1. Zeta functions of prehomogeneous vector spaces

PREHOMOGENEOUS VECTOR SPACES

HIROSHI SAITO

1. Zeta functions of prehomogeneous vector spaces Let (G, ρ, X) be a prehomogeneous vector space defined over an al- gebraic number field F. Let S be its sigular set. Assume that S is a hypersurface of X. This conditions is satified if (G, ρ, X) is irreducible regular reduced. Then S is a disjoint union of F irreducible subsets S_i, i = 1,2, . . . , n, of codimension 1. For each S_i, there exist an F- irreducible polynomial P_i defining S_i and χ_i ∈X(G)_F which satisfies

Pi(ρ(g)x) =χi(g)Pi(x),

where X(G) is the group of characters of G and X(G)_F is its subset consisiting of elements defined over F. Let X⁰ =X\S and let

X^∗(F) = { x∈X⁰(F) | X(G⁰_x) = {1} }. We fix x∈X^∗(F) and set Π_x=G_x/G⁰_x.

LetAbe the adele ring ofF and letS(X(A)) be the space of Schwartz- Bruhat functions onX(A). Then for Φ∈ S(A) ands= (s₁, s₂, . . . , s_n)∈ C, the zeta function of (G, ρ, X) is defined by

Z(Φ, s) =

∫

G(A)/G(F)

∏n

i=1

|χi(g)|^s_Aⁱ( ∑

x∈X^∗(F)

Φ(gx)) dg.

On the convergence of this zeta function, we can prove([6])

Theorem 1.1. The zeta function Z(Φ, s) converges absolutely if Re(si) is sufficiently large for i= 1,2, . . . , n.

Under a more restricted condition, we can give a precise result. It is known that there exist integers m, d₁, d₂, . . . , d_n such that (detρ)^m = χ^d₁¹· · ·χ^d_nⁿ. We set κi =di/m.

Theorem 1.2. Assume G⁰_x is semisimple and Πx is abelian. Then Z(Φ, s) converges absolutely if Re(si)> κi for i= 1,2, . . . , n.

These zeta functions usually do not have an Euler product and it is difficult to transform the integral defining the zeta functions into local ones. We try to decomposes Z(Φ, s) into a sum of subseries which are finite sums of Euler products using Galois cohomology.

By the definition of Π_x, we have an exact sequence 1 −−−→ G⁰_x −−−→ G_x −−−→ Π_x −−−→ 1,

1

and an exact sequence derived from it

H¹(F, G⁰_x) −−−→ H¹(G, G_x) −−−→^µ H¹(F,Π_x).

By composing the natural map

X⁰(F)−→G(F)\X⁰(F)−→H¹(F, G_x) with µ, we obtain

φx: X⁰(F)−→H¹(F,Πx).

Set

X⁰(F,α) =˜ φ⁻_x¹( ˜α) for α ∈H¹(F,Π_x). Then we obtain a disjoint union

X⁰(F) = ∪

˜

α∈H¹(F,Πx)

X⁰(F,α).˜

We define a patial zeta function associated to the class ˜α ∈ H¹(F,Π_x) by

Z(Φ, s; ˜α) =

∫

G(A)/G(F)

∏n

i=1

|χ_i(g)|^s_Aⁱ( ∑

x∈X^∗(F,˜α)

Φ(gx))dg.

Here X^∗(F,α) =˜ X⁰(F,α)˜ ∩X^∗(F). Then we have Z(Φ, s) = ∑

˜

α∈H¹(F,Πx)

Z(Φ, s,α).˜

These partial zeta functions are easier to treat and we can give an explicit expression of these partial zeta functions by local orbital zeta functions.

2. Explicit formula

To give an explicit expression of partial zeta functions by local orbital zeta functions, we introduce some notation. Let H be an reductive al- gebraic group defined over a field F of characteristic 0. Let ˜H_der be the simply connected covering of H_der, and let p denote the natural map of H˜_dertoH_der. LetT be a maximal torus ofHand setT_sc =p⁻¹(H_der∩

T).

We define the algebraic fundamental group of H by π₁(H) =X∗(T)/p_∗(X∗(T_sc)).

