Zeta functions of prehomogeneous vector spaces and weakly spherical homogeneous spaces

Fumihiro Sato

Department of Mathematics, Rikkyo University Nishi-Ikebukuro, Toshimaku, Tokyo 171, Japan

0. Let G be a reductive ℚ-group and H a ℚ-subgroup of G. We consider the homogeneous space X = H\G. Let P be a proper parabolic ℚ-subgroup of G. We call X = H\G P -spherical (resp. weakly spherical) if the P -action is prehomogeneous (resp. if it is P - spherical for some P). One can associate a family of zeta functions with a weakly spherical homogeneous space and it is a conjecture that the associated zeta functions satisfy certain functional equations similar to those satisfied by Eisenstein series.

If G = GL(n), then the notion of weakly spherical homogeneous space is closely related to the notion of prehomogeneous vector space and it is quite fruitful to investigate zeta functions of prehomogenous vector spaces from the view point of weakly spherical homogeneous spaces.

The aim of this note is to explain this new point of view through an example. First we give the functional equations satisfied by the zeta functions attached to the weakly spherical space Spin(10)\GL(16). Then we explain briefly how the result on Spin(10)\GL(16) can be used to obtain the explicit functional equation of the zeta functions attached to the prehomogeneous vector space (Spin(10) × GL(3), half-spin ⊗ □). This space is one of the most complicated ones among irreducible regular prehomogeneous vector spaces and the explicit formula for the functional equation of the zeta function has not been known before.

1. The starting point is the following simple observation:

Lemma 1.1 Let P_{e_{1},...,e_{r}} be the standard (namely, upper triangular) parabolic subgroup of GL(n) corresponding to the partition n = e_{1} + ⋯ + e_{r}. Let H be a connected ℚ-subgroup of GL(n) Then the following are equivalent:

(1) (H × P_{e_{1},...,e_{r−1}}, M(n, m)) (m = e_{1} + ⋯ + e_{r−1}) is a prehomogeneous vector space.

(2) X = H\GL(n) is P_{e_{1},...,e_{r}}-spherical.

In the case of H = Spin(10), we have the following.

Lemma 1.2 (Kimura [KKO]) We identify H = Spin(10) with the image of the halfspin representation in GL(16). Let P be a standard parabolic subgroup of GL(16). Then, X = Spin(10)\GL(16) is P -spherical if and only if P contains one of P_{1,1,1,13}, P_{1,1,13,1}, P_{1,13,1,1}, and P_{13,1,1,1}.

Combining these two lemma, we see that (Spin(10) × P_{1,1,1}, half-spin ⊗ □) is a prehomogeneous vector space. Since the calculation of the explicit functional equation for (Spin(10) × GL(3), half-spin ⊗ □) can easily be reduced to the calculation for (Spin(10) × P_{1,1,1}, half-spin ⊗ □), we concentrate our attention to this prehomogeneous space and the weakly spherical homogeneous space Spin(10)\GL(16).

2. Let P be one of P_{1,1,1,13}, P_{1,1,13,1}, P_{1,13,1,1}, and P_{13,1,1,1}. Denote by Ω_{P} the open P -orbit in X. Let X (P) be the group of rational characters of P and X_{H}(P) the subgroup of X (P) of characters corresponding to relative P -invariants on X. Then there exist 4 algebraically independent relative invariants on X and X_{H}(P) is of finite index in X (P). For χ ∈ X_{H}(P), we fix a relative invariant f^{χ} satisfying

f^{χ}(xp) = χ(p)f^{χ}(x) (x ∈ X, p ∈ P).

We may assume that

f^{χψ} = f^{χ}f^{ψ} (χ, ψ ∈ X_{H}(P)).

Put

E(P) = Hom(X_{H}(P), {±1}).

For ϵ ∈ E(P), put

Ω_{P,ϵ} = \{x ∈ Ω_{P,ℝ} | sgn f^{χ}(x) = \frac{f^{χ}(x)}{|f^{χ}(x)|} = ϵ(χ) (∀χ ∈ X_{H}(P))\} .

Then

Ω_{P,ℝ} = ⋃_{ϵ∈E(P)} Ω_{P,ϵ}

gives the decompositon into connected components.

We put

𝔞_{P,ℂ}^{∗} = X_{H}(P) ⊗ ℂ = X (P) ⊗ ℂ.

Choose a system of generators {χ_{1}, χ_{2}, χ_{3}, χ_{4}} of X_{H}(P) so that f^{χ_{i}} is regular everywhere on X. For λ = ∑_{i=1}^{4} λ_{i}χ_{i} ∈ 𝔞_{P,ℂ}^{∗} (Re(λ_{i}) > 0) and ϵ ∈ E(P), the function

|f(x)|_{ϵ}^{λ} = \{ \begin{matrix}∏_{i=1}^{4} |f^{χ_{i}}(x)|^{λ_{i}} & if x ∈ Ω_{P,ϵ}, \\ 0 & otherwise. \end{matrix}

does not depend on the choice of system of generators and is extended to a distribution with meromorphic parameter λ ∈ 𝔞_{P,ℂ}^{∗}.

Let N be the unipotent radical of P and put

Δ_{P} (p) = d(pnp^{−1})/dn ∈ X (P),

where dn is the invariant gauge form on N. We define δ = δ_{P} ∈ 𝔞_{P,ℂ}^{∗} by

δ = δ_{P} = \frac{1}{2}Δ_{P} .

Now we can define the zeta functions E_{ϵ}(P; x, λ) for x ∈ X_{ℚ} and ϵ ