Euler Products Robert P. Langlands

EULER PRODUCTS ^{†}

James K. Whittemore Lectures in Mathematics given at Yale University

CONTENTS

1. Introduction

2. Some Euler products

3. Spherical functions

4. The formula of Gindikin and Karpelevich 5. A review of Eisenstein series

6. Examples

† First published in 1971 as a Yale Mathematical Monograph by Yale University Press

1. INTRODUCTION

The first undisguised automorphic forms met by most of us are the modular forms. A modular form of weight k is an analytic function on the upper half-plane which satisfies

f ( \frac{az + b}{cz + d} ) = (cz + d)^{k}f(z)

for all integral matrices

( \begin{matrix}a & b \\ c & d \end{matrix} )

of determinant 1. Actually there is a supplementary condition, of no importance to us, on its rate of growth as Im(z) → ∞. Nowadays the Siegel modular forms are met soon afterwards. A Siegel modular form of weight k is a complex analytic function on the space of complex n × n symmetric matrices with positive definite real part which satisfies

f((AZ + B)(CZ + D)^{−1}) = det(CZ + D)^{k}f(Z)

for all integral 2n × 2n symplectic matrices

( \begin{matrix}A & B \\ C & D \end{matrix} ) .

For some purposes it is best to consider not f but an associated function ϕ_{f} on the group G of real 2n × 2n symplectic matrices defined by

ϕ_{f}(g) = det(Ci + D)^{−k}f((Ai + B)(Ci + D)^{−1})

if

g = ( \begin{matrix}A & B \\ C & D \end{matrix} ) .

If Γ is the group of integral matrices in G then ϕ_{f} is a function on Γ\G. The functions ϕ_{f} associated to those f which have a finite norm in the Petersson metric can be characterized in terms of the representations of G. Associating to each h in G the operator λ(h) on L^{2}(Γ\G) defined by

(λ(h)ϕ)(g) = ϕ(gh) ,

we obtain a representation of G on L^{2}(Γ\G). There is a representation π_{k} of G on a Hilbert space H_{k} and a distinguished subspace H_{k}^{0} of H_{k} such that ϕ is a ϕ_{f} if and only if there is a G-invariant map of H_{k} to L^{2}(Γ\G) which takes H_{k}^{0} to the space generated by ϕ.

This is the first hint that it might not be entirely unprofitable to study automorphic forms in the context of the theory of group representations. The Hecke operators, which play a major role in the study of modular forms, provide a second. To see this we have to introduce the adèle group of 2n × 2n symplectic matrices. It will be convenient to change the notation a little. If R is a commutative ring let G_{R} be the group of 2n × 2n symplectic matrices with entries from R. Thus the groups Γ and G become G_{ℤ} and G_{ℝ}. If p is a prime, finite or infinite, let ℚ_{p} be the corresponding completion of ℚ. The adèle ring 𝔸 is the set of elements {a_{p}} in ∏_{p} ℚ_{p} such that a_{p} is integral for all but a finite number of p. The diagonal map defines an imbedding of ℚ in 𝔸. There is a standard topology on G_{𝔸} which turns it into a locally compact group with G_{ℚ} as a discrete subgroup. Let

𝔸^{0} = ℝ × ∏_{p finite} ℤ_{p} .

It is known that G_{𝔸} = G_{ℚ}G_{𝔸^{0}}. Any function ϕ on G_{ℤ}\G_{ℝ} can be regarded as a function on G_{ℚ}\G_{𝔸} if one sets ϕ(g) = ϕ(g′_{2}) when g = g_{1}g_{2}, g_{1} ∈ G_{ℚ}, g_{2} ∈ G_{𝔸^{0}}, and g′_{2} is the projection of g_{2} on G_{ℝ}. The functions so obtained are characterized by their right invariance under

U = 1 × ∏_{p finite} G_{ℤ_{p}} .

If f is a function, with compact support, on

G^{0} = {1 × ∏_{p finite} G_{ℚ_{p}}} ∩ G_{𝔸} ,

which is invariant on the left and right under U, and if λ(f)ϕ is defined by

λ(f)ϕ(g) = ∫_{G^{0}} ϕ(gh)f(h) dh ,

then λ(f)ϕ is invariant on the right under U if ϕ is. Thus the operators λ(f), the Hecke operators, act on the functions on G_{ℤ}\G_{ℝ}. If ϕ belongs to a subspace H of L^{2}(G_{ℚ}\G_{𝔸}) which is invariant and irreducible under the action of G_{𝔸} then ϕ is an eigenfunction of all the Hecke operators and the corresponding eigenvalues are determined by the equivalence class of the representation of G_{𝔸} on H.

The theory of modular forms and the operators λ(f) is far from complete. Indeed very little attempt has been made, so far as I can see, to understand what the goals of the theory should be. Once it is put in the above form it is clear that the concepts at least admit of extension to any reductive algebraic group defined over a number field. It may be possible to give some coherency to the subject by introducing the simple principle, implicit in the work of Harish-Chandra, that what can be done for one reductive group should be done for all.

The simplest reductive group over ℚ is GL(1). Since G_{𝔸} is abelian, the space G_{ℚ}\G_{𝔸} is a locally compact abelian group C, the group of idèle classes of ℚ. According to the Plancherel theorem for abelian groups the characters of C can be used to decompose L^{2}(G_{ℚ}\G_{𝔸}) under the action of G_{𝔸} and the characters of C must be regarded as the basic automorphic forms.

Since ℚ_{p}^{×}, the multiplicative group of ℚ_{p}, is contained in G_{𝔸}, each character χ of C defines a character χ_{p} of ℚ_{p}^{×}. If p does not divide f_{χ}, the conductor of χ, χ_{p} is trivial on the units of ℚ_{p}^{×}. It is customary to associate to χ the L-series

L(s, χ) = ∏_{p ∤ f_{χ}} \frac{1}{1 − \frac{χ_{p}(p)}{p^{s}}}

and the function

ξ(s, χ) = (\frac{π}{f_{χ}} )^{−(s+α+β)/2} Γ (\frac{s + α + β}{2} ) L(s, χ)

if

χ_{∞}(x) = (sgn x)^{α}|x|^{β} .

The number α is 0 or 1. The function ξ(s, χ) is known to be meromorphic in the entire complex plane and to satisfy

ξ(s, χ) = ϵ(χ)ξ(1 − s, χ^{−1})

where ϵ(χ) is a constant of absolute value 1.

For a general group L^{2}(G_{ℚ}\G_{𝔸}) decomposes not into a direct integral of one-dimensional spaces but into a direct integral of Hilbert spaces on each of which G_{𝔸} acts irreducibly. Can we associate to each of these Hilbert spaces an Euler product with the same analytic properties as the functions L(s, χ)? In these lectures I would like to present a little evidence, far from conclusive, that the answer is “yes”. Let me observe that each power of a character is also a character, thus to χ is associated the whole collection {L(s, χ^{n}) | n ∈ ℤ}, and that ℤ parametrizes not only the powers of χ but also the rational representations of GL(1).

Before beginning the substantial part of these lectures let me make, without committing myself, a further observation. The Euler products mentioned above are defined by means of the Hecke operators. Thus they are defined in an entirely different manner than those of Artin or Hasse-Weil. An assertion that an Euler product of the latter type is equal to one of those associated to an automorphic form is tantamount to a reciprocity law (for one equation in one variable in the case of the Artin L-series and for several equations in several variables in the case of the Hasse-Weil L-series).

2. SOME EULER PRODUCTS

Suppose 𝔤 is a split semisimple Lie algebra over ℚ and G its adjoint group. Fix a split Cartan subalgebra 𝔥 of 𝔤 and choose a Chevalley basis for 𝔤 with respect to 𝔥. Let M be the lattice generated by the Chevalley basis over ℤ. If p is a finite prime let G_{ℤ_{p}} be the stabilizer of M ⊗ ℤ_{p} in G_{ℚ_{p}}. If p is the infinite prime let

G_{ℤ_{p}} ⊆ G_{ℚ_{p}} ≡ G_{ℝ}

be the maximal compact subgroup of G_{ℝ} corresponding to the involution associated to the Chevalley basis. Let

U = ∏_{p} G_{ℤ_{p}} ⊆ G_{𝔸} ,

be the adèle group of G. Fix a Borel subgroup B containing T, the Cartan subgroup with Lie algebra 𝔥. It is known that

G_{ℚ_{p}} = B_{ℚ_{p}}G_{ℤ_{p}}

for each p. For this and various other facts about reduction theory over local fields see the article by F. Bruhat in the collection Algebraic Groups and Discontinuous Subgroups. As a consequence G_{𝔸} = B_{𝔸}U. Moreover B_{𝔸} = B_{ℚ}B_{ℝ}(B_{𝔸} ∩ U); hence G_{𝔸} = B_{ℚ}G_{ℝ}U and G_{ℚ} = B_{ℚ}G_{ℤ} if G_{ℤ} is the stabilizer of M. In particular any function on G_{ℚ}\G_{𝔸}/U is determined by its restriction to G_{ℝ}.

Let L be the space of all square integrable functions on G_{ℚ}\G_{𝔸} which are invariant under right translations by elements of U. Let P be a parabolic group containing B and let N be its unipotent radical. If ϕ lies in L then

∫_{N_{ℚ}\N_{𝔸}} ϕ(ng) dn

vanishes for almost all g in G_{𝔸} if and only if

∫_{N_{ℤ}\N_{ℝ}} ϕ(ng) dn

vanishes for almost all g in G_{ℝ}. If these integrals vanish for almost all g for all choices of P except G itself we say that ϕ is a cusp form. The set of cusp forms is a closed subspace L_{0} of L.

If p is a prime, finite or infinite, let H_{p} be the algebra of all compactly supported regular Borel measures on G_{ℚ_{p}} invariant under left and right translations by elements of G_{ℤ_{p}}. Multiplication is given by convolution. If μ lies in H_{p}, define the operator λ(μ) on L_{0} by

λ(μ)ϕ(g) = ∫_{G_{ℚp}} ϕ(gh) dμ(h) .

If μ is the measure associated with an L^{1} function f we shall sometimes write λ(f) instead of λ(μ). Moreover f will be regarded as an element of H_{p}. If p is finite all the measures in H_{p} are absolutely continuous with respect to Haar measure. There is an orthonormal basis ϕ_{1}, ϕ_{2}, … of L_{0} such that each ϕ_{i} is, for all p, an eigenfunction of λ(μ) for all μ in H_{p}.

Fix one element ϕ of this basis. If μ belongs to H_{p} let λ(μ)ϕ = χ_{p}(μ)ϕ. The map μ → χ_{p}(μ) is a homomorphism of H_{p} into the complex numbers. Let me remind you of the standard method of obtaining all such homomorphisms. Let V be the unipotent radical of B. Now V_{ℚ_{p}}\B_{ℚ_{p}} is isomorphic to T_{ℚ_{p}}. Thus any homomorphism w of T_{ℚ_{p}}/T_{ℤ_{p}} into the complex numbers determines a homomorphism of B_{ℚ_{p}} into the complex numbers which we again denote by w. If b belongs to B let ξ(b) be the determinant of the restriction of Ad(b) to 𝔳, the Lie algebra of V. If g lies in G_{ℚ_{p}} then g can be written as a product bk with b in B_{ℚ_{p}} and k in G_{ℤ_{p}}. Set

ψ_{w}(g) = w(b)|ξ(b)|^{\frac{1}{2}} .