Here X∗ is the group of cocharacters. We can define an action of Γ = Gal( ¯F /F). We define

A(H) = (π₁(H)_Γ)_tor.

We fix a∈X⁰(F,α) and let˜ λ_a be the canonical map ofY_a =G/G⁰_a to G/Ga =X⁰. Let ιa: A(G⁰_a)→ A(G) be the natural map, and let Ker\ιa

be the charater group of Kerι_a. Then for each ε ∈Ker\ι_a and a place v of F, we can define a function ε_v onY_a(F_v) and by

˜

εv(wv) = 1

|Π_a(F_v)|

∑

λa(zv)=wv

εv(zv)

a function on X⁰(F_v,α˜_v). With these notation, we can state our result as follows([5], [6]).

Theorem 2.1. Assume G satisfies the Hasse principle for H¹(F, G).

Then for Φ = ∏

vΦ_v ∈ S(X(A)), one has

Z(Φ, s,α) =˜ τ(G⁰_a)ker¹(G⁰_a)

|Π_a(F)||Kerι_a|L(1, χ_X(G⁰_a))) ∑

ε∈Ker\ιa

∏

v

Z(Φ_v, s; ˜α_v,ε˜_v), where

Z(Φ_v, s; ˜α_v,ε˜_v) =

∫

X⁰(Fv,˜αv)

˜ ε_v(x_v)|

∏n

i=1

P_i(x_v)|^si−κiΦ_v(x_v)dx_v, τ(G⁰_a) is the Tamagawa number of G⁰_a andker¹(G⁰_a)is the number of the kernel of the Hasse map.

We give an application of this theorem. Let (G, ρ, X) be an irreducible regular reduced prhomogeneous vector space. Such prehomogeneous vec- tor spaces are classfied into 29 types by M. Sato and Kimura. It has an irreducible relative invariant P. Assume Kerρ = {1}. Let H be the connected component of

{g ∈GL(X) |P(gx) = cP(x) forc∈F¯ }.

If H = G, (G, ρ, X) is called saturated and nonsaturated otherwise. In other words, nonsaturated prehomogeneous vector space is obtained by the restriction of the group from a prehomogeneous vector space (H,ρ, X˜ ) with a bigger group H and the same vector spaceX. There exist 7 cases in 29 types of irreducible regular reduced prehomoeneous vector spaces.

We can compare zeta functions of (G, ρ, X) and those of (H,ρ, X˜ )([7]).

For Φ ∈ S(X(A)), we denote by Z_G(Φ, s) the zeta function of (G, ρ, X) and by Z_H(Φ, s) that of (H,ρ, X˜ ).

Proposition 2.2. Except in the case of type 17, for Φ∈ S(X(A)), one has

2Z_G(Φ, s) = Z_H(Φ, s).

In the case of type 17, 4Z_G(Φ, s)−Z_H(Φ, s) is a finite sum of products of Riemann zeta function.

The above difference can be given explicitly.

3. Examples

In this section, we give some examples of zeta functions of irreducible regular reduced prehomogeneous vector spaces. Let Π_x be as before.

Then it is known that Π_x is isomorphic to S_i for i= 1,2,3,4,5. In this section, we discuss some examples of cases with Π_x=S_i fori= 1,2. The group Πx seems to play an important role for the shape of zeta functions.

To simplify the description, we assume that F = Q, G splits and Kerρ={1}. The zeta functions depends on Φ and we take Φ_v to be the characteristic function ofX(Zp). But even for such Φ_p, the calculation of Z_v(Φ_v, s, ε_v) is difficult for some primesp. Hence the following exmaples should be understood up to a finite number of Euler factors. OurZ(Φ, s) is a sum of products of infinite parts and finite parts. We are interested in finite parts, and we have a zeta function for each G(R)\X⁰(R), more exactly for each class of H¹(R, G⁰_x)(cf. [5]). In the following we give our zeta only for one class of H¹(R, G⁰_x). For the calcuation of local zeta functions, we refer to [4], [8] and their references.

