 (O ν ). Moreoever, we have the Iwasawa decomposition SL 2 (F ν ) = K B. For more information see [10], §2. Lemma 2.6. We have the following: (2) ϕ a (K) ⊂K ν , for any a ∈∆ W .
(3) We can choose the elements of Y i for i = 1, . . . , l + 1, so that they are the images through ϕ α i of elements in K. In particular, we may assume that Y i ⊂K ν for i = 1, . . . , l + 1.
(2) We know that (see [10], §2 ), and so an arbitrary element of K will be a finite product of these matrices. As a result, the image of an element in K through the homomorphism ϕ a will be a finite product of χ a (s) and χ −a (s ′ ) with s, s ′ ∈ O ν . It suffices to show that χ a (s) and χ −a (s ′ ) are elements ofK ν . If a = α + nδ, then −a = −α − nδ and we can express χ a (s) = χ α (sX n ) and χ −a (s ′ ) = χ −α (s ′ X −n ).
Since the coefficients s and s ′ are each elements of O ν , we have that χ a (s) and χ −a (s ′ ) are elements ofK ν .
(3) Recall that Y i is a set of coset representatives for (B ν w iBν )/B ν , for w i ∈ S. By Lemma 2.4 we can choose these representatives to be of the form χ α i (s)w i for s ∈ F ν , but then ). However, by the Iwasawa decomposition for SL 2 (F ), we know that ( −s 1 −1 0 ) = kb for k ∈ K and b ∈ B. Thus, we have χ α i (s)w i = ϕ α i (k)ϕ α i (b) and by part (1) of this lemma we know that the coset So we can take our representatives for (B ν w iBν )/B ν to be of the form ϕ α i (k) for some k ∈ K. Finally, part (2) of this lemma implies that we can choose our Y i to be a subset ofK ν . Now we can prove that for any ν ∈ V, the affine Kac-Moody groupĜ ν has an Iwasawa decomposition.
Proof. We have already established a Bruhat decomposition forĜ ν ; in other words, Since this is a disjoint union, it suffices to show that eachB ν wB ν decomposes in the desired way. It follows from Corollary 2.3 and part (iii) of Lemma 2.6 that for each w ∈Ŵ we havê B ν wB ν ⊂K νBν , and thusĜ ν =K νBν . We have already observed thatB ν =Ĥ ν ⋉Û ν , and so we obtain the Iwasawa decomposition:Ĝ ν =K νĤνÛν .
The Iwasawa decomposition of an element is not uniquely determined. See Remark 3.2 and Corollary 3.11.

Defining the Eisenstein Series
For the remainder of this paper, we set F to be a global function field of genus g. Let V denote the set of all places of F , and for ν ∈ V let F ν denote the completion of F with respect to ν. As in the previous section, we let | · | ν denote the corresponding non-Archimedean absolute value on F ν , and we define the local ring O ν and its maximal ideal P ν as before. We fix a uniformizer π ν ∈ O ν , so π ν generates the ideal P ν . Finally let the integer q ν denote the cardinality of the residue field O ν /P ν .
In this section, we will use an adèlic approach and define our Eisenstein series on the group G A , a restricted direct product of affine Kac-Moody groups over the completions of the field F .
3.1. The Adèlic Approach. Using the completions F ν , we define the adèle ring A as the restricted direct product: The units of this ring, the group of idèles A × , can also be realized as a restricted direct product: We setĜ ν =Ĝ Fν and define the groupĜ A as the following restricted direct product: In order to define our Eisenstein series onĜ A , we must first establish an Iwasawa decomposition for this group. With the appropriately defined subgroups ofĜ A , this will be a direct result of Theorem 2.10. To this end, we distinguish the following subgroups ofĜ A : where the restricted direct products are with respect toĤ ν ∩K ν andÛ ν ∩K ν , respectively.
Theorem 3.1. We have the following Iwasawa decomposition for the groupĜ A : Remark 3.2. In [2], Garland develops an Iwasawa decomposition for the groupĜ R and establishes that the decomposition of an element is unique. In our setting, the Iwasawa decomposition of an element is not unique. This is potentially problematic because we will use this decomposition to define the Eisenstein series. However, we will see in Proposition 3.10 that due to the structure ofĤ A this is not an issue.

3.2.
The Structure and Topology of the Torus. We fixed a normal weight λ ∈ D, and hence aĝ e -module V λ Fν . It is a highest weight module, so we let v λ denote the highest weight vector. In [2] we see that the representation space V λ Fν decomposes into a direct sum of its weight spaces Moreover, every weight of V λ Fν is of the form µ = λ − l+1 i=1 k i α i , for k i ∈ Z ≥0 . Using this unique expression, we define the depth of µ as We fix a basis B of V λ Z by choosing basis vectors {v λ , v 1 , . . . , v n , . . . } in V λ Z and ordering them so that , and i < j, then we have dp(µ) ≤ dp(µ ′ ), and (2) for each weight µ of V λ Z , the basis vectors of V λ µ,Z appear consecutively. A basis of V λ Z that satisfies these conditions is called coherently ordered. It is important to note that since we chose our basis vectors from V λ Z , the basis B serves as a basis for V λ F as well as V λ Fν for every ν ∈ V. The advantage to fixing such a basis is that with respect to B we can view the elements ofĤ ν as (infinite) diagonal matrices which are scalar matrices when we restrict to a weight space. In addition, the elements ofÛ ν are strictly upper triangular block (infinite) matrices where the blocks are determined by the weight spaces of V λ Z . For more information, see [2].
By this observation, we can clearly see that elements ofĤ ν commute with each other, andĤ ν normalizes the subgroupÛ ν . Since this holds for all ν ∈ V, we obtain the same results forĤ A andÛ A .
Because the definition of our Eisenstein series depends on it, we are interested in studying the structure ofĤ A . Let h ∈Ĥ A . Considering the local components, we let As a result, we may write h ∈Ĥ A as the following product: We will prove the following: Then we have ord ν (s i,ν ) = 0 for i = 1, . . . , l + 1.
Remark 3.4. We have included the superscript λ above since we must consider different λ's in the proof below.
Proof. As a consequence of Lemma 15.7 and Theorem 15.9 in [2], we know that for each fundamental weight Λ i there exist a positive integer m i and a surjective group homomorphism and this homomorphism is characterized by the fact that it maps χ λ α (σ) to χ m i Λ i α (σ). As a result, Then by [2] we know that Since elements ofK ν preserve the subspace V λ Oν , we have ord ν (s i,ν ) ≥ 0. Moreover, sinceĤ λ ν ∩K λ ν is a group, we know (h λ is also in the intersection. Applying the argument above to (h λ ν ) −1 , we find ord ν (s i,ν ) ≤ 0. Thus we have ord ν (s i,ν ) = 0 for each i = 1, . . . , l + 1.
Corollary 3.6. The subgroupĤ A ≤Ĝ A may be realized in the following way: Proof. Proposition 3.3 shows that for almost all ν ∈ V, we have s i,ν ∈ O × ν . As a result, these infinite tuples s i are actually elements of A × . Remark 3.7. We have a surjective group homomorphism ϑ : The group (A × ) l+1 inherits the product topology induced by the usual topological structure of A × , and we give the spaceĤ A the quotient topology induced by the map ϑ. We will use the map ϑ again in Section 5.

