Zeta functions of Jordan algebras representations.

Dehbia ACHAB.

Université Pierre et Marie Curie, Institut de mathématiques, Tour 46-00, 5^{ieme} etage , 4, place Jussieu 75252 Paris cedex 05.

0.Introduction. Riemann zeta function has been generalized by Epstein as follows : let 𝔖 be a symmetric positive matrix of order k, Epstein zeta function is defined by

ζ_{1}(𝔖, s) = ∑_{g∈Z^{k}−\{0\}} \frac{1}{(g′𝔖g)^{s}}, Re(s) > \frac{k}{2}

where g′ is the adjoint of g. In [15], Kœcher has generalized Epstein zeta function as follows

ζ_{m}(𝔖, s) = ∑_{𝔘∈[Z^{k×m}/GL(m,Z),rank(𝔘)=m]}Det(𝔘′𝔖𝔘)^{−s}, (k ≥ m).

Kœcher zeta series converges absolutly and is analytic in the half-plane Re(s) > \frac{k}{2} , it admits an analytic continuation as a meromorphic function on C and satisfies to the functional equation

R_{m}(𝔖, s) = |𝔖|^{− \frac{m}{2}} R_{m}(𝔖^{−1}, \frac{k}{2} − s)

where R_{m}(𝔖, s) is a product of ζ_{m}(𝔖, s) and some gamma factor, more precisely

R_{m}(𝔖, s) = π^{m( \frac{m−1}{4} −s)}Γ(s)Γ(s − \frac{1}{2}) … Γ(s − \frac{m−1}{2})ζ_{m}(𝔖, s)

Γ(s) being the usual Euler gamma function. Later, in [15], A.Krieg studied the Kœcher zeta function for the hermitian matrices with quaternionic coefficients defined by

ζ(s) = ∑_{A∈GL(m,O)\[M(m,k,O)|rank(A)=m]} Det(AĀ^{t})^{−s}

where O is the ring of Hurwitz integers.

This work is situated in a more general context. In fact, we define the Kœcher zeta series associated to a self-adjoint Euclidean Jordan algebra representation and we obtain the above zeta series as particular cases of it. More precisely, let V be a Euclidean simple Jordan algebra of dimension n and rank m, E a Euclidean space of dimension N, ϕ a regular self-adjoint representation of V in the space Sym(E) of symmetric morphisms of E. Let Q be the quadratic form associated to ϕ, Ω the symmetric cone associated to V and G(Ω) its automorphism group

G(Ω) = {g ∈ GL(V) | g(Ω) = Ω}.

(H_{1}) We assume that V and E have Q-structures V_{Q} and E_{Q} respectively and that ϕ is defined over Q.

Typeset by AMS-TEX

Let L be a lattice in E_{Q}.

We define the zeta series associated to ϕ and L by the following

ζ_{L}(s) = ∑_{l∈Γ_{∘}\L′}[det(Q(l))]^{−s}, ∀s ∈ C

where L′ = {l ∈ L | det(Q(l)) ≠ 0} and Γ_{∘} is some arithmetic subgroup of GL(E) which we will precise.

Recall that the primitive rank of a Jordan algebra is the cardinality of a maximal complete system of primitive orthogonal idempotents. A Jordan algebra is said to be split if its rank equals its primitive rank. (H_{2}) We assume that V_{Q} is split. The fundamental results in this work are :

Theorem 1. Under the assumptions (H_{1}) and (H_{2}), the zeta series converges absolutly for Re(s) > \frac{N}{2m}.

Theorem 2. If the arithmetic subgroup Γ_{∘} is self-adjoint, then the zeta function ζ_{L} admits an analytic continuation as a meromorphic function on the whole plane C and satisfies to the functional equation

ζ_{L}(\frac{N}{2m} − s) = vol(L)π^{\frac{N}{2} −2ms} \frac{Γ_{Ω}(s)}{Γ_{Ω}( \frac{N}{2m} −s)}ζ_{L^{⋆}}(s).

where Γ_{Ω}(s) is the Kœcher-Gindikin gamma function of the symmetric cone Ω and L^{⋆} is the dual lattice of L.

This article is composed of three parts ; the first consisting in the proof of the theorem 1 by using reduction theory, the second is an adaptation of the classical method to prove theorem 2 and the last one gives some examples. Part I. Construction and convergence of the zeta series. Let V be a simple Euclidean Jordan algebra with unity e, of dimension n and rank m, E a Euclidean space of dimension N, ϕ a representation of V in the space Sym(E) of self-adoint endomorphisms of E such that

∀x, y ∈ V, ϕ(xy) = \frac{1}{2}(ϕ(x)ϕ(y) + ϕ(y)ϕ(x)),

and Q : E → V the quadratic form associated to ϕ determined by

(Q(ξ) | x)_{V} = (ϕ(x)ξ | ξ)_{E} ∀x ∈ V, ∀ξ ∈ E.