Case 1. Π_x ≃S₁, Kerι_x ={1}

There exist 8 types of irreducible regular reduced prehomogeneous vec- tor spaces which satisfy the condition.

type G ρ X

(1) GLm×GLm Λ1 ⊗Λ1 V(m)⊗V(m)

(3) GL_2m Λ₂ Λ²V(2m)

(6) GL₇ Λ₃ V(35)

(13) Sp_n×GL_2m Λ₁ ⊗Λ₁ V(2n)⊗V(2m) (16) * GL₁×Spin₇ Λ₁⊗spin V(1)⊗V(8)

(19)* GL₁×Spin₉ Λ₁⊗spin V(1)⊗V(16) (20) Spin₁₀×GL₂ half spin⊗Λ₁ V(16)⊗V(2) (27) GL₁×E₇ Λ₁ ⊗Λ₁ V(1)⊗V(27)

Here ∗ indicates unsaturated prehomoegeous vector spaces. In all these cases, the zeta functions have an Euler product and they are given by products of the Riemann zeta function. For example, in the case of type (6), the zeta function is equal to

ζ(s−1)ζ(s−3)ζ(s−5)ζ(2s−4)ζ(2s−7) up to a constant.

Case 2. Π_x ≃S₁, Kerι_x ≃Z/2Z.

There exist 8 types of irreducible regular redueced prehomogeneous vector spaces satisfying this condition. In the following tables, in the case of type (14), m, nare integers satisfying n≥2m≥2.

type G ρ X

(2) GL_n n, odd 2Λ₁ S²V(n)

(10)** SL5×GL3 Λ2⊗Λ1 V(10)⊗V(3) (15) SO_n×GL_m, n even,m odd Λ₁⊗Λ₁ V(n)⊗V(m) (18)* Spin7×GL3 spin⊗Λ1 V(8)⊗V(3) (21)** Spin₁₀×GL₃ half spin⊗Λ₁ V(16)⊗V(3)

Except in the cases of type (10) and (21), the zeta functions are given by a sum of two products of the Riemann zeta function. For example, in the case (2), it is equal to

C1ζ(s−(n−1)/2)

[n/2]∏

l=1

ζ(2s−(2l−1)) +C2ζ(s)

[n/2]∏

l=1

ζ(2s−2l).

Here C₁, C₂ can be given explicitly. The functional equation changes these Euler products. This example is given in Ibukiyama and Saito[3].

In the cases of (10) and (21), local zeta functions have not been cal- culated, hence the zeta functions are unknown, and the b-functions in these cases are rather complicated.

Remark 1. It is known that the b-function deterimines the Γ-factor of the functional equation. In the case of type (10), the b-function is

{(s+ 1)(s+ 3/2)(s+ 2)}³{(s+ 4/3)((s+ 5/3)}²(s+ 5/4)(s+ 7/4)

= (s+ 1)(s+ 5/4)(s+ 6/4)(s+ 7/4)

× {(s+ 1)(s+ 4/3)(s+ 5/3)}³{(s+ 3/2)(s+ 2)}³

= (s+ 5/4)(s+ 6/4)(s+ 7/4)(s+ 8/2)

× {(s+ 4/3)(s+ 5/3)((s+ 6/3)}²{(s+ 1)(s+ 3/2)}³.

Hence the Γ-factor is

Γ(4s−3)Γ(3s−2)²Γ(2s−2)³ or

Γ(4s−4)Γ(3s−3)²Γ(2s−1)³.

The zeta functions ζ(4s−3)ζ(3s−2)²ζ(2s−2)³ and ζ(4s −4)ζ(3s− 3)²ζ(2s−1)³ interchange with the above Γ-factors under s 7−→ κ−s.

We can make a similar observation also in the case of type (25).

Case 3. Π_x ≃S₂

In this case, we need another type of Dirichlet series to describe zeta functions. Let m be an odd integer ≥ 3, and let Zm(s) be the zeta

function of the quardatic form in m variables(namely, the Dirichlet se- ries associated to the Eisensein seies of weight m/2). For odd integers m, m^′ ≥3, let

Z_m,m7(s) =ζ(2s+ 2−(m+m^′)/2)

∑∞ n=1

a_nb_nn^−s, where

Z_m(s) =

∑∞ n=1

a_nn^−s, Z_m′(s) =

∑∞ n=1

b⁻_n^s.