3.3.
Characters. Defining our Eisenstein series onĜ A requires that we specify a character of the subgroupĤ A . Let | · | be the idèlic norm. We define a character by fixing a linear functional χ ∈ĥ * and setting By our definition of χ, we see that h χ = 1 for any h ∈Ĥ A ∩K, since |s i | = 1 for each i = 1, . . . , l + 1.
Fix χ ∈ĥ * , and define Φ χ :Ĝ A −→ C × to be the function induced by the character χ onĤ A and the Iwasawa decomposition forĜ A . In other words, if g = k h u is an element ofĜ A , then we set We noted earlier that the Iwasawa decomposition forĜ A is not unique, so we need to prove that this function is well-defined.
Proof. With respect to the coherently ordered basis B, the elements ofK ν are matrices with elements from the ring O ν and the elements ofB ν are upper triangular block matrices. Thus, we can view b ∈B ν ∩K ν as an upper triangular infinite block matrix with entries from O ν . By the definition ofB ν , we know that b = h u for h ∈Ĥ ν and u ∈Û ν . In fact, with respect to B, the matrix h will be diagonal with the same diagonal entries that appear in the matrix b. In particular, h ∈Ĥ ν ∩K ν which also implies that u ∈Û ν ∩K ν .
In particular, Φ χ is a well-defined function fromĜ A into C × .

Proof. We begin by noting that if
By Lemma 3.9, k ′−1 k =hū withh ∈Ĥ A ∩K andū ∈Û A ∩K. Using this information, we see SinceĤ A normalizesÛ A , we can express k ′ h ′ u ′ = k ′h hu 1 , for some u 1 ∈Û A . Finally, we observe that by Remark 3.8 we have Proof. From the proof of the above proposition, we obtain 3.4. An Important Automorphism. As in [4], we need to extend the groupĜ A by an automorphism related to the degree operator D that appears in the associated affine Kac-Moody Lie In [4], Garland uses e −rD ∈ Aut(V λ R ) for r > 0 to extend the groupĜ R ; however, in our case we are considering the restricted direct productĜ A . For this reason, we first define local automorphisms η mν D ν ∈ Aut(V λ Fν ) for each ν ∈ V, and then work with a product of the local automorphisms. For each ν ∈ V and integer m ν ∈ Z, we define η mν D ν to be the automorphism of V λ Fν defined by the conditions acts as scalar multiplication on the weight spaces, we can consider this automorphism as being a diagonal block matrix with respect to the coherently ordered basis B, and as such the automorphism will commute withĤ ν and normalizeÛ ν . Moreover, note that if we chose m ν = 0, then η mν D ν is the identity map. We fix a tuple m = (m ν ) ν∈V such that m ν ∈ Z and m ν = 0 for all but a finite number of ν. By doing so, we fix the associated automorphism η mD defined as the product We will define the Eisenstein series onĜ A η mD for our fixed automorphism η mD . In particular, we will consider Φ χ as a function onĜ A η mD by setting Φ χ (gη mD ) = Φ χ (g).

Defining the Eisenstein Series. For each completion
From this map, we see that there is a natural injection i ν :Ĝ F ֒→Ĝ ν for each ν ∈ V, and we may define the diagonal embedding i :Ĝ F ֒→ ν∈VĜ ν by The image of the map i is not entirely contained in the groupĜ A . To see this clearly, we construct an example of an element fromĜ F that does not diagonally embed intoĜ A . Example 3.13. Let F = F q (T ). It is well known that all but one of the places (the "infinite" place corresponding to 1 T ) are indexed by monic, irreducible polynomials in F q [T ]. We let f ν (T ) denote the polynomial associated with the place ν. Set where we set the coefficient of X i to be 1 fν i for some ν i that has not previously appeared in the expansion. Clearly, we have 1 fν ∈ F q (T ) for all ν, so σ ∈ F ((X)). However, by design σ / for an infinite number of ν ∈ V, and as a result i ν (χ α (σ)) is not an element ofK ν for an infinite number of ν. Therefore, i(χ α (σ)) / ∈Ĝ A .
Keeping this example in mind, we consider the subgroupΓ F defined bŷ By an abuse of notation,Γ F will be considered as a subgroup ofĜ F as well asĜ A , where in the latter case we consider the elements as being diagonally embedded. Note that that h χ = 1 for any h ∈Ĥ A ∩Γ F and χ ∈ĥ * . We also have the subgroupsĤ F ,Û F andB F of the groupĜ F .
In the definition of the Eisenstein series,Γ F /(Γ F ∩B F ) will be the coset space over which we index our sum. Before continuing, we first establish the following facts about Φ χ . Lemma 3.14. Let g, β ∈Ĝ A , and γ ∈Γ F ∩B F ; then (2) Using an argument similar to that of Lemma 3.9, we can show that for any γ ∈Γ F ∩B A , we have a decomposition γ = h 1 u 1 with h 1 ∈Ĥ A ∩Γ F and u 1 ∈Û A . Using our Iwasawa decomposition, express g = khu. Then where the second to last equality holds because h 1 ∈Ĥ A ∩Γ F .
(3) As before we let γ = h 1 u 1 with h 1 ∈Γ F ∩Ĥ A and u 1 ∈Û A . Since η mD commutes witĥ We know Φ χ is right invariant byÛ A , and in light of part (1) of this lemma, the following equalities hold: Due to part (3) of the previous lemma, the following definition of the Eisenstein series E χ on the spaceĜ A η mD is well-defined.
whenever the series converges to a complex number; otherwise we define E χ (gη mD ) = ∞.
The goal of this paper is to prove the convergence of the series E χ . Later, we will see that after some reductions we can consider E χ (gη m D) as a function on the spaceĤ A ×Û A /(Û A ∩Γ F ). In the next three sections, we will prove the following theorem: Theorem 3.17. Let χ ∈ĥ * such that Re(χ(h α i )) < −2 for i = 1, . . . , l + 1, and let m = (m ν ) ν∈V be a tuple such that m ν ∈ Z ≥0 and 0 < ν m ν < ∞. Then the Eisenstein series To prove this theorem, we first assume that χ is a real character, so χ :Ĥ A → R >0 . As a result, the Eisenstein series E χ takes values in R >0 ∪ {∞}. This assumption is not very restrictive because for any complex character χ, the series E χ is dominated by E Re(χ) . Hence, we can apply the dominated convergence theorem for the complex case after we consider the real character χ.
As in Corollary 2.3, we see thatĜ F has the Bruhat decomposition into the following disjoint unionĜ then the Bruhat decomposition above allows us to express our Eisenstein series as This is simply a regrouping of the sum in Definition 3.15. As with E χ , we can consider each E χ,w as a function on the spaceĤ A ×Û A /(Û A ∩Γ F ). In order to prove Theorem 3.17, it suffices to show that for h varying in an arbitrary compact set ofĤ A and for real χ.
If we establish the convergence (3.20), then Note that the last expression is nothing but the constant term of the Eisenstein series E χ .
and call E # χ the constant term of the Eisenstein series E χ .
In the next section, we will calculate the integrals for w ∈Ŵ . In Section 5, we establish the convergence (3.20) when χ is a real character and h varies in a compact set ofĤ A . As a result, we will obtain the almost everywhere convergence of the Eisenstein series and a concrete description of its constant term.