For x ∈ V, we denote by L(x) the multiplication endomorphism, L(x) : V → V, y ↦ xy and by P (x) the quadratic representation of V, i.e P (x) = 2L(x)^{2} − L(x^{2}). Let Ω be the symmetric cone associated to V and G(Ω) the automorphism group of Ω,

G(Ω) = {g ∈ GL(V) | g(Ω) = Ω}.

In the sequel, we assume that ϕ is regular, that is, ∃ξ ∈ E such that det(Q(ξ)) ≠ 0; then Q(E) = Ω̄ where Ω̄ is the closure of Ω. We assume too that ϕ(e) = id_{E}.

(H_{1}) Assume that V, E and ϕ are defined over Q, that is there exists a Q-Jordan subalgebra V_{Q} of V, and a Q-subspace E_{Q} of E such that

V = V_{Q} ⊗_{Q} R, E = E_{Q} ⊗_{Q} R,

and for each x ∈ V_{Q} we have ϕ(x) ∈ Sym(E)_{Q}. Let L be a lattice in the space E_{Q}. In all the sequel, we denote by det the determinant in the Jordan algebra and by Det the usual determinant of matrices. 1.1. Arithmetic subgroups associated to ϕ and L. Let H = {(h̃, h) ∈ GL(V) × GL(E) | Q(h.ξ) = h̃.Q(ξ), ∀ξ ∈ E}. H is non empty because, for each invertible x ∈ V,

Q(ϕ(x)ξ) = P (x)Q(ξ).

It is clear that H is an algebraic subgroup of GL(V) × GL(E). As ϕ is defined over Q then it is the same for H. If π_{1} and π_{2} are the projections of H, then the groups π_{1}(H) and π_{2}(H) are algebraic, defined over Q, we denote them by G(ϕ) and F (ϕ) respectively. Notice that G(ϕ) ⊆ G(Ω) and that π_{2} is injective. Denote by F and G the identity connected components of F (ϕ) and G(Ω) for the ordinary topologies respectively. Consider the map

ρ :F (ϕ) → G(ϕ)
f ↦ f̃

ρ is well defined because π_{2} is injective and we have :

1.1.1 Proposition. ρ satisfies to the following :

(1) ρ(F) = G.

(2) ρ is a surjective Q-morphism of algebraic groups.

(3) The groups F (ϕ) and G(ϕ) are self-adjoint ρ(h^{⋆}) = ρ(h)^{⋆}. So they are reductive. Moreover |Detρ(h)| = |Det(h)|^{\frac{2n}{N}}.

Proof.

(1) Denote by 𝔣 and 𝔤 the Lie algebras of F and G respectively. It suffices to show that the differential dρ : 𝔣 → 𝔤 is surjective. We start by showing the following lemma :

1.1.2 Lemma. For each x ∈ V we have ϕ(x) ∈ 𝔣 and dρ(ϕ(x)) = 2L(x).

Proof of the lemma. For x ∈ V, ξ ∈ E, t ∈ R,

Q(exp(ϕ(tx)).ξ) = Q(ϕ(exp(tx)).ξ)
= P (exp(tx)).Q(ξ) = exp(2tL(x)).Q(ξ)

then ρ(exp(tϕ(x))) = exp(2tL(x)), and,

dρ(ϕ(x)) = \frac{d}{dt}|_{t=0}[ρ(exp(tϕ(x)))] = \frac{d}{dt}|_{t=0}[exp(2tL(x))] = 2L(x). □ As 𝔤 is generated by the L(x), x ∈ V, then the lemma shows that dρ is surjective.

(2) As ρ = π_{1} ∘ i_{2} where i_{2} is the injection F (ϕ) → H, f ↦ (f̃, f), then ρ is clearly a morphism of algebraic groups. Moreover, as

Q(fξ) = ρ(f)Q(ξ)∀ξ ∈ E ⇔ f^{⋆}ϕ(x)f = ϕ(ρ(f)^{⋆}x) ∀x ∈ V,

then, if (e_{i})_{1≤i≤n} is a basis of V_{Q}, and (ϵ_{α})_{1≤α≤N} a basis of E, we have

∑_{β,γ=1}^{N} f_{βα}ϕ(e_{i})_{βγ}f_{γδ} = ϕ(∑ ρ(f)_{ij}e_{i}) = ∑_{j=1}^{n} ρ(f)_{ij}ϕ(e_{i})_{αδ}.

The above formula shows that the coefficients of ρ(f) are polynomials of degree 2, with rational coefficients, in the coefficients of f.

(3) Let h ∈ F (ϕ). We know that Q(hξ) = ρ(h)Q(ξ) ∀ξ ∈ E ⇔ h^{⋆}ϕ(x)h = ϕ(ρ(h)^{⋆}x) ∀x ∈ V

then, for each invertible x ∈ V, we have

h^{−1}ϕ(x^{−1})h^{⋆}^{−1} = ϕ((ρ(h)^{⋆}x)^{−1}).

As (ρ(h)^{⋆}x)^{−1} = ρ(h)^{−1}x^{−1}, we find

h^{−1}ϕ(x^{−1})h^{⋆}^{−1} = ϕ(ρ(h)^{−1}x^{−1}) ∀x ∈ I(V)

where I(V) is the set of all invertible elements of V. It follows that

h^{−1}ϕ(x)h^{⋆}^{−1} = ϕ(ρ(h)^{−1}x) ∀x ∈ V, i.e ρ(h^{⋆})^{−1} = (ρ(h)^{⋆})^{−1}.

The last assertion is a direct consequence of the properties

Det(ϕ(x)) = det(x)^{\frac{N}{m}} and det(gx) = Det(g)^{\frac{m}{n}} det(x)

for x ∈ V and g ∈ G(Ω). □

Now consider the arithmetic subgroup Γ_{∘} of F (ϕ) defined by

Γ_{∘} = {f ∈ F (ϕ) | f(L) = L}.

As ρ is a surjective Q-morphism of algebraic groups, then Γ = ρ(Γ_{∘}) is an arithmetic subgroup of G(ϕ). Moreover,

∀γ ∈ Γ_{0}, det(Q(γ.ξ)) = det(Q(ξ)), ∀ξ ∈ E.

1.2 Reduction theory. The following hypothesis is essential to use in this context reduction theory and to obtain some Minkowski inequality. (H_{2}) In all the sequel, we assume that V_{Q} is a split Jordan algebra, that is, its primitive rank (which is the cardinality of maximal system of primitive orthogonal idempotents), equals its rank. As rank(V) = m, then the assumption (H_{2}) implies that there exists in V_{Q} a complete system of orthogonal primitive idempotents {c_{1}, …, c_{m}} which we will fix along this paper. The corresponding Peirce decomposition V = ⊕_{i≤j}V_{ij} is defined over Q, that is

V_{Q} = ⊕_{i≤j}V_{ij}_{Q}.

Let P be the subgroup of G(ϕ) defined by

P = {g ∈ G(ϕ) | (gx_{ij})_{ij} = λ_{ij}x_{ij} ∀i, j (gx_{ij})_{kl} = 0 ∀(k, l) < (i, j)} where the λ_{ij} are reals, and for each y in V, the y_{ij} are the Peirce components of y, with respect to the Jordan frame {c_{1}, …, c_{m}}. The order on the pairs (i, j) is the lexicographic one.

1.2.1 Proposition. P is a Borel subgroup of G(ϕ) defined over Q.

Proof. As the Peirce decomposition is defined over Q, then there exists some basis of V_{Q} whose each element lies in some V_{ij}_{Q}. An element of G(ϕ) lies in P iff its matrix in such a basis is upper triangular. □

Now consider the subgroup A of G defined by A = {P (a) | a = ∑_{i=1}^{m} a_{i}.c_{i}, a_{i} > 0 ∀i, 1 ≤ i ≤ m}.

1.2.2 Proposition. A is a maximal Q-split algebraic torus of P.

(cf. [15], chapter 2, proposition 3.5). We denote by N the unipotent radical of P,

N = {t ∈ P | λ_{ij} = 1 ∀i, j},

We know (cf.[15]), that

N = {n(z) | z ∈ ⊕_{j<k}V_{jk}}

where

n(z) = τ(z^{(1)}) … τ(z^{(m−1)})

z^{(j)} = ∑_{k=j+1}^{m} z_{jk}, τ(z^{(j)}) = exp(2z^{(j)}□c_{j})

the operation □ being defined by

x□y = L(xy) + [L(x), L(y)].

(cf [15], Chapter 6, Theorem 6.3.6).

Let K be the maximal compact subgroup of G(ϕ) defined by

K = {g ∈ G(ϕ) | g.e = e}

Where e is the unity of V. Then we have the Iwasawa decomposition of G(ϕ), G(ϕ) = N.A.K

1.2.3 Definition. A Siegel set of G(ϕ) (with respect to K, N, A) is a the cartesian product 𝔖_{t,u} = N_{u}.A_{t}.K with

A_{t} = {P (a) ∈ A | a_{i} ≤ ta_{i+1}, ∀ ≥ 1i ≤ m − 1, a = ∑_{i=1}^{m} a_{i}c_{i}}

N_{u} = {n(z) ∈ N | ∥z_{kj}∥ ≤ u},

where t, u are two positive constants.

1.2.4 Proposition. There exist positive constants t and u and some finite subset B of G(ϕ)_{Q} such that

G(ϕ) = Γ.B.𝔖_{t,u},

moreover, as Ω = G(ϕ)/K, then

Ω = ΓB.N_{u}.A_{t}.e.