The function ψ_{w} is well defined and any other function ψ on G_{ℚ_{p}} satisfying

(1)  ψ(bgk) = w(b)|ξ(b)|^{\frac{1}{2}} ψ(g)

for all b, g, and k is a scalar multiple of ψ_{w}. If μ lies in H_{p} define λ(μ)ψ_{w} by

(λ(μ)ψ_{w})(g) = ∫_{G_{ℚp}} ψ_{w}(gh) dμ(h) .

The function λ(μ)ψ_{w} satisfies (1) so there is a scalar χ_{w}(μ) such that

λ(μ)ψ_{w} = χ_{w}(μ)ψ_{w} .

The map μ → χ_{w}(μ) is a homomorphism of H_{p} into ℂ. All homomorphisms of H_{p} into the complex numbers which are continuous in the weak topology are obtained in this way. The homomorphism χ_{w} equals χ_{w′} if and only if there is a σ in the Weyl group such that w(t) = w′(t^{σ}) for all t in T_{ℚ_{p}}.

Suppose p is finite. If L is the lattice generated by the roots of 𝔥 there is a homomorphism λ from \overline{T} = T_{ℚ_{p}}/T_{ℤ_{p}}, or from T_{ℚ_{p}}, to ^{c}L = Hom(L, ℤ) such that |ξ_{α}(t)| = p^{λ(t)(α)} if α is a root. Here ξ_{α} is the character of T associated to α. If α is a root let H_{α} be, in the language of Chevalley, the copoid attached to α. Let α_{1}, …, α_{n} be the simple roots. The matrix

(A_{ij}) = (\frac{(α_{i}, α_{j})}{(α_{i}, α_{i})})

is the Cartan matrix of 𝔤. The matrix

(\frac{(H_{α_{i}}, H_{α_{j}})}{(H_{α_{i}}, H_{α_{i}})})

in the transpose of (A_{ij}) and the Cartan matrix of another Lie algebra ^{c}𝔤. The lattice ^{c}L′ generated by the roots of a split Cartan subalgebra ^{c}𝔥 of ^{c}𝔤 can be identified with the lattice in 𝔥_{ℝ} generated by H_{α_{1}}, …, H_{α_{n}} in such a way that the roots of ^{c}𝔥 correspond to the elements H_{α}. Moreover ^{c}L can be regarded as a lattice in 𝔥_{ℝ}. It contains ^{c}L′ and can in fact be regarded as the lattice of weights of ^{c}𝔥 so 𝔥_{ℝ} ⊇ ^{c}L ⊇ ^{c}L′. In the same way ^{c}𝔥_{ℝ} may be identified with Hom (^{c}L, ℝ) so ^{c}𝔥_{ℝ} ⊇ L′ ⊇ L if L′ is the lattice of weights of 𝔥. Let cG be the simply connected group with Lie algebra ^{c}𝔤 and let ^{c}T be the Cartan subgroup corresponding to c𝔥. There is an isomorphism σ → ^{c}σ of the Weyl group of T in G with that of ^{c}T in ^{c}G such that

cσ(λ(t)) = λ(σt), t ∈ T_{ℚ_{p}} .

If w is a homomorphism of \overline{T} into the complex numbers there is a unique point g in^{c}T_{ℂ} such that w(t) = ξ_{λ}(g) for all t. Here λ = λ(t) and ξ_{λ} is the rational character of ^{c}T associated to λ. Thus associated to each homomorphism of H_{p} into the complex numbers is an orbit of the Weyl group in ^{c}T_{ℂ} or, as I prefer, a semisimple conjugacy class in ^{c}G_{ℂ}.

The automorphic form ϕ determined for each p a homomorphism χ_{p} of H_{p} into ℂ. If p is a finite prime let {g_{p}} be the conjugacy class in ^{c}G_{ℂ} corresponding to χ_{p}. If π is a finite dimensional complex representation of ^{c}G_{ℂ}, we can consider the Euler product

(2)  ∏_{p} \frac{1}{det (1 − \frac{π(g_{p})}{p^{s}})} = L(s, π, ϕ) .

As we shall see, this Euler product is absolutely convergent for Re (s) sufficiently large. I do not know in general what the analytic properties of the function L(s, π, ϕ) are. I will show, however, that for all but three of the simple groups there is at least one nontrivial representation for which L(s, π, ϕ) is meromorphic in the whole complex plane. For some groups there are several such representations.

Let me first introduce a Γ-factor to go with L(S, π, ϕ). If p is the infinite prime there is a homomorphism λ of

\overline{T} = T_{ℚ_{p}}/T_{ℚ_{p}} ∩ G_{ℤ_{p}}

into 𝔥_{ℝ} = Hom (L, ℝ) such that |ξ_{α}(t)| = e^{λ(t)(α)} if α is a root. Since L is a lattice in ^{c}𝔥_{ℝ}, every homomorphism of \overline{T} into C is of the form

w(t) = e^{λ(t)(X)}

for some X in ^{c}𝔥_{ℂ}. Thus to every homomorphism of H_{p} into ℂ there is associated an orbit of the Weyl group in ^{c}𝔥_{ℂ} or a semisimple conjugacy class in ^{c}𝔤_{ℂ}. If χ_{p} is the homomorphism associated to the automorphic form ϕ, let {X} be the associated conjugacy class and let Γ(s, π, ϕ) be the inverse of

π^{trace^{(\frac{s−π(X)}{2})}}det (\frac{s − π(X)}{2}) e^{γ trace(\frac{s−π(X)}{2})} ∏_{n=1}^{∞} \{det (I + \frac{s − π(X)}{2n}) e^{−trace^{(\frac{s−π(X)}{2n})}}\}

where γ is Euler’s constant. The function Γ(s, π, ϕ) can be expressed as a product of Γ-functions. Set

ξ(s, π, ϕ) = Γ(s, π, ϕ)L(s, π, ϕ).

The functional equation to expect is

ξ(s, π, ϕ) = ξ(1 − s, π̃, ϕ)

if π̃ is the representation contragredient to π.

3. SPHERICAL FUNCTIONS

Each g_{p} that occurs in the expression on the left of (2) can be chosen to lie in ^{c}T_{ℂ}. To see that the product converges in a half-plane it would be enough to show that for all λ in ^{c}L and all p

|ξ_{λ}(g_{p})| ≤ λ̄(ρ)

where λ̄ is that element in the orbit of λ under the Weyl group which lies in the positive Weyl chamber and ρ is one-half the sum of the positive roots. We can associate to each g in ^{c}T_{ℂ} a point μ = μ(g) in ^{c}𝔥_{ℂ} so that^{∗} ξ_{λ}(g) = p^{λ(μ)} for all λ in ^{c}L. The point μ is not uniquely determined by g but its real part is. If μ_{p} = μ(g_{p}) we have to show that

Re λ(μ_{p}) ≤ λ̄(ρ) .

The class {g_{p}} is associated to the homomorphism χ_{p} of H_{p} into ℂ determined by ϕ. This homomorphism has the property, not shared by all homomorphisms, that

|χ_{p}(f)| ≤ c ∫_{G_{ℚp}} |f(g)| dg

for all f in H_{p}. The factor c is a fixed constant. To prove this we must recall that ϕ is a cusp form and hence bounded. If M is a bound for ϕ and if ϕ(g_{0}) ≠ 0 then

|χ_{p}(f)ϕ(g_{0})| = | ∫_{G_{ℚp}} ϕ(g_{0}h)f(h) dh | ≤ M ∫_{G_{ℚp}} |f(h)| dh

and the assertion follows.

Lemma. Suppose μ in ^{c}h_{ℂ} is associated to the homomorphism χ_{μ} of H_{p} into ℂ. If there is a constant c such that

|χ_{μ}(f)| ≤ c ∫_{G_{ℚp}} |f(g)| dg

for all f, then

Re λ(μ) ≤ λ̄(ρ)

for all λ.

If w is the homomorphism defined by w(t) = p^{λ(t)(μ)}, set

ψ_{μ}(g) = ψ_{w}(g) .

By definition

χ_{μ}(f)ψ_{μ}(g) = ∫_{G_{ℚp}} ψ_{μ}(gh)f(h) dh .

If

ϕ_{μ}(g) = ∫_{G_{ℤp}} ψ_{μ}(kg) dk ,

∗ At the infinite prime we would take μ = X with X as above. then ϕ_{μ}(1) = 1, and ϕ_{μ}(k_{1}gk_{2}) = ϕ_{μ}(g) if k_{1} and k_{2} lie in G_{ℤ_{p}}. Moreover it is easily verified that

χ_{μ}(f)ϕ_{μ}(g) = ∫_{G_{ℚp}} ϕ_{μ}(gh)f(h) dh .

If μ satisfies the assumption of the lemma, take f_{g} to be the characteristic function of the double coset G_{ℤ_{p}}gG_{ℤ_{p}}. Then χ_{μ}(f_{g}) is equal to χ_{μ}(f_{g})ϕ_{μ}(1) = ϕ_{μ}(g) times the measure of (G_{ℤ_{p}}gG_{ℤ_{p}}). It follows immediately that |ϕ_{μ}(g)| ≤ c for all g.

To prove the lemma, it will be necessary to study the asymptotic behavior of ϕ_{μ} for general values of μ. Let T_{ℚ_{p}}^{−} be the set of t in T_{ℚ_{p}} for which −λ(t) lies in the positive Weyl chamber. Since

G_{ℚ_{p}} = G_{ℤ_{p}}T_{ℚ_{p}}^{−}G_{ℤ_{p}}

it is sufficient to study the function ϕ_{μ} on T_{ℚ_{p}}^{−}.

So far we are free to choose Haar measures in any manner we like. We so choose them on V_{ℚ_{p}}, T_{ℚ_{p}} and G_{ℤ_{p}} that the total measures of V_{ℤ_{p}}, T_{ℤ_{p}} and G_{ℤ_{p}} are 1. Then we choose the Haar measure on G_{ℚ_{p}} so that

∫_{G_{ℚp}} f(g) dg = ∫_{V_{ℚp}} ∫_{T_{ℚp}} ∫_{G_{ℤp}} f(vtk)p^{−2λ(t)(ρ)} dv dt dk .

Choose an f in H_{p} and let C be a compact set in V_{ℚ_{p}} such that the support of f is contained in CT_{ℚ_{p}}G_{ℤ_{p}}. There is a constant c_{f} > 0 such that if |ξ_{α}(t)| ≤ c_{f} for α > 0, then

tCt^{−1} ⊆ G_{ℤ_{p}} .

Choose a t satisfying this condition. Then

χ_{μ}(f)ϕ_{μ}(t) = ∫_{V_{ℚp}} ∫_{T_{ℚp}} ϕ_{μ}(tvs)f(vs)p^{−2λ(s)(ρ)} dv ds

= ∫_{C} ∫_{T_{ℚp}} ϕ_{μ}(tvt^{−1}ts)f(vs)p^{−2λ(s)(ρ)}dv ds

= ∫_{C} ∫_{T_{ℚp}} ϕ_{μ}(ts)f(vs)p^{−2λ(s)(ρ)}dv ds .