Calculating the Constant Term of the Eisenstein Series
In this section, we simply state the existence and properties of the measures necessary for our calculation, leaving the details to Appendix A. Constructing these measures involves taking the projective limit of a family of measures. For now we will also refrain from showing that E χ is a measurable function, a topic that we will address in Section 6.

Definition and Preliminary Calculation.
From Appendix A, we have an invariant probability measure du on the spaceÛ A /(Û A ∩Γ F ). As was discussed at the end of the previous section, we now turn our attention to calculating the expression We first calculate the integrals LetÛ −,F be the subgroup ofĜ F consisting of the elements that are strictly lower triangular block matrices with respect to our coherently ordered basis Note that this definition works over F ν as well, so the notationsÛ −,ν andÛ w,ν are clear. Finally, we setÛ −,A andÛ w,A to be the expected restricted direct products. The Bruhat decomposition has this refinement:Ĝ Moreover, every element of u ∈Û w,F is of the form As a result, we can choose the coset representatives ofΓ F (w)/Γ F (w)∩B F to be {bw} for b ∈Û w,F . Thus, We have the following decompositions: This decomposition, along with the fact thatÛ w,F ⊂Γ F , implies that So we can consider the set of b ∈Û w,F as a set of coset representatives for Here we consider the measure du ′ as the measure induced from du and the projection Using the decomposition (4.3) forÛ A , we observe that integrating over this coset is the same as first integrating overÛ In Appendix A we see that the measure du ′ decomposes into measures du 1 and du 2 on these spaces, respectively. Using these measures and decompositions, we can manipulate our integral (4.5) to be . As a result, the integral (4.6) becomes Since the values Φ χ (gη mD u 1 w) no longer depend on u 2 and the measure du 2 has a total measure of 1, this equals The following proposition summarizes our results from this subsection: Proposition 4.7. For g ∈Ĝ A and w ∈Ŵ , we have

4.2.
Further Calculation. We continue our computation by further manipulating the integral in Proposition 4.7. Fix an Iwasawa decomposition g = khu. Since η mD normalizesÛ A , we have The decomposition (4.3) allows us to write and we obtain that . For any u 1 ∈Û w,A and a fixed u + ∈Û A ∩ wÛ A w −1 , the map that sends u 1 toπ(u + u 1 (u + ) −1 ) is a unimodular change of variables, so the integral in Proposition 4.7 becomes However, note that u ′ − ∈Û w,A remains fixed as u 1 ranges overÛ w,A , and since du 1 isÛ w,Atranslation invariant, our integral may now be expressed as Continuing, we note that and since w −1 hη mD w ∈Ĥ A , we obtain from Lemma 3.14 part (1) Since (w −1 hη mD w) χ = (hη mD ) wχ , the integral (4.8) becomes (4.9) (hη mD ) wχ Set∆ w =∆ W,+ ∩ w∆ W,− . Then applying the change of variables has the following effect on our integral in (4.9): where ρ ∈ĥ * such that ρ(h α i ) = 1 for i = 1, . . . , l + 1. Therefore, Finally, since we may assume w ∈K, theK-left invariance of Φ χ allows us to rewrite our integral (4.10) as where du − is the Haar measure induced by conjugating by w −1 .
Using Definition 3.22, our results from this section appear in the following proposition: Proposition 4.11. For any g ∈Ĝ A , we have

4.3.
Calculating the Local Integrals. In this subsection, we shift our focus away from the globalĜ A and into the local pieces ofĜ ν . We aim to calculate some local integrals that will help us determine the value of the integral in Proposition 4.11. We will see in the next subsection that the integral in Proposition 4.11 may be expressed as a product of the local integrals that we discuss in this section.
As before we use∆ w to denote∆ W,+ ∩w∆ W,− . Then∆ w is a finite set of affine Weyl roots that we can explicitly describe. If w = w ir . . . w i 1 is a minimal expression in terms of the generators ofŴ , then by setting β j = w ir . . . w i j+1 α i j , we have∆ w = {β 1 , . . . , β r }. Using these roots we can completely describeÛ For more information see ( [4], §6) and ( [3], §6). Moreover, each element in this group is uniquely expressed in this way, so if we set U β i ,ν = {χ β i (s) | s ∈ F ν }, then we have thatÛ w,v uniquely decomposes into U βr,ν . . . U β 1 ,ν .
In §13 of [2], we see that for any β ∈∆ W the effect of conjugation by w is with uniqueness of expression. Calculating these roots we find: For convenience we set γ j = w i 1 . . . w i j−1 α i j and conclude Remark 4.14. Each of the spaces U −γ i ,ν is isomorphic to F ν , so we can define a measure on these spaces using this isomorphism and the usual Haar measure µ ν on F ν . The measure du − on U −,w,ν may now be considered the product measure induced by the µ ν . For more information, see Appendix A.
Recall that we fixed χ ∈ĥ * , so for every ν ∈ V we can define a function Φ χ :Ĝ ν → R >0 in precisely the same way we defined Φ χ onĜ A . If g = khu and h Remark 4.15. Using the same argument of Proposition 3.10, we may conclude that this map is well-defined.
It is our goal in this subsection to prove Proposition 4.16 which calculates the value of the local integral involving the function Φ χ onĜ ν . As before, we let h α denote the co-root corresponding to α ∈∆ W , and in particular, h α i for i = 1, . . . , l + 1 denote the simple co-roots corresponding to the simple roots α i . Finally, let ρ be the element ofĥ * defined by ρ(h α i ) = 1 for i = 1, . . . , l + 1.
Proposition 4.16. Assume χ(h α i ) < −2 for i = 1, . . . , l + 1. Then for any ν ∈ V and w ∈Ŵ The above identity is an affine analogue of the Gindikin-Karpelevich formula ( [13]). The proof is by induction on the length of w ∈Ŵ . Our first step is to consider a local integral over a unipotent subgroup of SL 2 (F ν ).
For each ν ∈ V and a ∈∆ W , we have a unique group homomorphism ϕ a from SL 2 (F ν ) intô G ν by Lemma 2.5. Moreover, SL 2 (F ν ) has an Iwasawa decomposition into KAU ([10]), where We can define a real character on A by fixing a real number κ and setting a κ = ( a 0 0 a −1 ) κ = |a| κ ν . Using this character, we define the functionΦ κ : where k ∈ K, a ∈ A and u ∈ U .
Remark 4.17. The Iwasawa decomposition of SL 2 (F ν ) is not unique for an element. However in [10], we see that , and note that this group is isomorphic to the additive group F ν , and so we define a measure dũ − on U −,ν to be the Haar measure µ ν on F ν normalized so that O ν has a total measure of 1. Then the following lemma is well-known.
Lemma 4.18. If we fix a real number κ < −2, then for any ν ∈ V we have Observe that the map ϕ a of Lemma 2.5 provides an isomorphism between U −,ν and U −a,ν . Moreover, the measures dũ − and du − are identified under this isomorphism. As a result, we can equate the following integrals assuming that we choose κ ∈ R correctly.
Lemma 4.19. Fix χ ∈ĥ * to be real valued such that χ(h α i ) < −2. Then for any ν ∈ V and a ∈∆ W , As a result, in an Iwasawa decomposition for u − , we may take itsĤ ν -component to be of the form We claim that we must set κ = χ(h a ). Indeed, with this choice, we havẽ Now the identity (4.20) is a direct result of Lemma 4.18 and our choice of κ.
Armed with Lemma 4.19, we are prepared to prove Proposition 4.16: Proof of Proposition 4.16. We assume χ(h α i ) < −2 for i = 1, . . . , l + 1, and we want to show that for any ν ∈ V and w ∈Ŵ As mentioned previously, the proof is by induction on l(w), the length of the Weyl group element.
Base case: To prove the base case, we assume that l(w) = 1, and therefore that w = w i for some i = 1, . . . , l + 1. We begin by observing that∆ by Lemma 4.19 and the fact that ρ(h α i ) = 1.
Induction step: Now suppose we choose w ∈Ŵ with reduced expression w ir . . . w i 1 , and that our proposition holds for all w ′ ∈Ŵ such that l(w ′ ) < l(w). Specifically, we set w ′ = w i r−1 . . . w i 1 . At the beginning of this subsection, we showed that U −,w,ν = U −γr,ν U −,w ′ ,ν , and so our integral breaks into (4.21) The measure du − decomposes naturally by Remark 4.14. Let the element u −,1 have the Iwasawa decomposition k 1 h 1 u 1 . Recall that by definition Φ χ is left invariant byK ν and right invariant byÛ ν . In light of these observations and part (1) of Lemma 3.14, we can make the following manipulations: The integral in (4.21) now becomes We wish to show u 1 u −,2 u −1 1 ∈ w ′−1Û ν w ′ , and note that it suffices to prove that w ′ (u 1 u −,2 u −1 For any u ∈ U γr we have w ′ uw ′−1 ∈ U w ′ ·γr = U α ir , and so both w ′ u 1 w ′−1 and w ′ u −1 1 w ′−1 are elements of U α ir ⊂Û ν . Hence, The decomposition (4.3) provides us with the unique group decomposition Let π ν be the projection from w ′−1Û ν w ′ ։ U −,w ′ ,ν , which exists by the decomposition above. Then ). The map from U −,w ′ ,ν to itself defined by u −,2 → π ν (u 1 u −,2 u −1 1 ) is a unimodular change of variables, so the integral (4.22) becomes Now applying the change of variables h 1 u −,2 h −1 1 → u −,2 has the following effect on our integral: Our calculations so far have proven the following result regarding our local integrals: By our inductive hypothesis,

Moreover, observe that
and so by Lemma 4.19 we obtain Putting all of these calculations together we see In our next step, we use Proposition 4.16 to evaluate Û −,w,A Φ χ (u − )du − .
Note that the spaceÛ −,w,A can be identified with the product of ℓ(w) copies of A. Since we have assumed that χ is real, we can apply the monotone convergence theorem to see that where we use the relation (A.9) and the following remark there. In the expression above, we take S to range over the finite subsets of V. Now by Proposition 4.16, this is equal to Let ζ F (s) denote the zeta function associated to the function field F (see [17]). Then whenever Re(s) > 1. Since we have assumed χ(h α i ) < −2, we have that −(χ + ρ)(h a ) > 1 for any a ∈∆ W,+ . As a result, we obtain that .
We set c(χ, w) = q ℓ(w)(1−g) Finally, we obtain the main result of this section: Theorem 4.23. For any g = khu ∈Ĝ A and χ ∈ĥ * such that χ(h α i ) < −2 for i = 1, . . . , l + 1, we have We saw at the end of Section 3 that in order to prove the almost everywhere convergence of the series E χ , it suffices to show for h varying in compact sets ofĤ A , when χ is a real character. In this section, we showed that the series (4.25) is the same as the series (4.24). So to establish the almost everywhere convergence of the Eisenstein series E χ , we direct our attention to proving that the series (4.24) converges for h varying in compact sets ofĤ A . We will prove this result in the next section.
We also note that Theorem 4.23 closely resembles Garland's result for the constant term of the Eisenstein series overĜ R ( [4]) with the Riemann zeta function replaced by the zeta function of the function field.

Convergence of the Constant Term
In this section, we prove the convergence of the series (4.24) by showing that if h varies in a compact subset ofĤ A , then the series (4.24) is bounded above by a theta series in R. In order to establish this result, we require an additional condition on the tuple m = (m ν ) ν∈V , which determines the automorphism η mD . Specifically, we will further assume that ν∈V log(q ν )m v > 0; this will be important for the calculation in section 5.4.
We will approach this proof by treating the factor (hη mD ) w(χ+ρ)−ρ in subsections 5.2 through 5.5, and the factor c(χ, w) in subsection 5.6 . In the final subsection, we will combine the results to finish proving the convergence of the series (4.24), which is indeed the constant term E # χ of E χ . We begin with considering the compact subsets ofĤ A .
For the rest of the section, we fix a compact subset C ofĤ A along with positive real numbers r and R that satisfy the conditions of Lemma 5.1. We wish to prove that is bounded by a theta series in R as h varies in C. Clearly In the next four subsections, we focus on finding a bound for (hη mD ) w(χ+ρ) .

Preliminary Calculation.
In Section 3, we fixed the automorphism η mD where m = (m ν ) ν∈V is an infinite tuple of integers such that m ν = 0 for almost all ν ∈ V. However, we could replace D with any element ofĥ e to obtain a similar automorphism. To be more precise, let n = (n ν ) ν∈V be an infinite tuple of integers where n ν = 0 for almost all ν ∈ V. For h ∈ĥ e and ν ∈ V, we define the local automorphism η nν h ν to (i) preserve each weight space V λ µ,Fν , and (ii) act on each weight space by η nν h µ,Fν . Then we consider the product of these local automorphisms and define the global automorphism η nh := ν∈V η nν h ν . Note that this automorphism only affects a finite number of places. An element χ ∈ĥ * can be considered an element of (ĥ e ) * by setting χ(D) = 0. With this in mind, for χ ∈ĥ * we define One can easily see that for w ∈Ŵ we have If s = (s ν ) ν∈V ∈ A × then ord(s) := (ord ν (s ν )) ν∈V is a tuple of integers with the property that ord ν (s ν ) = 0 for almost all ν. Then we see that Moreover, since χ(D) = 0 we also have (η mD ) χ = 1. As a notational convenience, we setχ := χ + ρ and consider the factor for some w ∈Ŵ . Let h = h α 1 (s 1 ) . . . h α l+1 (s l+1 ). Then by our observation in (5.4) we have .
As a result, the factor (5.5) becomes .

(5.6)
For the remainder of this subsection, we will focus on calculating The affine Weyl group ofĝ e can be decomposed into a semi-direct product of the classical Weyl group and a group of translations:Ŵ = W ⋉ T . In particular, T = {T H | H ∈ h Z }. For more information see [2], [4], or [11]. Using this product decomposition, we may express w −1 = w 1 T H , for some w 1 ∈ W , and H ∈ h Z . Moreover, we note that for any H ∈ h Z the translation T H does not affect the imaginary co-root h δ , and we recall that the classical Weyl group leaves h δ and D invariant.
In order to calculate the element ofĥ e in (5.7), we will use this decomposition of w −1 and a particular result from [4] which we summarize in the following lemma: Lemma 5.8 (Garland,[4]). Let h ′ ∈ h and T H be the translation element associated to H ∈ h Z . We have the following formula: where ( , ) is the normalized bilinear form on h.
We let w −1 = w 1 T H for w 1 ∈ W and T H ∈ T . For each ν ∈ V, we set where h ν ∈ h and e ν ∈ R. We begin our calculation by making these substitutions: After factoring out −m ν for m ν = 0, we have The translation T H does not affect the imaginary root, so if we apply Lemma 5.8 we see this equals where the last equality follows from the fact that the classical Weyl group element w 1 does not affect D or h δ . One can check that the same formula holds for m ν = 0.
We continue to break up this product by writing it as a product of three factors that we obtain by grouping the parts appearing in the exponent of 1 qν . Sinceχ(D) = 0, the factor containing m ν D disappears from our computation. In particular, we have the following proposition.
Proposition 5.11. The term (hη mD ) w(χ) appearing in E # χ (gη mD ) is the product of the following three factors: Remark 5.12. Note the calculations in this subsection work for any w ∈Ŵ .
In this subsection, we calculated the term (hη mD ) w(χ) explicitly and showed that it is a product of the factors (i), (ii), and (iii) above. In the next three subsections, we will work with each factor individually showing that if we let h vary in a compact set C, we have an upper bound that depends on w ∈Ŵ . .

First notice that since we set
by our observation in (5.4). Hence it is our goal to prove there exists an upper bound for the factor h w −1 1χ for any w 1 ∈ W , the classical Weyl group.
Since the classical Weyl group is a finite group and we fixedχ ∈ĥ, we know there exist real numbers m 1 and M 1 such that m 1 < w −1 1χ (h α i ) < M 1 , for all i = 1, . . . , l + 1 and w 1 ∈ W . Moreover, we assumed in the beginning of the section that we are choosing h to vary in a compact set C ofĤ A such that for any h = h α 1 (s 1 ) . . . h α l+1 (s l+1 ) ∈ C we have r < |s i | < R for each i. Combining these, we see that there exists a positive constant M such that < M for all i = 1, . . . , l + 1 and w 1 ∈ W . Hence, we conclude: Lemma 5.14.
for all w 1 ∈ W .

5.4.
Bounding Factor (ii) of the Product. We now turn our focus to factor (ii), so we consider the infinite product .
However, recall that in the definition of the automorphism η mD we specified m = (m ν ) ν∈V where m ν ∈ Z, m ν = 0 for all but a finite number of ν. Now we further assume that ν (log q ν ) m ν > 0. Let S = {ν ∈ V | m ν = 0}. Then our infinite product above reduces to the finite product: .

(5.17)
We will treat each product in (5.17) separately, beginning with the factor involvingχ(w 1 H). If we write H = (H, H) 1/2 for H ∈ h, we can prove the following lemma: Lemma 5.18. There exists N 1 > 0 such that for any H ∈ h Z and w 1 ∈ W , Then this is clearly a compact set of h, where the topology on h is the metric topology induced by the norm · . The linear mapχ : B 0 → R is a continuous map, and hence is bounded. Since W preserves B 0 , we can find a real number N 1 such that We set N 1 = exp ν∈S (log q ν ) m ν N 1 , and obtain the desired inequality (5.19).
Next we consider the other factor from (5.17), specifically .

5.5.
Bounding Factor (iii) of the Product. We finally consider factor (iii). In particular, we find a bound for the infinite product .
Since we set h ν + e ν h δ = l+1 i=1 ord ν (s i,ν )h α i , and (h δ , H) = 0 for all H ∈ h Z , we can make the following changes As a result, the infinite product in (5.22) becomes As before, we let B 0 = {H ∈ h | H = 1}. For each i = 1, . . . , l+1, we consider the continuous maps determined by the bilinear form Since we are only considering l + 1 different images of the compact set B 0 in R, we can find numbers n and N such that n < (h α i , H) < N for all H ∈ B 0 . Using the same argument as in Lemma 5.18, we conclude that for any H ∈ h Z and i = 1, 2, . . . , l + 1, we have Since we assumed that h varies in the compact set C ⊂Ĥ A , we know that r < |s i | < R for i = 1, . . . , l + 1. It is now straightforward to prove the following lemma.
Lemma 5.23. There exists a constant N 3 such that for any H ∈ h Z we have . Now we collect the results of subsections 5.2 through 5.5 and summarize them in a proposition. Recall that Proposition 5.11 proved that (hη mD ) wχ is the product of the factors (i), (ii) and (iii). By combining this with the results of Lemmas 5.14, 5.18, 5.20 and 5.23, we obtain: Lemma 5.25. For w ∈Ŵ , we write w −1 = w 1 T H , where w 1 ∈ W and H ∈ h Z . We have the following upper bound for the factor (hη mD ) wχ : In subsection 5.7, we wish to bound the constant term by a theta series. To this end, we rewrite N We set σ 1 = log(N 1 N (l+1) 3 ) and σ 2 = − log N 2 . Since N 2 < 1, we have σ 2 > 0. We state our final result in the following proposition: Proposition 5.26. For w ∈Ŵ , suppose w −1 = w 1 T H where w 1 ∈ W and H ∈ h Z . There exist constants M, σ 1 , and σ 2 that do not depend on w ∈Ŵ , such that σ 2 > 0 and 5.6. Bounding the Zeta Functions. In this subsection, we focus on bounding the c(χ, w) factor in the constant term E # χ . Recall the definition: .
Standard techniques involving zeta functions establish the following lemma: Lemma 5.28. Let s ∈ C and ε > 0. Then for all s such that Re(s) ≥ 1 + ε we have where M ε is a positive constant. In particular, we can take M ε = q (1−g) (ζ F (1 + ε)) 2 .
Each w ∈Ŵ can be expressed as w = w 1 T H for w 1 ∈ W and some H ∈ h Z , and we have ℓ(w) ≤ ℓ(w 1 ) + ℓ(T H ). However, since the classical Weyl group is finite, ℓ(w 1 ) can only be as large as the length of the longest element of W . Thus for each w ∈Ŵ , In (8.17) of [4], we see that there exists a postive constant σ 3 such that ℓ(T H ) ≤ σ 3 H . In light of these observations, we have the following proposition:  is bounded above by a theta series. Hence the constant term of the Eisenstein E # χ is absolutely convergent for these compact sets.
By substituting the results of Propositions 5.26 and 5.30, we see where #(W ) is the cardinality of the finite Weyl group W . It is essential to note that σ 2 > 0, so this theta series converges. As a result of this computation, we have proven the following theorem.
Remark 5.33. Combining the above theorem with (3.21), Definition 3.22 and Theorem 4.23, we have proved the identities and established convergence of the constant term E # χ .

Convergence of the Eisenstein Series
In this section, we will use the results of the previous sections to prove the convergence of the Eisenstein series E χ . Recall that in Section 4, we skipped the proof of the measurability of E χ with respect to du. In the next subsection, we prove this fact. 6.1. Measurability of the Eisenstein Series. The constant term of the Eisenstein series E χ is defined to be the following integral: It is the purpose of this subsection to prove that the map u → E χ (gη mD u) is a du-measurable function. Observe however, that since Φ χ is left invariant byK, and η mD normalizesÛ A , it is enough to show that for a fixed h ∈Ĥ A the map u → E χ (hη mD u) is a du-measurable function.
As in Section 4, we express the Eisenstein series as the sum In turn, each E χ,w (hη mD u) is also a sum of particular values for Φ χ . Recall that we setΓ F (w) = Γ F ∩ (B F wB F ) and defined and we saw in Section 4 that we may take the coset representatives γ above to be of the form {bw} where b ∈Û w,F . Now with respect to our coherently ordered basis,Û w,F is a finite-dimensional space for each w ∈Ŵ , and as such we can choose our coset representatives ofΓ F (w)/(Γ F (w)∩B F ) to come from this finite-dimensional space.
In Appendix A, we construct the measure du by expressingÛ A /(Û A ∩Γ F ) as a projective limit of compact spaces. Part of this construction is important for this discussion, so we briefly state some definitions from that section. A . Since we may choose our coset representatives γ ∈Γ F (w)/(Γ F (w) ∩B F ) so that they come from a finite-dimensional space, there exists an s large enough so that for any u ∈Û A,s and γ as above, we have uγ = γu ′ for some u ′ ∈Û A . Then we observe that Using Φ χ and γ ∈Γ F (w)/(Γ F (w) ∩B F ), we define the function ψ γ fromÛ A to R >0 by setting By the observation (6.2) in the previous paragraph, ψ γ defines a function on the finite dimensional spaceÛ A /Û A,s for s large enough. We will see in Appendix A that we may considerÛ A /Û A,s as embedded into the group of upper triangular (s + 1) × (s + 1) block matrices with entries from A. Most importantly, for any γ ∈Γ F (w)/(Γ F (w) ∩B F ) the function ψ γ can be written as a composition of continuous maps and hence measurable on the spaceÛ A /Û A,s . Then it follows from the definition of the measure du in Appendix A that the function ψ γ is a measurable function onÛ A /(Û A ∩Γ F ). For a fixed h ∈Ĥ A and w ∈Ŵ , we define the function The function ψ γ is a positive, measurable function for every γ above, and so ψ w (u) is also a measurable function onÛ A /(Γ F ∩Û A ). To see this, we view (6.4) ψ w = sup{finite sums of ψ γ }, and note that this is measurable. Likewise, for a fixed h ∈Ĥ A we can consider E χ as a function from the quotientÛ A /(Û A ∩Γ F ) to the positive real numbers by sending u → E χ (hη mD u). Since E χ can be expressed as the sum over the affine Weyl group of the positive, measurable functions ψ w , we conclude: Lemma 6.5. For any h ∈Ĥ A , the function E χ is a du-measurable function into the positive real numbers.

6.2.
Convergence of the Series. The Eisenstein series E χ onĜ A η mD can be considered as the function fromĤ A ×Û A /(Û A ∩Γ F ) to R >0 defined by (6.6) (h, u) → E χ (h η mD u).
Moreover, Theorem 5.32 proved that the constant term is absolutely convergent for h varying in compact sets ofĤ A . However, this also tells us that the Eisenstein series E χ is integrable with respect to du for h varying in any compact subset ofĤ A . Hence the series E χ is convergent almost everywhere onĤ A ×Û A /(Û A ∩Γ F ), sinceĤ A is locally compact. Moreover, we can prove the following proposition: Proposition 6.7. Let χ ∈ĥ * be a real character such that χ(h α i ) < −2 for i = 1, . . . , l + 1, and let m = (m ν ) ν∈V be a tuple such that m ν ∈ Z ≥0 and 0 < ν m ν < ∞. Then the series E χ (hη mD u) (absolutely) converges to a positive real number for all (h, u) ∈Ĥ A ×Û A /(Û A ∩Γ F ).
Proof. The remarks in the previous paragraph tell us that the series E χ (hη mD u) converges for all (h, u) ∈Ĥ A ×Û A /(Û A ∩Γ F ) off a set of measure zero. Assume that E χ (hη mD u) = ∞ for some (h, u). We claim that there exists a subset U ′ ⊂Û A of positive measure such that hη mD u ′ (hη mD ) −1 ∈Û A ∩K for all u ′ ∈ U ′ . If the claim is true, we will have for all u ′ ∈ U ′ . Since the set U ′ u has positive measure, it is a contradiction. Now we prove the claim. We write u ′ = (u ′ ν ) and (13.10) of [2]. Recall that m ν ≥ 0 for all ν and ord ν (s i,ν ) = 0 for almost all ν. Because of these conditions on m ν and ord ν (s i,ν ), it is straightforward to construct the set U ′ of the claim.
Having established this important proposition, the following theorem is a simple consequence of the dominated convergence theorem and the fact that E Re(χ) dominates E χ for any complex character χ.

Functional Equations for the Constant Term
In this section, we will establish meromorphic continuation of the constant term of the Eisenstein series and prove their functional equations. We will begin by stating some results for the zeta function of the global function field F . 7.1. Background on the Zeta Function of a Function Field. We refer the reader to [17] for specifics on the definition of the zeta function associated to a global function field F . The following result from [17] describes the functional equation for ζ F (s).
Theorem 7.1. Let F be a global function field in one variable over a finite constant field F q . Suppose F is of genus g. Then there exists a polynomial L F (u) ∈ Z[u] of degree 2g such that for Re(s) > 1. Moreover, (7.2) provides an analytic continuation of ζ F to the complex plane. If we set ξ F (s) = q (g−1)s ζ F (s), then we have the functional equation Using the functional equation for ξ F (s), we prove the following lemma: Lemma 7.4. Let ζ F (s) be the zeta function associated to F , and ξ F (s) = q (g−1)s ζ F (s) its completed form. Then the following identities hold: Proof. These computations follow by replacing ζ F (s) with q s(1−g) ξ F (s) and using the functional equation of ξ F (s). For example, we consider calculation for the second identity: The first identity follows in the same manner, and third one is simply the product of the other two identities.
Let F ⊂ C ε be the set of all χ ∈ C ε such that the function ζ F (−(χ+ρ)(ha)) ζ F (−(χ+ρ)(ha)+1) has a pole for some a ∈ Ξ ε . Then F is contained in the union of a countable, locally finite family of hyperplanes.
Lemma 7.7. Suppose that B is a bounded open subset of (ĥ e C ) * whose closure is contained in the set C ε . Assume that the function ζ F (−(χ+ρ)(ha)) ζ F (−(χ+ρ)(ha)+1) has no pole for any χ ∈ B and for any a ∈ Ξ ε . Then there are positive constants M ′ ε and M ε such that for each w ∈Ŵ we have Proof. Since Ξ ε is a finite set and independent of w, this lemma is proved by a slight modification of the proof of Corollary 5.29.
Proof. Assume that B is a bounded open subset of (ĥ e C ) * whose closure is contained in the set C ε \ F, where the set F is defined above. Consider a compact subset C ofĤ A . If we replace Corollary 5.29 with Lemma 7.7, all the other arguments in the proof of Theorem 5.32 remain valid to prove that the infinite sum w∈Ŵ (hη mD ) w(χ+ρ)−ρ c(χ, w) converges uniformly and absolutely as χ varies over B and h varies over C. Since ε was arbitrary, the theorem follows. 7.3. Proving Functional Equations. We wish to prove functional equations for the constant term of the Eisenstein series E χ onĜ A . The following property of the function c(χ, w) is essential: Proposition 7.9. With c(χ, w) defined as above and w, w ′ ∈Ŵ , we have where w • χ is the usual shifted action ofŴ onĥ * , i.e. w • χ = w(χ + ρ) − ρ.
We will prove this proposition at the end of this section. First, we show that this proposition leads to the following functional equation for the constant term of the Eisenstein series.
Proof. For the first assertion, we only need to consider the simple reflections w i ∈Ŵ . We have Since we assumed Re χ(h δ ) < −h ∨ , the the first assertion follows. Now we fix an arbitrary element w of the affine Weyl group. Then for anyw ∈Ŵ we set w ′ =ww −1 so thatw = w ′ w. Now by Proposition 7.9, we have In order to establish this functional equation for E # χ , it suffices to prove Proposition 7.9.
We proceed by an induction argument, and first consider the following cases.
Case 1: Suppose that w i is a simple reflection, w ∈Ŵ satisfies ℓ(ww i ) = 1 + ℓ(w), and w −1 = w ir . . . w i 1 is a reduced expression. Then since∆ W,+ ∩w −1∆ W,− =∆ w −1 the discussion in Section 4.3 shows∆ . This is our desired result, so we move on to our second case.
Case 2: Suppose w i is a simple reflection and w ∈Ŵ such that ℓ(ww i ) = ℓ(w) − 1. If this is the case, then w has a reduced expression w = w i 1 . . . w ir where w ir = w i . If we set w ′ = w i 1 . . . w i r−1 , then w = w ′ w i and w ′ = ww i , and ℓ(w ′ w i ) = 1 + ℓ(w ′ ). As such, we can apply the result of Case 1 to this situation withχ = w i • χ and we see So by solving for the factor c(χ, ww i ), we see that In order to prove our result for this case, it suffices to show that Observe that .
In Lemma 7.4, we calculated the value of this product of zeta functions, so we conclude that As a result, we see c(χ, ww i ) = c(w i • χ, w)c(χ, w i ), the desired result. We now consider the general case.
General Case: The proof is by induction on the length of w ′ . If ℓ(w ′ ) = 1, then the proofs of Case 1 and Case 2 secure our result. Now suppose the result holds for ℓ(w ′ ) ≤ k. We consider w ′′ ∈Ŵ such that ℓ(w ′′ ) = k + 1, and write w ′′ = w ′ w i for some simple reflection w i and with ℓ(w ′′ ) = ℓ(w ′ ) + 1. By Cases 1 and 2 and the induction hypothesis, we obtain . This completes our proof of Proposition 7.9.

Appendix A. Measures
In this appendix, we will describe various measures that are important for our calculation of the constant term of an Eisenstein series. The first subsection addresses the technique of constructing a measure by means of a projective limit of a family of measures.
A.1. The projective limit construction of a measure. We begin by stating a result of [1]. Let {U (s) , π s s ′ } s∈Z >0 be a projective family of compact spaces, and equip each U (s) with a regular, Borel, probability measure du (s) . If s > s ′ , then by our assumption we have the map π s s ′ : U (s) ։ U (s ′ ) . We say {du (s) } s∈Z >0 is a consistent family of measures with respect to the projections π s s ′ if for any measurable set X ′ ⊂ U (s ′ ) we have du (s) ((π s s ′ ) −1 (X ′ )) = du (s ′ ) (X ′ ), for any s > s ′ .
Corollary A.2. Under the conditions of Theorem A.1, if we further assume that each measure du (s) is translation invariant, then du is also translation invariant.
Proof. Our measure du is a Borel measure, so it is enough to show that this property holds for any open set Y ⊂ lim ← − s U (s) . However, the projective limit of topological spaces inherits the coarsest topology such that the canonical projections π (s) are all continuous. It is a standard result that a basis for this topology consists of the sets (π (s) ) −1 (X (s) ) for an open set X (s) ⊂ U (s) . Hence, the translation invariance of the measure du is a result of part (2) of Theorem A.1, and the invariance of the measure du (s) .
A.2. Measure on the Arithmetic Quotient. As mentioned in Section 4, we need to define a measure on the quotient spaceÛ A /(Û A ∩Γ F ). The main result of [1] allows us to define this measure du by expressingÛ A /(Û A ∩Γ F ) as a projective limit of compact spacesÛ We first note that any element u ∈Û

(s)
A is an infinite tuple (u ν ) ν∈V . In [2] we see that each u ν is in the subgroup of (s + 1) × (s + 1), strictly upper triangular block matrices with entries from F ν , where the blocks are determined by the weight spaces of V λ Z . Moreover, we know that for all but a finite number of ν, our entries are from O ν . As such, we may identifyÛ (s) A with n copies of A, for some n. We giveÛ Proof. The first part follows from the local compactness of A and the homeomorphism φ (s) . In order to prove the second part, we first check that {Û

(s)
A , π s s ′ } forms a projective system. Suppose we have s > s ′ . Then V λ Fν ,s ′ ⊂ V λ Fν ,s for all ν ∈ V, as a result we have thatÛ A,s ⊂Û A,s ′ and obtain a unique surjective map π s s ′ :Û A . In order to prove the isomorphism, we only need to show that Φ is an injective map.
Finally, we obtain the main result of this appendix in the following proposition.
Proposition A.7. There exists a unique,Û A -left invariant, probability measure du on the arithmetic quotientÛ A /(Û A ∩Γ F ).
Proof. This follows from Lemma A.6 and Theorem A.1.
A.3. Measures on other spaces. Before we discuss measures on other spaces, we briefly recall some constructions from Section 4. With respect to our coherently ordered basis B, we fixÛ −,F to be the group of strictly lower triangular block matrices and setÛ w,F to beÛ F ∩ wÛ −,F w −1 . This definition works for all of our fields F ν , and so we can definê U w,A := ν∈V ′Û w,ν with respect toÛ w,ν ∩K ν .
By the decomposition (4.3), we know (A.8)Û A =Û w,A (Û A ∩ wÛ A w −1 ), and we have the covering projection We define the measure du ′ onÛ A /(Γ F ∩Û F ∩ wÛ F w −1 ) to be the one induced from the measure du onÛ A /(Û A ∩Γ F ) through the projection π ′ .
We construct the measure du 2 by expressing this space as a projective limit of compact spaces equipped with a Haar probability measure. We omit the details and note that the construction will be very similar to our proofs regarding the construction of the measure du onÛ A /(Û A ∩Γ F ). Using Theorem A.1, the measure du 2 is a Haar probability measure.
To construct du 1 , we recall that every element of u ∈Û w,A takes the form u = a∈∆ W,+ ∩w −1∆ W,− χ a (s a ) , for s a ∈ A.
For more information, see Section 4.3. The set∆ W,+ ∩ w −1∆ W,− is finite and of size ℓ(w). As in Lemma A.3, we have a homeomorphism fromÛ w,A to A ℓ(w) . The product measure induced from ℓ(w) copies of the Haar measure µ on A becomes a Haar measure onÛ w,A , where we normalize µ so that µ(A/F ) = 1. We set du 1 to be this measure.
If we consider the Haar measure µ ′ ν on F ν with the normalization µ ′ ν (O ν ) = 1 for each ν ∈ V, we obtain from 2.1.3 of [18] that If we let du ′ 1,ν be the measure onÛ w,Fν , then we have A similar relation holds between du − forÛ −,w,A and the product of measures on local components in Section 4.4.