Proof. It is a direct consequence of theorem 13.1 of [15], page 90, applicated to the Q-reductive group G(ϕ) under the action of the arithmetic subgroup Γ. □

1.2.5 Proposition. (Minkowski inequality) Let x = ∑_{i=1}^{m} x_{i}c_{i} + ∑_{i<j} x_{ij} be the Peirce decomposition of x ∈ V. For positive reals t, u, there exists a positive constant C_{t,u} such that, for each x ∈ 𝔖_{t,u}.e ,

∏_{i=1}^{m} x_{i} ≤ C_{t,u}.det(x).

Proof. Let x = n.P (a)(e) be an element of the Siegel set 𝔖_{t,u}.e of the symmetric cone Ω, i.e.

n = τ(z^{(1)}) … τ(z^{(m−1)})

with

z^{(j)} = ∑_{k=j+1}^{m} z_{jk}, ∥z_{jk}∥ ≤ u

and a = ∑_{i=1}^{m} a_{i}c_{i} such that

a_{i} ≤ ta_{i+1} ∀i, 1 ≤ i ≤ m − 1.

The Peirce componants of x are as follows

x_{j} = a_{j}^{2} + \frac{1}{2} ∑_{k=1}^{j−1} a_{k}^{2}∥z_{kj}∥^{2}

x_{jk} = a_{j}^{2}z_{jk} + 2 ∑_{l=1}^{j−1} a_{l}^{2}z_{lj}z_{lk}.

So we find the following inequality :

x_{j} ≤ a_{j}^{2} + \frac{1}{2}a_{j}^{2} ∑_{k=1}^{j−1} t^{2(j−k)}∥z_{kj}∥^{2}

≤ a_{j}^{2}(1 + \frac{1}{2} ∑_{k=1}^{j−1} t^{2(j−k)}∥z_{kj}∥^{2} ≤ a_{j}^{2}(1 + \frac{1}{2}u^{2} ∑_{k=1}^{j−1} t^{2(j−k)}).

Otherwise, as det(x) = ∏_{j=1}^{m} a_{j}^{2}, we find

∏_{j=1}^{m} x_{j} ≤ C_{t,u}det(x)

where

C_{t,u} = (1 + \frac{1}{2}u^{2} ∑_{k=1}^{m−1} t^{2(j−k)})^{m}. □

1.3 Convergence of the zeta series ζ_{L}. The zeta series associated to the representation ϕ and the lattice L is defined by

ζ_{L}(s) = ∑_{l∈Γ_{∘}\L′} det(Q(l))^{−s}, s ∈ C

where L′ is the set L′ = {l ∈ L | det(Q(l)) ≠ 0}.

1.3.1 Theorem. Under the assumptions (H_{1}) (section 1.1) and (H_{2}) (section 1.2), the zeta series ζ_{L}(s) converges absolutly for Re(s) > \frac{N}{2m}. Proof. For a ∈ Ω, we set

ν(a) = #{l ∈ L′ | Q(l) = a}
ϵ(a) = #{γ ∈ Γ | γ(a) = a}
μ(a) = \frac{ν(a)}{ϵ(a)}.

Assume s real. By the proposition 1.2.4, there exists a Siegel set 𝔖_{t,u} and a finite subset B of G(ϕ)_{Q} such that Ω = ΓB.𝔖_{t,u}.e and then

ζ_{L}(s) = ∑_{a∈Γ\Q(L′)} μ(a)det(a)^{−s} ≤ ∑_{a∈Q(L′)∩B.𝔖_{t,u}.e}ν(a).det(a)^{−s}.

Before getting to the proof of the theorem, we will show the following lemma : 1.3.2 Lemma. The series S = ∑_{a∈Q(L′)} ν(a)(∏_{i=1}^{m} a_{i}^{−s}) converges for s > \frac{N}{2m}.

Proof of the lemma. Let E_{i} = ϕ(c_{i})E. We have E = ⊕_{i=1}^{m}E_{i} and this decomposition is defined over Q. Then we can find lattices R_{i} ⊆ (E_{i})_{Q} such that

L ⊆ R = ⊕_{i=1}^{m}R_{i}.

For ξ ∈ E, denote by ξ_{i} = ϕ(c_{i})ξ ∈ E_{i}. The series S becomes

S = ∑_{l∈L′}(∏ _{i=1}^{m}∥l_{i}∥^{−2s})

then

S ≤ ∑_{i=1}^{m} ∑_{l_{i}∈R_{i}−{0}} ∏_{i=1}^{m} ∥l_{i}∥^{−2s} = ∏_{i=1}^{m}( ∑_{l_{i}∈R_{i}−{0}} ∥l_{i}∥^{−2s}).

Each one of these series is an Epstein zeta series which converges for s > \frac{1}{2}dim(E_{i}) = \frac{N}{2m}. □

Let’s now return to the proof of the theorem. For each equivalence class of Q(L′) modulo Γ, we choose a representant of the form a = bα, b ∈ B, α ∈ 𝔖_{t,u}.e. By the Minkowski inequality, we have

det(a) = Det(b)^{\frac{m}{n}} det(α) ≥ Det(b)^{\frac{m}{n}} C_{t,u}^{−1} ∏_{i=1}^{m} α_{i}

and if s > 0, then

det(a)^{−s} ≤ M^{s} ∏_{i=1}^{m} α_{i}^{−s}, with M = C_{t,u}sup_{b∈B}Det(b)^{− \frac{m}{n}} .