Case 3a. In the followng cases, the zeta functions can be written by ψ_m and the Riemann zeta function.

type G ρ X

(2) GLn, n even 2Λ1 S²V(n) (14) GL₁×Sp₃ Λ₁ ×Λ₃ V(1)⊗V(14) (15) SOn×GLm n odd Λ1 ×Λ1 V(n)×V(m) (17) * Spin₇×GL₂ spin×Λ₁ V(8)⊗V(2)

(25)* GL1×G2 Λ1 ⊗Λ1 V(1)⊗V(7) (26)* G₂×GL₂ Λ₂ ⊗Λ₁ V(7)⊗V(2)

For example, in the case of type (14), the zeta function is given by([7]) Z₇(s)ζ(2s−3).

Case 3b. In the following cases, the zeta function can be written by Zm,m^′(s) and the Riemann zeta function.

type G ρ X

(5) GL₆ Λ₃ V(29)

(15) SO_n×GL_m n, m even Λ₁×Λ₁ V(n)×V(m) (22)* GL₁×Spin₁₁ Λ₁⊗half spin V(1)⊗V(32) (23) GL1×Spin12 Λ1⊗half spin V(1)⊗V(32) (29) GL₁×E₇ Λ₁⊗Λ₁ V(1)⊗V(36) For example, in the case of type (5), the zeta function is Z_7,5(s), and in the case of type (15) for n = 10, m= 4, the zeta function is equal to

C₁Z_7,5(s)ζ(2s−2)ζ(2s−7) +C₂ζ(2s−8)ζ(2s−6)ζ(2s−3)ζ(2s−1).

The zeta function for the prehomeogeneous vector spaces of type (15) was determined also by Ibukiyama and Katsurada as Koecher-Maass sreies of Siegel Eisenstein series[2].

Remark 2. We do not know how to distinguish the above two cases Case 3a and Case 3b.

The following two types (7) and (24) are not included the above Case 3a and Case 3b. Their local zeta functions have not been calculated, but it is possible to make a similar observation as in Remark 1.

type G ρ X

(7)** GL₈ Λ₃ V(56)

(24)** GL₁×Spin₁₄ Λ₁⊗half spin V(1)⊗V(64)

Remark 3. There exist 4 cases of S₄(type (4), (9), (12), (28)), and one case of S₄(type (18)) and one case ofS₅(type (11)).

type G ρ X Πx

(4) GL₂ 3Λ₁ V(4) S₃

(9)** SL6×GL2 Λ2×Λ1 V(15)⊗V(2) S3

(12)** SL3×SL₃×GL₂ Λ₁⊗Λ₁⊗Λ₁ V(3)⊗V(3)⊗V(2) S₃ (28)** GL1 ×E7 Λ1⊗Λ1 V(1)⊗V(54) S3

(8)** SL₃×GL₂ 2Λ₁⊗Λ₁ V(6)⊗V(2) S₄ (11)** SL₅×GL₄ Λ₂⊗Λ₁ V(10)⊗V(4) S₅

The zeta function in the case of type (4) was determined by Datskovski- Wright[1]. In the other cases, the global zeta functions have not been calculated.

References

[1] B. Datskovski and D. J. Wright, The adelic zeta function associated to binary cubic forms, Part II:Local thoery, J. reine angew. Math. 367(1986), 27-75.

[2] T. Ibukiyama and H Katsurada, Squared Moebius function for half-integral ma- trices and its application, J. Number Theory 86(2001), 78-117.

[3] T. Ibukiyama and H. Saito,On zeta functions associated to symmetric matrices, I:An explicit form of zeta functions, Amer. J. math., 117(1995), 1097-1155.

[4] J. Igusa,An Introductions to the theory of local zeta functions, A.M.S. 2000.

[5] H. Saito, Explicit form of the zeta functions of prehomogeneous vector spaces, Math.Ann. 315(1999), 587-615.

[6] , Convergence of the zeta fucntions of prehomoegeous vector spaces, preprint.

[7] ,Global zeta functions of Freudenthal quartics, preprint.

[8] ,Explicit formula of orbial p-adic zea functions associated to symmetric and hermitian matrices, Comment. Math. Univ. St. Pauli 48(1997), 175-216.

Graduate School of Human Environmental Studies, Kyoto University