Set ϕ̃_{μ}(t) = ϕ_{μ}(t)p^{−λ(t)(ρ)}. Replacing the integral over C by an integral over V_{ℚ_{p}} we see that

χ_{μ}(f)ϕ̃_{μ}(t) = ∫_{T_{ℚp}} ϕ̃_{μ}(ts)f̃(s) ds = ∫_{T} ϕ̃_{μ}(ts)f̃(s) ds

if f → f̃ is the Satake homomorphism of H_{p} into the group algebra H′_{p} of \overline{T} defined by

f̃(s) = p^{−λ(s)(ρ)} ∫_{V_{ℚp}} f(vs) dv .

If ψ_{1} and ψ_{2} are two functions on \overline{T} and c > 0, we shall say that ψ_{1} ≃_{c} ψ_{2} if ψ_{1}(t) = ψ_{2}(t) whenever |ξ_{α}(t)| ≤ c for all positive roots α. The set of equivalence classes forms a vector space W_{c}. Let W_{∞} be the injective limit of the spaces W_{c}. If f lies in H′_{p} and ψ is a function on \overline{T}, then we define λ(f)ψ by

λ(f)ψ(t) = ∫_{T} ψ(ts)f(s) ds .

We can also regard λ(f) as an operator on W_{∞}. If ϕ_{μ} is the image of ϕ̃_{μ} in W_{∞}, then λ(f̃)ϕ_{μ} = χ_{μ}(f)ϕ_{μ} for all f in H_{p}. Since H′_{p} is a finite module over the image of H_{p}, the space

W = {λ(f)Φ_{μ} | f ∈ H′_{p}}

is a finite-dimensional subspace of W_{∞}.

Choose t_{1}, …, t_{n} such that

|ξ_{α_{i}}(t_{j})| = p^{−δ_{ij}} .

The points λ(t_{1}), …, λ(t_{n}) are a basis of ^{c}L over Z. Let δ_{i} be the characteristic function of {t̄_{i}} and let S_{i} and N_{i} be respectively the semisimple and nilpotent parts of the restriction of λ(δ_{i}) to W. The matrices S_{i}, N_{i}, where 1 ≤ i ≤ n, all commute. Choose a basis Ψ_{1}, …, Ψ_{1} of W with respect to which S_{1}, …, S_{n} are in diagonal form. Let

S_{i}ψ_{j} = p^{γ_{ij}}ψ_{j} .

If a_{i} is the smallest integer satisfying N_{i}^{a_{i}+1} = 0, then

(S_{i} + N_{i})^{l_{i}} = ∑_{r_{i}=0}^{a_{i}} ( \begin{matrix}l_{i} \\ r_{i} \end{matrix} ) S_{i}^{l_{i}−r_{i}}N_{i}^{r_{i}} .

Let ψ_{1}, …, ψ_{l} be representatives of Ψ_{1}, …, Ψ_{l}, and choose c_{0} > 0 such that

ψ_{i}(tt_{k}) = ∑_{j} (s_{k}^{ji} + n_{k}^{ji})ψ_{j}(t)

if |ξ_{α}(t)| ≤ c_{0} for α positive. Choose t_{0} such that |ξ_{α}(t_{0})| ≤ c_{0} if α is positive. If

Ω_{0} = (ψ_{1}(t_{0}), …, ψ_{l}(t_{0})), Ω(t) = (ψ_{l}(t), …, ψ_{l}(t)) ,

and l_{1} ≥ 0, …, l_{n} ≥ 0 then

Ω (t_{0} ∏_{k=1}^{n} t_{k}^{l_{k}}) = Ω_{0} ∏_{k=1}^{n}(S_{k} + N_{k})^{l_{k}}

= Ω_{0}\{ ∑_{r_{1}=0}^{a_{1}} ⋯ ∑_{r_{n}=0}^{a_{r}} ( \begin{matrix}l_{1} \\ r_{1} \end{matrix} ) ⋯ ( \begin{matrix}l_{n} \\ r_{n} \end{matrix} ) S_{1}^{−r_{1}} … S_{n}^{−r_{n}}N_{1}^{r_{1}} ⋯ N_{n}^{r_{n}}\} ∏_{k=1}^{n} S_{k}^{l_{k}}

= Θ(l_{1}, …, l_{n}) ∏_{k=1}^{n} S_{k}^{l_{k}}

where Θ(l_{1}, …, l_{n}) is a row vector with entries which are polynomials in l_{1}, …, l_{n}. Choose μ_{j} such that λ(t_{i})(μ_{j}) = γ_{ij}, for 1 ≤ i ≤ n. If |ξ_{α}(t)| ≤ |ξ_{α}(t_{0})| for α positive, then

ψ_{j}(t) = p^{λ(t)(μ_{j})}θ_{j}(λ(t))

where θ_{j}(λ(t)) is a polynomial in λ(t). Thus there is a constant c_{1} and polynomials ξ_{1}(λ(t)), …, ξ_{1}(λ(t)) such that

ϕ̃_{μ}(t) ≃_{c_{1}} ∑_{j=1}^{1} p^{λ(t)(μ_{j})}ξ_{j}(λ(t)) .

For our purposes it will be necessary to know the relation between μ_{1}, …, μ_{1} and μ and to have a more or less explicit formula for the polynomials ξ_{j}(λ(t)). Let ^{c}Ω be the Weyl group of ^{c}T. For notational convenience let us use the map σ → ^{c}σ to identify ^{c}Ω and Ω, the Weyl group of T. Let

A = {μ ∈ ^{c}h_{ℝ} | σμ = tμ, σ, t ∈ Ω implies σ = t} .

Let t̄_{i} be the image of t_{i} in \overline{T} and let

t̄_{i}^{(1)}, …, t̄_{i}^{(b_{i})}

be the distinct elements in the orbit of t̄_{i} under Ω. Set

B = {μ ∈ ^{c}𝔥_{ℝ} | λ(t̄_{i}^{(k)})(μ) = λ(t̄_{j}^{(1)})(μ) implies i = j and k = 1} .

Define μ_{k_{1}}, …, k_{n} by the condition that

λ(t̄_{i}^{(k_{j})})(μ) = λ(t̄_{i})(μ_{k_{1}}, …, k_{n}) 1 ≤ i ≤ n

and set

C = {μ ∈ ^{c}𝔥_{ℝ} | σμ = μ_{k_{1}....k_{n}} implies σt̄_{i}^{(k_{i})} = t̄_{i}, 1 ≤ i ≤ n} .

The complements of A, B, and C are the union of a finite number of proper affine subspaces of ^{c}𝔥_{ℝ}. Thus there is a point μ^{0} in A ∩ B ∩ C. Choose t_{0} such that

(i) λ(t_{0})(σμ^{0}) = (t_{0})(τμ^{0}) implies σ = τ; (ii) λ(t_{0})(^{c}σμ^{0}) = (t_{0})(μ_{k_{1},...,k_{n}}^{0}) implies σt̄_{i}^{(k_{i})} = t̄_{i}, 1 ≤ i ≤ n.

Let S be the collection of points μ in ^{c}𝔥_{ℂ} satisfying (i) p^{λ(t_{0})(σμ)} = p^{λ(t_{0})(τμ)} implies σ = τ; (ii) p^{λ(t_{0})(σμ)} = p^{λ(t_{0})(μ_{k_{1},...,kn})} implies σt̄_{i}^{(k_{i})} = t̄_{i}, 1 ≤ i ≤ n;

(iii) p^{λ(t̄_{i}^{(k)})(μ)} = p^{λ(t̄_{j}^{(l)})(μ)} implies i = j and k = l. Then S is an open, dense, and connected subset of ^{c}𝔥_{ℂ}.

Suppose μ lies in S. Since the coefficients of the polynomial

p_{i}(X) = ∏_{j=1}^{b_{j}} (X − δ(t̄_{i}^{(j)}))

lie in the image of H_{p}, the equation

λ(p_{i}(X))Φ_{μ} = ∏_{j=1}^{b_{j}} (X − p^{λ(t_{i}^{(j)})(μ)})Φ_{μ}

is satisfied. It is satisfied not only by Φ_{μ} but by every element of W. Since p_{i}(δ(t̄_{i})) = 0, the minimal polynomial of the restriction of λ(δ(t̄_{i})) to W divides

∏_{j=1}^{b_{j}} (X − p^{λ(t̄_{i}^{(j)})(μ)}) .

Since this polynomial has no multiple root, N_{i} = 0 and

λ(δ_{t})Ψ_{j} = p^{λ(t)(μ_{j})}Ψ_{j}

for all t. Here δ_{t} is the characteristic function of {t̄}. The point μ_{j} must be equivalent modulo 2πiL/ log p to an element in the orbit, under Ω, of μ. If not, there is a t such that

p^{λ(t)(σμ)} ≠ p^{λ(t)(μ_{j})}

for any σ in Ω. This is impossible because the minimal polynomial of the restriction of λ(δ_{t}) is divisible by

X − p^{λ(t)(μ_{j})}

and divides

∏_{σ} (X − p^{λ(t)(σμ)}) .

Thus there is a constant c_{μ} and constants a_{σ}(μ) such that

ϕ̃_{μ}(t) ≃_{c_{μ}} ∑_{σ} a_{σ}(μ)p^{λ(t)(σμ)}.

We need to prove also that the constant c_{μ} can be chosen to be independent of μ. Since the coefficients of p_{i}(X) are independent of μ, there is a constant c_{1} such that

λ(p_{i}(X))ϕ̃_{μ} ≃_{c_{1}} ∏_{j=1}^{b_{i}} (X − p^{λ(t̄_{i}^{(j)})(μ)})ϕ̃_{μ}

for all μ. If

ϕ̃_{μ}(^{•} , k_{1}, …, k_{n}) = ∏_{i=1}^{n} \{ ∏_{{j_{i}≠k_{i} \atop 1≤j_{i}≤b_{i}}} (\frac{λ(δ_{i}) − p^{λ(t̄_{i}^{(j_{i})})(μ)}}{p^{λ(t_{i}^{(k_{i})})(μ)} − p^{λ(t_{i}^{(j_{i})})(μ)}} ) \} ϕ̃_{μ}

then

ϕ̃_{μ} = ∑_{k_{1}=1}^{b_{1}} ⋯ ∑_{k_{n}=1}^{b_{n}} ϕ̃_{μ}(^{•} , k_{1}, …, k_{n})

and, for some constant c_{2} which does not depend on μ,

λ(δ_{i})ϕ̃_{μ}(^{•} , k_{1}, …, k_{n}) ≃_{c_{2}} p^{λ(t_{i})(μ_{k_{1},...,kn)}} ϕ̃_{μ}, 1 ≤ i ≤ n.

If

p_{0}(X) = ∏_{σ} (X − p^{λ(t_{0})(σμ)})

there is a constant c_{3} such that

p_{0}(λ(δ_{t_{0}}))ϕ̃_{μ}(^{•} , k, …, k_{n}) ≃_{c_{3}} 0

and

p_{0}(λ(δ_{t_{0}}))ϕ̃_{μ}(^{•} , k_{1}, …, k_{n}) ≃_{c_{3}} p_{0}(p^{λ(t_{0})(μ_{k_{1},...,kn})})ϕ̃_{μ}(^{•} , k_{1}, …, k_{n}) .

Since μ lies in S.

ϕ̃_{μ}(^{•} , k_{1}, …, k_{n}) ≃_{c_{3}} 0

unless σt̄_{i}^{k_{i}} = t̄_{i}, 1 ≤ i ≤ n, for some σ in Ω. Then μ_{k_{1},...,k_{n}} = σμ. For a given σ, let k_{1}(σ), …, k_{n}(σ) be the indices satisfying σt̄_{i}^{(k_{i}(σ))} = t̄_{i} and set

ϕ̃_{μ}(^{•} , σ) = ϕ̃_{μ}(^{•} , k_{1}(σ), …, k_{n}(σ)).

There is a constant c_{4} such that

ϕ̃_{μ} ≃_{c_{4}} ∑_{σ} ϕ̃_{μ}(^{•} , σ)

and

ϕ̃_{μ}(·, σ) ≃_{c_{4}} a_{σ}(μ)p^{λ(t)(σμ)}

for all μ in S.

The next step is to evaluate the coefficients a_{σ}(μ). Since ϕ_{σμ} = ϕ_{μ}, the same is true of ϕ̃_{μ}. If μ lies in ∩_{σ}σS which is an open, dense, and connected set, then

∑_{τ} a_{τ} (μ)p^{λ(t)(μ)} ≃_{c_{4}} ∑_{τ} a_{τ} (σμ)p^{λ(t)(τσμ)}.

As a consequence a_{τσ}(μ) = a_{τ} (σμ). Thus it is enough to evaluate a(μ) = a_{σ_{0}}(μ) if σ_{0} is the element of the Weyl group which takes every positive root to a negative root. Since a(μ) is an analytic function, it is enough to evaluate it when Re μ lies in the interior of the positive Weyl chamber.

Suppose λ(t) lies in the interior of the positive Weyl chamber. Since

ϕ_{μ}(t^{n}) = ϕ_{μ}((σ_{0}t)^{n}) = ϕ̃_{μ}((σ_{0}t)^{n})p^{−nλ(t)(ρ)}

the relation

ϕ_{μ}(t^{n}) = ∑_{σ} a_{σ}(μ)p^{nλ(t)(σ_{0}^{−1}σμ−ρ)}

is valid for sufficiently large n and

lim_{n→∞} p^{−nλ(t)(μ−ρ)}ϕ_{μ}(t^{n}) = a_{σ_{0}}(μ) = a(μ)

because Re λ(t)(μ − σμ) > 0.

We shall evaluate this limit in another way and obtain a(μ). Recall that

ϕ_{μ}(t) = ∫_{G_{ℤp}} ψ_{μ}(kt) dk .

Following Harish-Chandra we study this integral by means of the following easily proved lemma.

Lemma. Suppose \overline{V} is the unipotent radical of the parabolic group opposed to B. If v̄ lies in \overline{V}_{ℚ_{p}}, let v̄ = v(v̄)t(v̄)k(v̄) with v(v̄) in V_{ℚ_{p}}, t(v̄) in T_{ℚ_{p}}, and k(v̄) in G_{ℤ_{p}}. Set λ(v̄) = λ(t(v̄)). There is a constant a such that if ψ is any integrable function on B_{ℤ_{p}}\G_{ℤ_{p}} then

a ∫_{G_{ℤp}} ψ(k) dk = ∫_{\overline{V}_{ℚp}} ψ(k(v̄))p^{λ(v̄)(2ρ)} dv̄ .

I ask you to bear in mind for a while that a must necessarily equal

∫_{\overline{V}_{ℚp}} p^{λ(v̄)(2ρ)} dv̄.

Using the lemma we see that

ϕ_{μ}(t) = \frac{1}{a} ∫_{\overline{V}_{ℚp}} ψ_{μ}(t^{−1}(v̄)v̄t)p^{λ(v̄)(2ρ)} dv̄

= \frac{1}{a} ∫_{\overline{V}_{ℚp}} ψ_{μ}(v̄t)p^{−λ(v̄)(μ−ρ)} dv̄

= \frac{p^{λ(t)(μ+ρ)}}{a} ∫_{V_{ℚp}} ψ_{μ}(t^{−1}v̄t)p^{−λ(v̄)(μ−ρ)} dv̄

= \frac{p^{λ(t)(μ−ρ)}}{a} ∫_{\overline{V}_{ℚp}} ψ_{μ}(v̄)p^{−λ(tv̄t^{−1})(μ−ρ)} dv̄

= \frac{p^{λ(t)(μ−ρ)}}{a} ∫_{V_{ℚp}} p^{λ(v̄)(μ+ρ)}p^{−λ(tv̄t^{−1})(μ−ρ)} dv̄ .

Thus

(3)  a(μ) = \frac{1}{a} lim_{n→∞} ∫_{\overline{V}_{ℚp}} p^{λ(v̄)(μ+ρ)}p^{−λ(t^{n}v̄t^{−n})(μ−ρ)} dv̄

if t lies in the interior of the positive Weyl chamber.

Lemma. If ν lies in the positive Weyl chamber, if λ(t) does also, and if v̄ lies in \overline{V}_{ℚ_{p}} then λ(v̄)(ν) ≤ 0 and λ(v̄)(ν) ≤ λ(tv̄t^{−1})(ν).

If g lies in G_{ℚ_{p}} and g = vsk, with v in T_{ℚ_{p}}, s in T_{ℚ_{p}}, and k in G_{ℤ_{p}} set λ(g) = λ(s). It is known that if t satisfies the condition of the lemma, then

λ(kt)(ν) ≤ λ(t)(ν) .

Since

λ(gt) = λ(g) + λ(kt) ,

we have

λ(g)(ν) + λ(t)(ν) ≥ λ(gt)(ν) .

Moreover

λ(t^{−1}gt)(ν) = −λ(t)(ν) + λ(gt)(ν) ≤ λ(g)(ν) .

The second assertion of the lemma follows. If v̄ lies in \overline{V}_{ℚ_{p}}, there is a t with λ(t) in the positive Weyl chamber such that tv̄t^{−1} lies in G_{ℤ_{p}}. Since λ(tvt^{−1}) is then zero, the first assertion follows from the second.

If R is any open half-space in ^{c}𝔥_{ℝ} which is bounded by a hyperplane passing through zero and if R̄ is its closure, let ∑_{R} be the set of roots lying in R, let ∑_{R}^{+} be the set of positive roots lying in R̄, and let ∑_{R}^{−} be the set of negative roots lying in R. Let n̄(R) be the Lie algebra spanned by the root vectors corresponding to roots in ∑_{R}^{−} and let \overline{N}(R) be the group with Lie algebra n̄(R). If p is any prime, finite or infinite, we consider

∫_{\overline{N}_{ℚp} (R)} ψ_{μ}(n̄) dn̄ = δ_{R}(μ) .

It has been shown by Gindikin and Karpelevich that, when p = ∞, this integral converges if Re μ(H_{α}) > 0 for every positive root α and is equal to

∏ ; −α∈∑_{R}^{−} \frac{π^{1/2} Γ (\frac{μ(H_{α})}{2})}{Γ (\frac{1+μ(H_{α})}{2})}

If X_{α} are the root vectors belonging to the Chevalley basis, the Haar measure is that associated to the form which takes the value 1 on

⋀ ; α∈∑_{R}^{−} X_{α} .

We shall imitate their proof and show that the integral converges in the same region when p is finite and is equal to

(4)  ∏ ; −α∈∑_{R}^{−} \frac{1 − \frac{1}{p^{μ(H_{α})+1}}}{1 − \frac{1}{p^{μ(H_{α})}}} .

For the moment we shall assume this and complete our evaluation of the limit (3). Choose ϵ > 0 such that Re (μ) − ϵρ lies in the positive Weyl chamber. Then

Re {λ(v̄)(μ + ρ) − λ(t^{n}v̄t^{−n})(μ − ρ)}

is the sum of

Re {(λ(v̄) − λ(t^{n}v̄t^{−n}))(μ − ϵρ) + λ(t^{n}v̄t^{−n})(ρ − ϵρ)}

and

Re {λ(v̄)(ρ + ϵρ)} .

It follows from the lemma that the first expression is less than or equal to zero. Since

∫_{\overline{V}_{ℚp}} p^{Re \{λ(v̄)(ρ+ϵρ)\}} dv̄

is finite, we can take the limit under the integral sign in (3) to obtain

\begin{aligned}a(μ) & = \frac{1}{a} ∫_{\overline{V}_{ℚp}} p^{λ(v̄)(μ+ρ)} dv̄ \\ & = \frac{1}{a} ∏_{α>0} \frac{1 − \frac{1}{p^{μ(H_{α})+1}}}{1 − \frac{1}{p^{μ(H_{α})}}} \end{aligned}

Thus there is a constant c such that if |ξ_{α}(t)| ≥ c for α positive then

(5)  ϕ_{μ}(t) = \frac{1}{a} ∑_{σ} \{ ∏_{α>0} \frac{1 − \frac{1}{p^{σμ(H_{α})+1}}}{1 − \frac{1}{p^{σμ(H_{α})}}} \} p^{λ(t)(σμ−ρ)} .

Let

∏_{α>0} (1 − \frac{1}{p^{ν(H_{α})+1}} ) = ∑_{s̄∈\overline{T}} bs̄p^{λ(s̄)(ν)} .

Only a finite number of the coefficients are not zero. If ^{c}α is the root of ^{c}𝔥 corresponding to α the formula (5) may be written

(6)  ϕ_{μ}(t) = \frac{1}{a} ∑_{S̄} bs̄\{\frac{∑_{σ} sgn σ p^{(λ(t̄s̄)+^{c}ρ)(σμ)}}{∏_{_{c}α>0} (p^{\frac{^{c}α(μ)}{2}} − p^{− \frac{^{c}α(μ)}{2}} )}\}p^{−λ(t)ρ}.

This formula is valid for all μ. The relation of this formula to the Weyl character formula need not be pointed out to the knowledgeable reader.

I do not know if it is valid for all t. However it is valid for t = 1. To prove this we show that the right side is 1 when t = 1. First of all, it follows from the formulae for δ_{R}(μ) that

a = ∏_{α>0} \frac{1 − \frac{1}{p^{ρ(H_{α})+1}}}{1 − \frac{1}{p^{ρ(H_{α})}}} = ∏_{^{c}α>0} \frac{1 − \frac{1}{p^{^{c}α(ρ)+1}}}{1 − \frac{1}{p^{^{c}α(ρ)}}} .

Now bs̄ is zero unless

λ(s̄) = ∑_{^{c}α∈w} ^{c}α

where w is a subset of the positive roots of ^{c}𝔥. Then ^{c}ρ + λ(s̄) is either singular or in the orbit of ^{c}ρ under cΩ. To prove this^{∗} we recall that Kostant has shown in lemma 5.9 of his paper on the Borel-Weil theorem that every element in the orbit of ^{c}ρ + λ(s̄) is of the form ^{c}ρ + λ(s̄′) with

λ(s̄′) = − ∑_{^{c}α∈w′}^{c} α

and suppose that ^{c}ρ + λ(s̄) lies in the positive Weyl chamber. If it is nonsingular, it equals ^{c}ρ + λ with λ in the positive Weyl chamber. Then λ = λ(s̄). It follows immediately that λ = λ(s̄) = 0. If bs̄ is not zero, the corresponding term in brackets on the right side of (5) is zero when t = 1 and λ(s̄) +^{c}ρ is singular and is ±1 when t = 1 and λ(s) +^{c}ρ is in the orbit of ^{c}ρ. In any case if, for brevity, we denote the right side of (6) by Θ_{μ}(t), then Θ_{μ}(1) is independent of μ. Thus

Θ_{μ}(1) = Θ_{ρ}(1) = \frac{1}{a} ∑_{σ} \{ ∏_{α>0} \frac{1 − \frac{1}{p^{σρ(H_{α})+1}}}{1 − \frac{1}{p^{σρ(H_{α})}}} \}.

Suppose σ ≠ 1. Then, for some simple root α_{0}, σα_{0} = −β_{0} is negative and

σρ(H_{β_{0}}) = ρ(H_{α_{0}}) = −1

and the corresponding term in the above sum is zero. Thus

Θ_{μ}(1) = \frac{1}{a}\{ ∏_{α>0} \frac{1 − \frac{1}{p^{ρ(H_{α})+1}}}{1 − \frac{1}{p^{ρ(H_{α})}}} \} = 1 .

∗ [Added 1970] I now notice that I made the matter unnecessarily complicated.

Since Θ_{μ}(t) is a linear combination of products of exponentials and polynomials in λ(t) it cannot vanish in an open cone without vanishing identically. This the last formula shows it cannot do.

We are now in a position to show that if ϕ_{μ} is bounded then

Re λ(μ) ≤ λ̄(ρ)

for all λ. We may suppose that Re μ lies in the positive Weyl chamber. Then

Re λ(μ) ≤ Re λ̄(μ) ,

and we need only consider λ lying in the positive Weyl chamber. It will be simpler to consider only λ lying in the interior of the positive Weil chamber. The assertion for points on the boundary can be obtained by a simple limiting argument. Then if Re (σμ) ≠ Re μ, for some σ in Ω, Re λ(σμ) < Re λ(μ).

Let w be the set of simple roots α for which Re ^{c}α(μ) = 0. Let ∑_{0}^{+}(w) be the set of all positive roots which are linear combinations of the elements of w, and let ∑^{+}(w) be the other positive roots. Let G_{1} be the subgroup of G corresponding to the Lie algebra generated by the root vectors associated to the elements of ∑^{+}(w) and their negatives and let Ω_{1} ⊆ Ω be the Weyl group of G_{1}. If σ belongs to Ω and Re σ(μ) = Re μ, then σ belongs to Ω_{1}. Set Θ′(t) equal to

\frac{1}{a} ∑_{σ∈Ω_{1}} \{ ∏ ; α∈∑^{+}(w) \frac{1 − \frac{1}{p^{σν(H_{α})+1}}}{1 − \frac{1}{p^{σν(H_{α})}}} \} \{ ∏ ; α∈∑_{0}^{+}(w) \frac{1 − \frac{1}{p^{σν(H_{α})+1}}}{1 − \frac{1}{p^{σν(H_{α})}}} \}p^{λ(t)(σν−ρ)} .

Since λ(t) = λ_{1} + λ_{2} with λ_{1} = λ(t_{1}) for some t_{1} in the adjoint group of G_{1} and λ_{2}(α) = 0 for α in w, we can write Θ′(t) as the product of

\frac{1}{a}\{ ∏ ; α∈∑^{+}(w) \frac{1 − \frac{1}{p^{ν(H_{α})+1}}}{1 − \frac{1}{p^{ν(H_{α})}}} \}

and

\{ ∑_{σ∈Ω_{1}} ( ∏ ; α∈∑_{0}^{+}(w) \frac{1 − \frac{1}{p^{σν(H_{α})+1}}}{1 − \frac{1}{p^{σν(H_{α})}}} )p^{λ_{1}(σν−ρ)}\}p^{λ_{2}(ν−ρ)} .

Applying the previous discussion to G_{1} instead of G, we see that Θ′_{ν} is analytic at μ. Moreover Θ′_{ν}(t) does not vanish, as a function of t, for λ(t) in an open cone. Set Θ′′_{ν}(t) = Θ_{ν}(t) − Θ′_{ν}(t); Θ′′_{ν}(t) is also analytic at μ. As a function of t, Θ′′_{μ}(t) is a linear combination of terms of the form p(λ(t))p^{λ(t)(μ′−ρ)} where μ′ is an element in the orbit of μ with Re μ′ ≠ Re μ and p(λ(t)) is a polynomial in λ(t). Thus, if λ(t) lies in the interior of the positive Weyl chamber,

lim_{n→∞} p^{−nλ(t)(μ−ρ)}Θ′′_{μ}(t^{n}) = 0

and

lim_{n→∞} p^{−nλ(t)(μ−ρ)}ϕ_{μ}(t^{n}) = lim_{n→∞} p^{−nλ(t)(μ−ρ)}Θ′_{μ}(t^{n}) .

Suppose ϕ_{μ} were bounded and for some λ in the positive Weyl chamber Re λ(ρ − μ) were less than zero. Then there would exist a t such that Re λ(t)(ρ − μ) < 0 and Θ′_{μ}(t^{n}) did not vanish identically. Then

lim_{n→∞} p^{−nλ(t)(μ−ρ)}Θ′_{μ}(t^{n}) = 0 .

Spherical functions 19 On the other hand if t is fixed

p^{−nλ(t)(μ−ρ)}Θ′_{μ}(t^{n}) = ∑_{k=0}^{q} ϕ_{k}(n)n^{k}

where ϕ_{k}(m) is a linear combination of exponentials e^{iαm} with α real. We can suppose ϕ_{q}(n) ≢ 0. Certainly

lim_{n→∞} ϕ_{q}(n) = 0 .

Let

ϕ_{q}(n) = ∑_{j=1}^{r} a_{j}e^{iα_{j}n}

with α_{1}, …, α_{r} real and incongruent modulo 2π. Then

0 = lim_{N→∞} \frac{1}{N} ∑_{n=1}^{N} ϕ_{q}(n)e^{−iα_{j}n} = a_{j} .

This is a contradiction.

4. THE FORMULA OF GINDIKIN AND KARPELEVICH

To complete the proof of the lemma and to prepare ourselves for the next stage of the argument, we must evaluate the functions δ_{R}(μ) in closed form. The argument is an induction on the number of elements in ∑_{R}^{−}. Since δ_{R}(μ) is certainly 1 when ∑_{R}^{−} is empty, we can start immediately with the induction step. Let C_{R}^{−} and C_{R}^{+} be the convex cones with vertex at the origin generated by ∑_{R}^{−} and ∑_{R}^{+} respectively. Let

D_{R}^{+} = {λ ∈ 𝔥_{R} | λ(μ) ≥ 0 for all μ ∈ C_{R}^{+}},
D_{R}^{−} = {λ ∈ 𝔥_{R} | λ(μ) ≥ 0 for all μ ∈ C_{R}^{−}}.

If, as before, ^{c}ρ = \frac{1}{2} ∑_{α>0} ^{c}α, then ^{c}ρ lies in the interior of D_{R}^{+} and, if ∑_{R}^{−} is not empty, in the exterior of D_{R}^{−}. If

R = {μ | λ_{0}(μ) ≥ 0},

then λ_{0} lies in the intersection of D_{R}^{+} with the interior of D_{R}^{−}. Joining ^{c}ρ to λ_{0}, we pass through a point of the boundary of D_{R}^{−} which lies in the interior of D_{R}^{+}. Since D_{R}^{−} is polygonal, there is a point λ_{1} near this boundary point which lies inside an n − 1 dimensional side of D_{R}^{−} and in the interior of D_{R}^{+}. Then ∑_{R}^{−} is the set of all negative roots satisfying λ_{1}(α) ≥ 0. There is exactly one negative root −α_{0} such that λ_{1}(−α_{0}) = 0. Let

S = {μ | λ_{1}(μ) > 0}.

Then ∑_{R}^{−} is the union of −α_{0} and ∑_{S}^{−}.

To establish the formula (4) we show that

δ_{R}(μ) = \frac{1 − \frac{1}{p^{μ(H_{α_{0}} )+1}}}{1 − \frac{1}{p^{μ(H_{α_{0}} )}}} δ_{S}(μ).

We shall also see that the integral defining δ_{R}(μ) converges if that defining δ_{S}(μ) does and

Re μ(H_{α_{0}}) > 0.

Let \overline{N}^{0} be the one parameter group generated by the root vector X_{−α_{0}} belonging to −α_{0}. Let G^{0} be the group corresponding to the Lie algebra spanned by X_{α_{0}}, X_{−α_{0}}, and H_{α_{0}}. As usual there is a mapping of SL(2) into G^{0} such that the image of

( \begin{matrix}1 & 0 \\ x & 1 \end{matrix} )

is exp xX_{−α_{0}} and the image of

( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )

is exp xX_{α_{0}}. The image of SL(2, ℤ_{p}) is contained in G_{ℤ_{p}}. If n̄_{1} = exp xX_{−α_{0}}, let a_{1} be the identity if x lies in ℤ_{p} and let a_{1} be the image of

( \begin{matrix}x^{−1} & 0 \\ 0 & x \end{matrix} )

if x is not in ℤ_{p}. Let n_{1} be the identity if x is in ℤ_{p} and let n_{1} be the image of

( \begin{matrix}1 & x^{−1} \\ 0 & 1 \end{matrix} )

if x is not in ℤ_{p}. In all cases n̄_{1} lies in n_{1}a_{1}G_{ℤ_{p}}. Thus, if n̄_{2} lies in \overline{N}_{ℚ_{p}}(S),

ψ_{μ}(n̄_{2}n̄_{1}) = ψ_{μ}(n̄_{2}n_{1}a_{1}) = ψ_{μ}(a_{1})ψ_{μ}(a_{1}^{−1}n_{1}^{−1}n_{2}n_{1}a_{1}) .

Consequently

\begin{aligned}∫_{\overline{N}_{ℚp} (R)} ψ_{μ}(n̄) dn̄ & = ∫_{\overline{N}_{ℚp}^{0}} \{ ∫_{\overline{N}_{ℚp} (S)} ψ_{μ}(n̄_{2}n̄_{1}) dn̄_{2}\} dn̄_{1} \\ & = ∫_{\overline{N}_{ℚp}^{0}} ψ_{μ}(a_{1})\{ ∫_{\overline{N}_{ℚp} (S)} ψ_{μ}(a_{1}^{−1}n_{1}^{−1}n̄_{2}n_{1}a_{1})dn̄_{2}\} dn̄_{1} . \end{aligned}

Let

𝔫 = ∑_{λ_{1}(α)>0} Q_{p}X_{α},  𝔞 = ∑_{{λ_{1}(α)>0 \atop α>0}} Q_{p}X_{α},  𝔟 = ∑_{{λ_{1}(α)>0 \atop α<0}} Q_{p}X_{α} .

Here 𝔫 is the direct sum of 𝔞 and 𝔟. If Q is a closed half-space contained in S, let Θ_{Q} be the set of roots contained in Q. Let the distinct collections of roots obtained in this way be, in decreasing order, Θ_{0}, Θ_{1}, …, Θ_{l}, Θ_{l+1} = ϕ, and set

𝔫_{k} = ∑_{α∈Θ_{k}} Q_{p}X_{α} .

The relations [𝔫, 𝔫_{k}] ⊆ 𝔫_{k+1} and 𝔫_{k} = 𝔞 ∩ 𝔫_{k} + 𝔟 ∩ 𝔫_{k} are clear; in particular, 𝔫 is nilpotent. The following rather complicated lemma is an easy consequence of the Campbell-Hausdorff formula.

Lemma. Suppose 𝔫 is a Lie algebra of nilpotent transformations of a vector space V over a field k of characteristic zero and N is the associated group of linear transformations. Suppose

𝔫 = 𝔫_{0} ⫌ 𝔫_{1} ⫌ … ⫌ 𝔫_{l+1} = {0}

is a decreasing sequence of ideals in 𝔫 and [𝔫, 𝔫_{k}] ⊆ 𝔫_{k+1}. Suppose that 𝔞 and 𝔟 are two subspaces of 𝔫 and

𝔫_{k} = 𝔫_{k} ∩ 𝔞 ⊕ 𝔫_{k} ∩ 𝔟

for each k. Set 𝔞_{k} = 𝔫_{k} ∩ 𝔞, 𝔟_{k} = 𝔫_{k} ∩ 𝔟, and choose 𝔞̃_{i}, 𝔟̃_{i} such that

𝔞_{k} = ∑_{i=k}^{l} ⊕ 𝔞̃_{i} and 𝔟_{k} = ∑_{i=k}^{l} ⊕ 𝔟̃_{i} .

Then every element of N can be written uniquely as

n = exp X_{0} exp X_{1} … exp X_{1} exp Y_{1} … exp Y_{0}

with X_{i} in 𝔞̃_{i}, Y_{i} in 𝔟̃_{i}. If X → X^{a} is an automorphism of 𝔫 leaving each 𝔫_{k} invariant let X_{k} +Y_{k} → X′_{k} +Y′_{k} be the induced transformation on

𝔫_{k}/𝔫_{k+1} ≃ 𝔞̃_{k} ⊕ 𝔟̃_{k}.

If

n^{a} = exp X′′_{0} … exp X′′_{1} exp Y′′_{1} … exp Y′′_{0},

then

\begin{aligned}X′′_{k} & = X′_{k} + f(X_{0}, …, X_{k−1}, Y_{0}, …, Y_{k−1}), \\ Y′′_{k} & = Y′_{k} + g(X_{0}, …, X_{k−1}, Y_{0}, …, Y_{k−1}) \end{aligned}

with some polynomial functions f and g.

In the case of concern to us, both 𝔞 and 𝔟 are subalgebras of 𝔫. The group corresponding to 𝔟 is \overline{N}_{ℚ_{p}}(S). There is a subgroup N(S) of G such that the group corresponding to 𝔞 is N_{ℚ_{p}}(S). Moreover N(S) is contained in V the unipotent radical of B. As a particular consequence of the lemma

N = N_{ℚ_{p}}(S)\overline{N}_{ℚ_{p}}(S).

If n̄_{2} lies in \overline{N}_{ℚ_{p}}(S) then a_{1}^{−1}n_{1}^{−1} n̄_{2}n_{1}a_{1} lies in N. Project it onto \overline{N}_{ℚ_{p}}(S) to obtain n̄′_{2}. It is an easy consequence of the lemma that the map n̄_{2} → n̄′_{2} is a one-to-one mapping of \overline{N}_{ℚ_{p}}(S) onto itself and that

dn̄_{2} = ∏_{{λ_{1}(α)<0 \atop α>0}} |ξ_{α}(a_{1})|^{−1}dn̄′_{2}.

Now ψ_{μ}(a_{1}) = p^{λ(a_{1})(μ+p)} and

ρ − ∑_{{λ_{1}(α)<0 \atop α>0}} α = \frac{1}{2}(α_{0} + ∑_{{λ_{1}(α)>0 \atop α>0}} α − ∑_{{λ_{1}(α)<0 \atop α>0}} α) = \frac{1}{2}(α_{0} + ∑_{λ_{1}(α)>0} α).

Since

∑_{λ_{1}(α)>0} λ(a_{1})(α) = 0,

we have

∫_{\overline{N}_{ℚp} (R)} ψ_{μ}(n̄) dn̄ = \{ ∫_{\overline{N}_{ℚp}^{0}} p^{λ(a_{i})(μ+ \frac{α_{0}}{2} )} dn̄_{1}\}\{ ∫_{\overline{N}_{ℚp} (S)} ψ_{μ}(n̄_{2}) dn̄_{2}\}.

The first integral is equal to

∫_{ℤ_{p}} 1 + ∫_{{x∈ℚp \atop |x|>1}} |x|^{−μ(H_{α_{0}} )−1} dx,

which is

\begin{aligned}1 + (1 − \frac{1}{p}) ∑_{n=1}^{∞} \frac{p^{n}}{p^{n(μ(H_{α_{0}} )+1)}} & = 1 + (\frac{1 − \frac{1}{p}}{p^{μ(H_{α_{0}} )}} ) \frac{1}{1 − \frac{1}{p^{μ(H_{α_{0}} )}}} \\ & = \frac{1 − \frac{1}{p^{μ(H_{α_{0}} )+1}}}{1 − \frac{1}{p^{μ(H_{α_{0}} )}}} \end{aligned}

if Re μ(H_{α_{0}}) > 0.

5. A REVIEW OF EISENSTEIN SERIES

Let Δ be the set of simple roots of 𝔥. If α_{0} belongs to Δ, there is a parabolic group P = P (α_{0}) of rank one which contains B associated to α_{0}. Let N be the unipotent radical of P. Then P is the semidirect product of N and a reductive group M. It is convenient to suppose that M contains T. Let A be the center of M and let ^{0}G = A\M. There is a map from P to ^{0}G. Furthermore, ^{0}G is the adjoint group of a split Lie algebra of rank one less than G. Its Dynkin diagram is obtained by deleting α_{0} from the Dynkin diagram of G.

Lemma. Each of the maps

P_{ℚ_{p}} → ^{0}G_{ℚ_{p}}, P_{ℤ_{p}} → ^{0}G_{ℤ_{p}}, P_{𝔸} → ^{0}G_{𝔸}

is surjective.

It is enough to verify this for the first two maps. Let ^{0}T be the image of T in ^{0}G. Using the Bruhat decomposition one readily shows that the map P_{ℚ_{p}} → ^{0}G_{ℚ_{p}} is surjective if the map T_{ℚ_{p}} → ^{0}T_{ℚ_{p}} is surjective. If ^{0}t lies in ^{0}T_{ℚ_{p}} then ξ_{β}(^{0}t) is given for β in Δ − {α}. There is certainly a t in T_{ℚ_{p}} such that ξ_{β}(t) = ξ_{β}(^{0}t) for these β; t is mapped to ^{0}t.

Suppose u in ^{0}G_{ℤ_{p}} is the image of p = vtk with v in V_{ℚ_{p}}, t in T_{ℚ_{p}}, k in G_{ℤ_{p}}. For the purposes of the lemma we may suppose that k = 1. If α is a root of ^{0}G, then |ξ_{α}(t)| = 1. Let t_{0} be such that ξ_{α}(t_{0}) = |ξ_{α}(t)| for each root α; t_{0} must lie in the center of M. Replacing t by t_{0}t, we may suppose that t lies in G_{ℤ_{p}} or, even better, that t is 1 and p = v. If p is infinite, v must lie in N_{ℚ_{p}} and if p is finite v must be congruent modulo N_{ℚ_{p}} to an element of G_{ℤ_{p}}. (See C. Chevalley, Séminaire Bourbaki, Exposé 219.)

If p belongs to P let χ(p) be the determinant of the restriction of Ad p to the Lie algebra of N. Every element of G_{𝔸} is a product g = bu with b in B_{𝔸} and u in U; set ξ_{s}(g) = ξ_{s}(b) = ∏_{p} |χ(b_{p})|^{s+ \frac{1}{2}} . The product is taken over all primes including the infinite one. The function ξ_{s} is well defined and is a function on P_{ℚ}\G_{𝔸}. Let ϕ be one of the basis elements for the cusp forms on ^{0}G_{ℚ}\^{0}G_{𝔸}. Of course ϕ is supposed to be invariant on the right under ^{0}U. Also ϕ may be lifted to a function on N_{𝔸}P_{ℚ}\P_{𝔸}. If g = bu, set

F (g, s, ϕ) = ξ_{s}(g)ϕ(b) .

This function is well defined. The sum

E(g, s, ϕ) = ∑_{γ∈P_{ℚ}\G_{ℚ}} F (γg, s, ϕ)

is called an Eisenstein series. It converges absolutely for Re s > \frac{1}{2} but the function on the left is actually a meromorphic function of s for all g. By the way, if g belongs to G_{ℝ},

E(g, s, ϕ) = ∑_{γ∈P_{ℤ}\G_{ℤ}} F (γg, s, ϕ) .

Suppose P′ = P (α′_{0}) is another parabolic group of rank one, and N′ is its unipotent radical. Then

∫_{N′_{ℚ}\N′_{𝔸}} E(ng, s, ϕ) dn

is for each g a meromorphic function of s. If Re s > \frac{1}{2}, it is equal to

∫_{N′_{ℚ}\N′_{𝔸}} ∑_{P_{ℚ}\G_{ℚ}} F (γng, sϕ) dn ,

which equals

∫_{N′_{ℚ}\N′_{𝔸}} ∑_{N_{ℚ}\G_{ℚ}/N′_{ℚ}} ∑_{γ^{−1}P_{ℚ}γ∩N′_{ℚ}\N′_{ℚ}} F (γδng, s, ϕ) dn

or

∑_{P_{ℚ}\G_{ℚ}/N′_{ℚ}} ∫_{γ^{−1}P_{ℚ}γ∩N′_{ℚ}\N_{𝔸}} F (γng, s, ϕ) dn.

Because of the Bruhat decomposition we can suppose that each γ is of the form γ = wγ′ with w in the intersection of G_{ℤ} and the normalizer of T and γ′ in P′_{ℚ_{p}}. Then a typical term equals*

∫_{w^{−1} P_{ℚ}w∩N′_{ℚ}\N′_{𝔸}} F (wng, s, ϕ) dn.

There is an order on the roots of ^{0}G such that the positive roots are those of the form wα where α is a positive root of G. Multiplying w on the left by an element in the normalizer of T in G_{ℤ} ∩ M, we may suppose this is the order induced from the original order on the roots of 𝔥. Let ^{0}∑_{+} be the roots of ^{0}G of the form wα where α is a positive root of G which is not a root of ^{0}G′ and let ^{0}∑_{0} be the roots of ^{0}G of the form wα where α is a positive root of G which is a root of ^{0}G′. If α belongs to ^{0}∑_{+} and β belongs to 0∑_{+} or to ^{0}∑_{0} and α + β is a root, then α + β belongs to ^{0}∑_{+}; on the other hand, if α and β both belong to ^{0}∑_{0} and α + β is a root, then α + β belongs to ^{0}∑_{0}. As a consequence, the group N′′ whose Lie algebra is the span of {X_{α}|α ∈ ^{0}∑_{+}} is the unipotent radical of a parabolic subgroup of ^{0}G. Since w^{−1}N′′w is contained in N′ and

w^{−1}P_{ℚ}w ∩ w^{−1}N′′_{ℚ}w = w^{−1}N′′_{ℚ}w,

our integral equals

∫_{(w^{−1}P_{ℚ}w∩N′_{ℚ})w^{−1}N′′_{𝔸}w\N′_{𝔸}} { ∫_{N′′_{ℚ}\N′′_{𝔸}} F (n_{1}wng, s, ϕ)dn_{1}} dn .