Set B = {b_{1}, …, b_{r}}, b_{j} = ρ(f_{j}), L_{j} = f_{j}^{−1}(L). If a = b_{j}α, then

ν(a) = #{l ∈ L′_{j} | Q(l) = α} = ν_{j}(α).

Otherwise, if f_{1}, f_{2} ∈ F (ϕ), then

ρ(f_{1}) = ρ(f_{2}) ⇔ ∀x ∈ V, f_{1}^{⋆}.ϕ(x).f_{1} = f_{2}^{⋆}.ϕ(x).f_{2} (∗)

so, if b ∈ G(ϕ), then x ∈ b^{−1}(Q(L′)) ⇔ ∃l ∈ L′, ∃f ∈ F (ϕ), x = b^{−1}.Q(l) = Q(fl).

Notice that f is not unique, but if x = ∑_{j=1}^{m} x_{j}.c_{j} + ∑_{k<l} x_{kl} is the Peirce decomposition

of x, then

x_{j} = Q(f.l)_{j} = (Q(f.l) | c_{j}) = (ϕ(c_{j})f.l | f.l) = (f^{⋆}.ϕ(c_{j}).f.l | l)

and x_{j} = ∥ϕ(c_{j})f.l∥^{2} does not depend on the choice of the antecedant f of b^{−1} by the map ρ. Finally, we find

ζ_{L}(s) ≤ M^{s} ∑_{j=1}^{r} ∑_{α∈Q(L′_{j})} μ_{j}(α) ∏_{i=1}^{m} α_{i}^{−s} = M^{s} ∑_{j=1}^{r} S_{j},

and the announced result is just a consequence of the above lemma. □

Part II. Analytic continuation and the functional equation. We use the classical method which consists to see the zeta series as the Mellin transform of the theta series associated to the representation ϕ and the lattice L, and the functional equation is a consequaence of the transformation formula of the theta series. Let’s first recall some results about zeta integrals.

2.1 Zeta integrals associated to ϕ. For each function f in the Schwartz space S(E), the zeta integral associated to the representation ϕ is defined by

Z(f, s) = ∫_{E}[detQ(ξ)]^{s}f(ξ)dξ, ∀s ∈ C.

2.1.1 Proposition. The zeta integral Z(f, s) converges absolutly for Re(s) > \frac{d}{2}(m − 1) − \frac{N}{2m} (d denotes the dimension of the subspaces V_{ij} for i ≠ j in the Peirce decomposition of V). It admits an analytic continuation as a meromorphic function on the whole plane C, and satisfies to the functional equation

Z(f̂, s − \frac{N}{2m}) = γ(s)Z(f, −s),

where

γ(s) = π^{\frac{N}{2} −2ms} \frac{Γ_{Ω}(s)}{Γ_{Ω}( \frac{N}{2m} −s)},

Γ_{Ω} being the Kœcher-Gindikin gamma function of the cone Ω, that is

Γ_{Ω}(s) = ∫_{Ω} e^{−tr(x)}det(x)^{s− \frac{n}{m}} dx,

and

f̂(ξ) = ∫_{E} e^{−2πi(ξ|η)}f(η)dη

is the Fourier transform of f.(cf [15], Chapter 16, Theorem 16.4.3).

2.2 Theta series associated to ϕ and L.

For each f ∈ S(E), the theta series associated to ϕ and L is defined by

Θ(x, f, L) = ∑_{l∈L} f[ϕ(x^{\frac{1}{2}} )l], ∀x ∈ Ω.

It is clear that this series converges absolutly for x ∈ Ω. 2.2.1 Proposition. (Transformation formula)

Θ(x^{−1}, f, L) = vol(L)^{−1}det(x)^{\frac{N}{2m}} Θ(x, f̂, L^{⋆}),

where f̂ is the Fourier transform of f and L^{⋆} is the dual lattice of L, that is

L^{⋆} = {b ∈ E | (b | a) ∈ Z, ∀a ∈ L},

and vol(L) = vol(E/L). Proof. It is a consequence of The Poisson summation formula. If ψ ∈ S(E), then

∑_{l∈L} ψ(l) = vol(L)^{−1} ∑_{l∈L^{⋆}} ψ̂(l).

If ψ(ξ) = f[ϕ(x^{− \frac{1}{2}} )ξ], then ψ̂(η) = Det(ϕ(x^{\frac{1}{2}} ))f̂[ϕ(x^{\frac{1}{2}} )η] = det(x)^{\frac{N}{2m}} f̂[ϕ(x^{\frac{1}{2}} )η]. □

2.3 Invariance property of theta series. 2.3.1 Lemma. If F is a K-invariant function defined on Ω̄, then there exists a kernel F′(x, y) defined on Ω̄ × Ω̄ such that

F′(x, e) = F (x),
F′(gx, y) = F′(x, g^{⋆}y),
F′(x, y) = F′(y, x), ∀x, y ∈ Ω̄, ∀g ∈ G(Ω)

Proof. The function F_{1} defined on Ω̄×G(Ω) by F_{1}(x, g) = F (g^{⋆}x) is right-invariant by K as a function of g. The function F′ defined by F′(x, g.e) = F_{1}(x, g) satisfies to the announced properties ; in fact, it is clear that F′(x, e) = F (x) and

F′(g_{1}.x, g.e)) = F_{1}(g_{1}.x, g) = F (g^{⋆}g_{1}.x)
= F ((g_{1}^{⋆}g)^{⋆}.x) = F_{1}(x, g_{1}^{⋆}g)
= F′(x, g_{1}^{⋆}g.e).

Moreover, as there exists k ∈ K such that

P (x^{\frac{1}{2}} )y = kP (y^{\frac{1}{2}} )x

(cf.[15], Chapter 14, Lemma 14.1.2), then F′(x, y) = F (P (x^{\frac{1}{2}} )y) = F (P (y^{\frac{1}{2}} )x) = F′(y, x). □

2.3.2 Proposition. Let F ∈ S(Ω̄), a K-invariant function on Ω̄ and let f defined by f(ξ) = F (Q(ξ)). If the arithmetic subgroup Γ_{∘} is self adjoint, then the theta series Θ(x, f, L) is Γ-invariant i.e.

Θ(γx, f, L) = Θ(x, f, L) ∀γ ∈ Γ.

Poof. Let F′ be the kernel of the lemma 2.3.1. We have

f(ϕ(x^{\frac{1}{2}} )ξ) = F (Q(ϕ(x^{\frac{1}{2}} )ξ)) = F (P (x^{\frac{1}{2}} )Q(ξ))

= F′(P (x^{\frac{1}{2}} )Q(ξ), e) = F′(Q(ξ), x),

and then,

Θ(x, f, L) = ∑_{a∈L} F′(x, Q(a)).

Moreover, for h in F (ϕ),

Θ(ρ(h)x, f, L) = ∑_{a∈L} F′(ρ(h)x, Q(a)) = ∑_{a∈L} F′(x, ρ(h)^{⋆}Q(a))

= ∑_{a∈L} F′(x, Q(h^{⋆}a)) = Θ(x, f, h^{⋆}L),

and, as we assumed that Γ_{∘}^{⋆} = Γ_{∘}, then Θ(x, f, L) is Γ-invariant. □

2.4 Mellin transform of theta series. The Mellin transform of the theta series Θ(x, f, L) is defined by

Ξ(s, f, L) = ∫_{Γ\Ω} Θ(x, f, L)det(x)^{s}d^{⋆}x, ∀s ∈ C

where d^{⋆}x is the G-invariant measure on Ω, d^{⋆}x = det(x)^{− \frac{n}{m}} dx, dx denoting the Euclidean measure on V.

(H_{3}) In all the sequel, we assume that N > m(m−1)d and then the image of the Euclidean measure on E under the quadratic form Q has a density with respect to the Euclidean measure of V.

2.4.1 Proposition. Let F ∈ S(Ω̄), K-invariant, null on ∂Ω. Let f be the function defined by f(ξ) = F (Q(ξ)). For s ∈ C, if Re(s) > max\{\frac{N}{2m}, (m − 1)\frac{d}{2}\}, then the integral Ξ(s, f, L) converges absolutly and satisfies to

Ξ(s, f, L) = \frac{Γ_{Ω}( \frac{N}{2m})}{π^{\frac{N}{2}}}ζ_{L}(s).Z(f, s − \frac{N}{2m}).

Proof. Assume f positive and s real, then

Θ(x, f, L) = ∑_{a∈Q(L′)} ν(a)F′(x, a) = ∑_{a∈Γ\Q(L′)} μ(a) ∑_{b∈Γ.a} F′(x, b), and as

∫_{Γ/Ω}[ ∑_{b∈Γ.a} F′(x, b)]det(x)^{s}d^{⋆}x = ∫_{Ω} F′(x, a)det(x)^{s}d^{⋆}x

= det(a)^{−s} ∫_{Ω} F (x)det(x)^{s}d^{⋆}x < ∞,

then

Ξ(s, f, L) = ∫_{Γ\Ω}[ ∑_{a∈Γ\Q(L′)} μ(a)( ∑_{b∈Γ.a} F′(x, b))]det(x)^{s}d^{⋆}x

= ∑_{a∈Γ\Q(L′)} μ(a) ∫_{Γ\Ω}[ ∑_{b∈Γ.a} F′(x, b)]det(x)^{s}d^{⋆}x

= ∑_{a∈Γ\Q(L′)} μ(a)det(a)^{−s} ∫_{Ω} F (x)det(x)^{s}d^{⋆}x

= ζ_{L}(s) ∫_{Ω} F (x)det(x)^{s}d^{⋆}x < +∞.

and we find

Ξ(s, f, L) = ζ_{L}(s). ∫_{Ω} F (x)det(x)^{s}d^{⋆}x.

Recall that for N > m(m − 1)d, the image of the measure dξ under Q is

dμ(x) = \frac{π^{\frac{N}{2}}}{Γ_{Ω}( \frac{N}{2m})}det(x)^{\frac{N}{2m} − \frac{n}{m}} dx,

(cf.[15], Chapter.16, Proposition.16.1.1), we find

Ξ(s, f, L) = \frac{Γ_{Ω}(\frac{N}{2m})}{π^{\frac{N}{2}}}ζ_{L}(s) ∫_{E} f(ξ)[det(Q(ξ))]^{s− \frac{N}{2m}} dξ

= \frac{Γ_{Ω}(\frac{N}{2m})}{π^{\frac{N}{2}}}ζ_{L}(s).Z(f, s − \frac{N}{2m}).

Recall the following lemma :

2.4.2 Lemma. Let f ∈ S(E) be a radial function, namely, there exists a function F defined on Ω̄, such that f(ξ) = F (Q(ξ)), then the Fourier transform f̂ of f is radial.

(cf.[15], Chapter 16, Proposition 16.2.5). Moreover, we can find a radial function f such that f and its Fourier transform f̂ vanish on the set

{ξ ∈ E | det(Q(ξ)) = 0}.

2.4.3 Proposition. Let F ∈ S(Ω̄), K-invariant, and f(ξ) = F (Q(ξ)). If f and f̂ vanish on the set {ξ ∈ E | det(Q(ξ)) = 0}, then Ξ(s, f, L)is an analytic function on its convergence domain, it admits an analytic continuation as analytic function on the whole C, and it satisfies to the functional equation

Ξ(\frac{N}{2m} − s, f̂, L^{⋆}) = vol(L)Ξ(s, f, L).

Proof. Set

Ω_{+} = {x ∈ Ω | det(x) ≥ 1},
Ω_{−} = {x ∈ Ω | det(x) ≤ 1}.

and we define

Ξ_{+}(s, f, L) = ∫_{Γ/Ω_{+}} Θ(x, f, L)det(x)^{s}d^{⋆}x
Ξ_{−}(s, f, L) = ∫_{Γ/Ω_{−}} Θ(x, f, L)det(x)^{s}d^{⋆}x.

The integral defining Ξ_{+}(s, f, L) converges for each s ∈ C and is analytic on the whole C. Indeed, for each positive constants A, B there exists a positive constant C such that

|F (y)| ≤ Cdet(y)^{−A}(1 + tr(y))^{−B},

and

|F′(x, a)| ≤ Cdet(x)^{−A}det(a)^{−A}(1 + (a | x))^{−B}.

If Re(s) ≤ A, and det(x) ≥ 1, then |det(x)^{s}| ≤ det(x)^{A}, and

|F′(x, a)det(x)^{s− \frac{n}{m}} | ≤ Cdet(a)^{−A}(1 + (a | x))^{−B}.

Otherwise,

∫_{Ω_{+}}(1 + (a | x))^{−B}dx ≤ det(a)^{− \frac{n}{m}} ∫_{Ω}(1 + tr(y))^{−B}dy.

So, if A > \frac{N}{2m}, then

∑_{a∈Γ\Q(L′)} μ(a)det(a)^{−A} ∫_{Ω_{+}}(1 + (a | x))^{−B}dx < ∞.

On an other side, as

Θ(x^{−1}, f, L) = vol(L)^{−1}det(x)^{\frac{N}{2m}} Θ(x, f̂, L^{⋆}),

then if Ξ_{−}(s, f, L) converges, i.e. if Ξ(s, f, L) converges, then

Ξ_{−}(s, f, L) = ∫_{Γ/Ω_{+}} Θ(x^{−1}, f, L)det(x)^{−s}d^{⋆}x

= vol(L)^{−1} ∫_{Γ/Ω_{+}} Θ(x, f̂, L^{⋆})det(x)^{\frac{N}{2m} −s}d^{⋆}x

= vol(L)^{−1}Ξ_{+}(\frac{N}{2m} − s, f̂, L^{⋆})

i.e.

Ξ_{−}(s, f, L) = vol(L)^{−1}Ξ_{+}(\frac{N}{2m} − s, f̂, L^{⋆}).

We deduce that Ξ_{−}(s, f, L) is analytic on its convergence domain and as Ξ_{+}(s, f, L) is analytic on C, then the above equation gives the analytic continuation of Ξ_{−}(s, f, L) as analytic function on C. It is also the same for the Mellin transform Ξ(s, f, L) which is given by

Ξ(s, f, L) = Ξ_{+}(s, f, L) + Ξ_{−}(s, f, L)

= Ξ_{+}(s, f, L) + vol(L)^{−1}Ξ_{+}(\frac{N}{2m} − s, f̂, L^{⋆}).