If wng = bk with b in B_{A} and k in U, the inner integral equals

ξ_{s}(b) ∫_{N′′_{ℚ}\N′′_{𝔸}} ϕ(n_{1}b) dn_{1}

which is zero if N′′ ≠ {1} because ϕ is a cusp form. Thus the integral vanishes identically unless every positive root of ^{0}G is of the form wα where α is a root of ^{0}G′. Then, if α is a positive root of ^{0}G′, wα is a linear combination of roots of ^{0}G and thus a root of ^{0}G. As a consequence, wM′w^{−1} = M. For these terms we can take γ′ = 1. If P = P′, then w = 1 is one possibility and the resulting integral is F (g, s, ϕ), which is for each g an entire function of s. The only other possibility is that wP′w^{−1} is the parabolic group opposed to P. This is the case we are interested in. Then w^{−1}P_{Q}w ∩ N′_{Q} = {1} and

∫_{N′_{𝔸}} F (wng, s, ϕ) dn

is for each g a meromorphic function of s in the whole complex plane.

* [Added 1970] This statement is, I now notice, an oversimplification. There should be a factor in front which depends on γ and g should be replaced by γ′g.

We have demanded that w lie in G_{ℤ}. We can also demand that it lie in G_{ℤ_{∞}}; this will make it easier to evaluate the integral for then w lies in U and

F (wng, s, ϕ) = F (wngw^{−1}, s, ϕ) .

It is enough to evaluate the integral for g = m′ in M′_{𝔸}. Set m = wm′w^{−1}; m lies in M_{𝔸}. A simple change of variable shows that the integral equals

∏_{p} |χ(m_{p})|^{−1} ∫_{N′_{𝔸}} F (mwnw^{−1}, s, ϕ) dn .

The product is taken over all primes including the one at infinity. If \overline{N} is the unipotent radical of the group opposed to P, this may also be written as

(7)  ∏_{p} |χ(m_{p})|^{−1} ∫_{\overline{N}_{𝔸}} F (mn̄, s, ϕ) dn̄.

The map T → ^{0}T determines a map 𝔥_{ℝ} → ^{0}𝔥_{ℝ}. Since we have more or less consistently viewed ^{c}𝔥_{ℝ} and c(^{0}𝔥)_{ℝ} as the duals of 𝔥_{ℝ} and ^{0}𝔥_{ℝ}, we can agree that this determines a map ^{c}(^{0}𝔥)_{ℂ} into ^{c}𝔥_{ℂ}. If χ_{p} is for each p the homomorphism of the Hecke algebra into the complex numbers determined by ϕ, let ^{0}μ_{p} be one of the elements in ^{c}(^{0}𝔥)_{ℂ} associated to χ_{p}. Its image in ^{c}𝔥_{ℂ} will again be denoted by ^{0}μ_{p}. If ν is the sum of those roots whose root vectors belong to the Lie algebra of N, we set μ_{p}(s) = ^{0}μ_{p} + sν. Denote this set of roots by ∑. If

M(s) = \{ ∏ ; α∈∑ \frac{π^{1/2} Γ (\frac{μ_{∞}(s)(H_{α})}{2})}{Γ (\frac{μ_{∞}(s)(H_{α})+1}{2})}\} ∏_{p finite} \{ ∏ ; α∈∑ \frac{1 − \frac{1}{p^{μ_{p}(s)(H_{α})+1}}}{1 − \frac{1}{p^{μ_{p}(s)(H_{α})}}} \} ,

the integral in the expression (7) is equal to

M(s)F (m, s, ϕ) .

This is not too difficult to prove. Observe first that if S_{k} is the set consisting of the finite prime and the first k finite primes the integral in (7) is equal to

lim_{k→∞} \begin{matrix}∫ \\ ∏ & _{p∈S_{k}} & \overline{N} & ℚ_{p} \end{matrix}F (mn̄, s, ϕ) dn̄ .

So to prove our assertion all we need to do is show that, if h lies in ∏_{q≠p} G_{ℚ_{q}} ∩ G_{𝔸} and m lies in M_{ℚ_{p}} then

∫_{\overline{N}_{ℚp}} F (hmn̄, s, ϕ) dn̄ = \{ ∏ ; α∈∑ \frac{π^{1/2} Γ (\frac{μ_{p}(s)(H_{α})}{2})}{Γ (\frac{μ_{p}(s)(H_{α})+1}{2})}\}F (hm, s, ϕ)

if p is the infinite prime and

∫_{\overline{N}_{ℚp}} F (hmn̄, s, ϕ) dn̄ = \{ ∏ ; π∈∑ \frac{1 − \frac{1}{p^{μ_{p}(s)(H_{α})+1}}}{1 − \frac{1}{p^{μ_{p}(s)(H_{α})}}} \}F (hm, s, ϕ)

if p is a finite prime.

Fix a prime, finite or infinite; fix h in

∏_{q≠p} G_{ℚ_{q}} ∩ G_{𝔸}

and consider the function F (hg, s, ϕ), g in G_{ℚ_{p}}.

If h = bu, b in B_{𝔸}, u in U, if b̄ is the projection of b on M_{𝔸}, and if g = n(g)m(g)k(g) with n(g) in N_{ℚ_{p}}, m(g) in M_{ℚ_{p}}, and k(g) in G_{ℤ_{p}} then it equals

ξ_{s}(h)ξ_{s}(m(g))ϕ(b̄m(g)).

If m belongs to M_{ℚ_{p}}, set ψ(m) = ϕ(b̄m). If it is convenient, we can regard ψ(m) as a function on ^{0}G_{ℚ_{p}}. We are reduced to evaluating

(8)  ∫_{\overline{N}_{ℚp}} ξ_{s}(mm(n̄))ψ(mm(n̄)) dn̄

if ψ is a function on ^{0}G_{ℚ_{p}} invariant under right translations by elements of ^{0}G_{ℤ_{p}} which is an eigenfunction of the operators λ(f) for f in ^{0}H_{p} associated to the homomorphism χ_{p} of ^{0}H_{p} into ℂ determined by ^{0}μ_{p}. Of course we assume that the integral converges absolutely. Recall that

λ(f)ψ(g) = ∫_{^{0}G_{ℚp}} ψ(gh)f(h) dh

if g belongs to ^{0}G_{ℚ_{p}}.

Let M = M_{ℚ_{p}} and let

K = G_{ℤ_{p}} ∩ P_{ℚ_{p}}/G_{ℤ_{p}} ∩ N_{ℚ_{p}}.

Define a measure μ on M/K by setting μ(E) equal to the measure of

\{n̄ ∈ \overline{N}_{ℚ_{p}} | m(n̄)E\} .

Suppose k lies in K and n̄ in \overline{N} equals n(n̄)m(n̄)k(n̄); let k be the coset of k̄. Since

k̄n̄k̄^{−1} = (k̄n(n̄)k̄^{−1})(k̄m(n̄)k̄^{−1})(k̄k(n̄)k̄^{−1}) ,

the sets {n̄|m(n̄) ∈ kE} and {k̄n̄k̄^{−1}|m(n̄) ∈ E} are the same and μ is left-invariant under K. Define a measure on M, again called μ, which is invariant under left and right translations by elements of K by setting

μ(E) = ∫_{M/K} { ∫_{K} χ_{E}(mk) dk} dμ(m)

if χ_{E} is the characteristic function of E.

The integral (8) is equal to

∫_{M} ξ_{s}(mm_{1})ψ(mm_{1}) dμ(m_{1}) .

If F is a subset of ^{0}G_{ℚ_{p}} = A_{ℚ_{p}} M_{ℚ_{p}} and E is the inverse image of F in M_{ℚ_{p}}, set

ν_{s}(F) = ∫_{E} ξ_{s}(m_{1}) dμ(m_{1}) .

Since, as we observed earlier, K maps onto ^{0}G_{ℤ_{p}}, ν_{s} is invariant on the left and the right under ^{0}G_{ℤ_{p}}. The integral (8) equals

ξ_{s}(m) ∫_{^{0}G_{ℚp}} ψ(m̄h) dν_{s}(h)

if m̄ is the image of m in ^{0}G_{ℚ_{p}}.

Let F_{1} ⊆ F_{2} ⊆ … be an increasing sequence of compact sets (we assume that ^{0}G_{ℤ_{p}}F_{i} ^{0}G_{ℤ_{p}} = F_{i}) whose union exhausts ^{0}G_{ℚ_{p}} and define the measure ν_{s}^{n} by ν_{s}^{n}(F) = ν_{s}(F ∩ F_{n}). Since ν_{s}^{n} belongs to H_{p}

∫_{^{0}G_{ℚp}} ψ(m̄h) dν_{s}(h) = lim_{n→∞} ∫_{^{0}G_{ℚp}} ψ(m̄h) dν_{s}^{n}(h) = ψ(m̄) lim_{n→∞} χ_{p}(ν_{s}^{n})

and the integral (8) equals

ξ_{s}(m)ψ(m) lim_{n→∞} χ_{p}(ν_{s}^{n}) .

To evaluate the limit, take ψ to be the function ψ_{^{0}μ_{p}} of Section 3; then

ξ_{s}(m(g))ψ_{^{0}μ_{p}}(m(g)) = ψ_{μ_{p}(s)}(g)

and

lim_{n→∞} χ_{p}(ν_{s}^{n}) = ∫_{\overline{N}_{ℚp}} ψ_{μ_{p}(s)}(n̄) dn̄.

The integral on the right can be evaluated by the formula of Gindikin and Karpelevich if Re s is sufficiently large. Retracing our steps, we see that the integral in (7) is indeed equal to M(s)F (m, s, ϕ) and conclude that M(s) is a meromorphic function in the whole complex plane.

J. Tits pointed out a way of expressing M(s) which is more convenient for our purposes. We observed that there was a map of ^{c}(^{0}𝔥)_{ℂ} into ^{c}𝔥_{ℂ}. It is easy to see that it is induced by an imbedding of ^{c}(^{0}𝔤) in c𝔤. Since ^{c}(^{0}G) is simply connected, there is an associated map of ^{c}(^{0}G) into ^{c}G. Let ^{c}𝔫 be the Lie algebra spanned by the root vectors belonging to positive roots of ^{c}G which are not roots of ^{c}(^{0}G). These are the roots ^{c}α corresponding to roots in ∑. Let H_{c_{α}} be the copoid attached to c_{α} and set

H_{0} = ∑ ; α∈∑ H_{c_{α}} .

Let 𝔫_{1}, …, 𝔫_{r} be the eigenspaces of ad (H_{0}) in ^{c}𝔫. Let a_{i} be the eigenvalue of ad (H_{0}) corresponding to 𝔫_{i}. Each of the subspaces 𝔫_{i} is invariant under ^{c}(^{0}G); let π_{i} be the representation of ^{c}(^{0}G) on 𝔫_{i}. If π̃_{i} is the representation contragredient to π_{i}, then M(s) can be written as

∏_{i=1}^{r} \frac{ξ(a_{i}s, π̃_{i}, ϕ)}{ξ(a_{i}s + 1, π̃_{i}, ϕ)} .

6. EXAMPLES

If r is 1, then

M(s) = \frac{ξ(a_{1}s, π̃_{1}, ϕ)}{ξ(a_{1}s + 1, π̃_{1}, ϕ)}

is meromorphic in the whole plane and

ξ(s, π̃_{1}, ϕ) = M (\frac{s}{a_{1}} ) ξ(s + 1, π̃_{1}, ϕ) .

Since we already know that ξ(s, π̃_{1}, ϕ) is analytic in a half-plane, we can conclude that it is meromorphic in the whole plane.

If r = 2 and ξ(s, π̃_{1}, ϕ) is known to be meromorphic in the whole plane, the same argument shows that ξ(s, π̃_{2}, ϕ) is meromorphic in the whole plane. Thus every time we can adjoin a point to the Dynkin diagram of ^{0}G to obtain the Dynkin diagram of a group of rank 1 greater, we can expect to find a representation π of ^{c}(^{0}G) for which ξ(s, π, ϕ) is meromorphic in the whole plane. Before listing the possibilities, there is one further remark I should make.

If we define the function ξ′_{s} in the same manner as ξ_{s}, the expression (7) is easily seen to equal

M(s)ξ′_{−s}(m′)ϕ(vm′w^{−1}) = M(s)ξ′_{−s}(m′)ϕ′(m′) .

Recall that m = vm′w^{−1}. Of course ϕ′ is a function on A′_{𝔸} M′_{𝔸} and thus a function on ^{0}G′_{𝔸}. It satisfies the same conditions as ϕ. Associated to it is a function

M′(s) = ∏_{i=1}^{r′} \frac{ξ(a′_{i}s, π̃_{i}, ϕ′)}{ξ(a′_{i}s + 1, π′_{i}, ϕ′)} .

But m′ → wm′w^{−1} defines an isomorphism of M′ with M and an isomorphism of ^{0}G′ with ^{0}G. Thus ϕ and ϕ′ are essentially the same. Moreover ^{c}(^{0}G′) and ^{c}(^{0}G) are isomorphic, such that a representation of ^{c}(^{0}G′) may be regarded as a representation of ^{c}(^{0}G). Recalling that the elements of the adjoint group of ^{c}G are orthogonal with respect to the Killing form and that the Killing form turns Ad w(^{c}𝔫′) into the dual of ^{c}𝔫, one sees readily that r = r′, that, with a suitable order, a′_{i} = a_{i}, and that π′_{i} is the contragredient of π_{i}. Thus

M′(s) = ∏_{i=1}^{r} \frac{ξ(a_{i}s, π_{i}, ϕ)}{ξ(a_{i}s + 1, π_{i}, ϕ)} .

It is known that M(s)M′(−s) = 1. This is implied by, but does not imply, the relation ξ(s, π_{i}, ϕ) = ξ(1 − s, π̃_{i}, ϕ).

In the examples we shall give the Dynkin diagram of G with the points belonging to the Dynkin diagram of ^{0}G labeled. We give the number r, the numbers a_{i}, and the highest weight λ_{i} of the representations π_{i} as a linear combination of the fundamental weights δ_{j}. In the examples considered, π_{i} is always irreducible. Do not forget that π_{i} is not a representation of ^{0}G but a representation of ^{c}(^{0}G).

\begin{aligned}(i) & 1_{o}  1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{n−1}  1_{o} ; α_{n} \\ & r = 1  a_{1} = n + 2  λ_{1} = δ_{n} \end{aligned}

\begin{aligned}(ii) & 1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{n−1}  1_{o} ; α_{n}  1_{o} \\ & r = 1  a_{1} = n + 2  λ_{1} = δ_{1} \end{aligned}

\begin{aligned}(iii) & 1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{m−1}  1_{o}  1_{o} ; α_{m} … 1_{o} ; α_{n} \\ & r = 1  a_{1} = n + 2  λ_{1} = δ_{1} + δ_{n} \end{aligned}

\begin{aligned}(iv) & 2_{o} ; α_{1}  2_{o} ; α_{2} … 2_{o} ; α_{n−1}  2_{o} ; α_{n}  1_{o} \\ & r = 1  a_{1} = 2(n + 1)  λ_{1} = 2δ_{1} \end{aligned}

\begin{aligned}(v) & 1_{o}  2_{o} ; α_{1}  2_{o} ; α_{2} … 2_{o} ; α_{n−1}  2_{o} ; α_{n} \\ & r = 1  a_{1} = 2(n + 1)  λ_{1} = 2δ_{n} \end{aligned}

(vi)  1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{n−1}  1_{o} ; α_{n}  2_{o}
r = 2  a_{1} = 2(n + 2)  λ_{1} = δ_{2}
a_{2} = n + 2 λ_{2} = δ_{1}

(vii)  2_{o}  1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{n−1}  1_{o} ; α_{n}
r = 2  a_{1} = 2(n + 2)  λ_{1} = δ_{n−1}
a_{2} = n + 2 λ_{2} = δ_{n}

\begin{aligned}(viii) & 1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{n−2}  1_{o} ; α_{n−1} \\ & r = 1  a_{1} = 2n  λ_{1} = δ_{2} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 1_{o} 1_{o} ; α_{n}

\begin{aligned}(ix) & 1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{n−2}  1_{o} ; α_{n−1} \\ & r = 1  a_{1} = 2n  λ_{1} = δ_{1} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 1_{o} 1_{o} ; α_{n}

(x)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; |1_{o} ; α_{3}  1_{o} ; α_{4}  1_{o} ; α_{5}

(xi)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; α_{3}  1_{o} ; |1_{o} ; α_{4}  1_{o} ; α_{5}  1_{o} ; α_{6}
r = 2  a_{1} = 14  λ_{1} = δ_{3}

a_{2} = 28 λ_{2} = δ_{6}

(xii)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; |1_{o} ; α_{3}  1_{o} ; α_{4}  1_{o} ; α_{5}  1_{o} ; α_{6}
r = 2  a_{1} = 14  λ_{1} = δ_{4}

a_{2} = 28 λ_{2} = δ_{1}

(xiii)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; α_{3}  1_{o} ; α_{4}  1_{o} ; |1_{o} ; α_{5}  1_{o} ; α_{6}  1_{o} ; α_{7}
r = 3  a_{1} = 51  λ_{1} = δ_{1}

a_{2} = 34 λ_{2} = δ_{6} a_{3} = 17 λ_{3} = δ_{3}

(xiv)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; |_{o} ; α_{3}  1_{o} ; α_{4}  1_{o} ; α_{5}  1_{o} ; α_{6}  1_{o} ; α_{7}
r = 3  a_{1} = 51  λ_{1} = δ_{1}

a_{2} = 34 λ_{2} = δ_{2} a_{3} = 17 λ_{3} = δ_{5}

(xv)  3_{o} ; α_{1}  1_{o}
r = 2  a_{1} = 10  λ_{1} = 0

a_{2} = 5 λ_{2} = 3δ_{1}

This example is particularly striking.

(xvi)  3_{o}  1_{o} ; α_{1}
r = 3  a_{1} = 3  λ_{1} = δ_{1}

a_{2} = 6 λ_{2} = 0 a_{3} = 9 λ_{3} = δ_{1}

(xvii)  2_{o}  2_{o} ; α_{1} … 2_{o} ; α_{2} ⋯ 2_{o} ; α_{n−1}  1_{o} ; α_{n}
r = 2  a_{1} = 2n + 1  λ_{1} = δ_{1}
a_{2} = 4n + 2 λ_{2} = 0

(xviii)  1_{o}  1_{o} ; α_{3}  2_{o} ; α_{2}  2_{o} ; α_{1}
r = 2  a_{1} = 22  λ_{1} = 0
a_{2} = 11 λ_{2} = δ_{3}

\begin{aligned}(xix) & 1_{o}  1_{o} ; α_{2}  2_{o} ; α_{1} \\ & r = 1  a_{1} = 4  λ_{1} = δ_{2} \end{aligned}

\begin{aligned}(xx) & 1_{o}  1_{o} ; α_{1} ⋯ 1_{o} ; α_{n−2}  1_{o} ; α_{n−1}  2_{o} ; α_{n} \\ & r = 1  a_{1} = 2(n + 1)  λ_{1} = δ_{1} \end{aligned}

(xxi)  1_{o} ; α_{1}  2_{o} ; α_{2}  2_{o}
r = 2  a_{1} = 5  λ_{1} = δ_{2}
a_{2} = 10 λ_{2} = 0

(xxii)  1_{o} ; α_{1}  1_{o} ; α_{2}  2_{o} ; α_{3}  2_{o}
r = 2  a_{1} = 16  λ_{1} = δ_{1}
a_{2} = 8 λ_{2} = δ_{3}

\begin{aligned}(xxiii) & 1_{o} ; α_{1}  1_{o} ; α_{2} … 1_{o} ; α_{n−2}  1_{o} ; α_{n−2} \\ & r = 1  a_{1} = 2n  λ_{1} = δ_{1} \end{aligned}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.} 1_{o} ; α_{n} 1_{o} ; α_{n}

\begin{aligned}(xxiv) & 1_{o} ; α_{1}  1_{o} ; α_{2}  {1 \atop o α_{5}} ; |1_{o} ; α_{3}  1_{o} ; α_{4}  1_{o} \\ & r = 1  a_{1} = 12  λ_{1} = δ_{5} \end{aligned}

\begin{aligned}(xxv) & 1_{o} ; α_{1}  1_{o} ; α_{2}  {1 \atop o α_{4}} ; |1_{o} ; α_{3}  1_{o} ; α_{5}  1_{o} \\ & r = 1  a_{1} = 12  λ_{1} = δ_{4} \end{aligned}

(xxvi)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; α_{3}  {1 \atop o α_{6}} ; |1_{o} ; α_{4}  1_{o} ; α_{5}  1_{o}
r = 2  a_{1} = 34  λ_{1} = 0
a_{2} = 17 λ_{2} = δ_{5}

(xxvii)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; α_{3}  {1 \atop o α_{5}} ; |1_{o} ; α_{4}  1_{o} ; α_{6}  1_{o}
r = 2  a_{1} = 34  λ_{1} = 0
a_{2} = 17 λ_{2} = δ_{6}

(xxviii)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; α_{3}  1_{o} ; α_{4}  {1 \atop o α_{7}} ; |1_{o} ; α_{5}  1_{o} ; α_{6}  1_{o}
r = 2  a_{1} = 46  λ_{1} = δ_{1}
a_{2} = 23 λ_{2} = δ_{7}

(xxix)  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; α_{3}  1_{o} ; α_{4}  {1 \atop o α_{6}} ; |1_{o} ; α_{5}  1_{o} ; α_{7}  1_{o}
r = 2  a_{1} = 46  λ_{1} = δ_{1}
a_{2} = 23 λ_{2} = δ_{6}

\begin{aligned}(xxx) & 1_{o}  1_{o} ; α_{1}  1_{o} ; α_{2}  {1 \atop o α_{6}} ; |1_{o} ; α_{3}  1_{o} ; α_{4}  1_{o} ; α_{5} \\ & r = 1  a_{1} = 18  λ_{1} = δ_{5} \end{aligned}

\begin{aligned}(xxxi) & 1_{o}  1_{o} ; α_{5}  1_{o} ; α_{4}  {1 \atop o α_{6}} ; |1_{o} ; α_{3}  1_{o} ; α_{2}  1_{o} ; α_{1} \\ & r = 1  a_{1} = 18  λ_{1} = δ_{1} \end{aligned}

(xxxii)  1_{o}  1_{o} ; α_{1}  1_{o} ; α_{2}  1_{o} ; α_{3}  {1 \atop o α_{7}} ; |1_{o} ; α_{4}  1_{o} ; α_{5}  1_{o} ; α_{6}