Moreover, it satisfies to the functional equation

Ξ(\frac{N}{2m} − s, f̂, L^{⋆}) = vol(L)Ξ(s, f, L). □

2.5 Analytic continuation and functional equation. From the above, we deduce

Ξ(\frac{N}{2m} − s, f, L) = \frac{Γ_{Ω}(\frac{N}{2m})}{π^{\frac{N}{2}}}ζ_{L}(\frac{N}{2m} − s)Z(f, −s)

= vol(L)Ξ(s, f̂, L^{⋆})

= vol(L)\frac{Γ_{Ω}(\frac{N}{2m})}{π^{\frac{N}{2}}}ζ_{L^{⋆}}(s)Z(f̂, s − \frac{N}{2m})

= vol(L)\frac{Γ_{Ω}(\frac{N}{2m})}{π^{\frac{N}{2}}}ζ_{L^{⋆}}(s)π^{\frac{N}{2} −2ms} \frac{Γ_{Ω}(s)}{Γ_{Ω}(\frac{N}{2m} − s)}Z(f, −s)

and finally, we have the theorem :

2.5.1 Theorem. Under the assumptions : (H_{3}) N > m(m − 1)d, (H_{4}) the arithmetic subgroup Γ_{∘} is self-adjoint, the zeta function ζ_{L}(s) admits an analytic continuation as a meromorphic function on the whole C and satisfies to the functional equation

ζ_{L}(\frac{N}{2m} − s) = vol(L)π^{\frac{N}{2} −2ms} \frac{Γ_{Ω}(s)}{Γ_{Ω}( \frac{N}{2m} −s)}ζ_{L^{⋆}}(s).

Remark. If Γ̃_{∘} is a finite-index subgroup of Γ_{∘}, then the zeta series defined by

ζ̃_{L}(s) = ∑_{l∈Γ̃_{∘}\L′} det(Q(l))^{−s}

has the same properties than ζ_{L}(s).

3. Examples. In this section we look at some examples of zeta functions. 3.1 Case of symmetric real matrices. Let V = Sym(m, R) the Jordan algebra with the product A ∘ B = \frac{1}{2}(AB + BA), then the symmetric associated cone is the cone Ω of positive definite symmetric real matrices. Let E = M(m, n, R) (with n ≥ m), and ϕ the representation

ϕ : V → Sym(E), x ↦ ϕ(x) : ξ ↦ xξ.

The associated quadratic form Q is given by Q(ξ) = ξξ′, ∀ξ ∈ E. (ξ′ is the adjoint of ξ.) Let V_{Q} = Sym(m, Q), then V_{Q} is a split Q-structure of V, and let E_{Q} = M(m, n, Q) and L the lattice L = M(m, n, Z). It is clear that ϕ is defined over Q, moreover, the arithmetic group GL(m, Z) is a finite-index subgroup of Γ_{∘}, where

Γ_{∘} = {f ∈ F (ϕ) | f(L) = L}.

and the zeta series is

ζ_{L}(s) = ∑_{A∈GL(m,Z)\M(m,n,Z),rank(A)=m} Det(AA′)^{−s},

which is the classical Kœcher zeta function. 3.2 Case of hermitian complex matrices. Let V = Herm(m, C) the Jordan algebra with the above product ”∘”, E = M(m, n, C) (with n ≥ m), and ϕ the representation

ϕ(x)ξ = xξ, ∀x ∈ V, ξ ∈ E.

The associated quadratic form is given by Q(ξ) = ξξ̄′. Let K be an imaginary quadratic field and O its ring of integers. Then V_{Q} = Herm(m, K) is a split Q-structure of V and the space E_{Q} = M(m, n, K) is a Q-structure of E. Let L be the lattice L = M(m, n, O), then it is clear that the representation ϕ is defined over Q, and the group GL(m, O) is of finite index in Γ_{∘}, and we obtain the zeta series

ζ_{L}(s) = ∑_{A∈GL(m,O)\M(m,n,O),rank(A)=m} Det(AĀ′)^{−s}.

and this case gives a new example of zeta function. 3.3 Case of hermitian quaternionic matrices. Let V = Herm(m, H) with the Jordan product, E = M(m, n, H) and ϕ the representation

ϕ(x)ξ = xξ, ∀x ∈ V, ξ ∈ E.

If O denotes the ring of Hurwitz integers, then L = M(m, n, O) is a lattice in E and the group GL(m, O) is of finite-index in the arithmetic subgroup Γ_{∘} associated to ϕ. The zeta series is the one studied by A.Krieg in [15], and is given by

ζ_{L}(s) = ∑_{A∈GL(m,O)\M(m,n,O),rank(A)=m} Det(AĀ′)^{−s}.

The case of zeta functions of representations of rank 2-Jordan algebras gives new examples of zeta functions and constitutes foritself an other article (cf. [1]).

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[15] I. Satake, private communication.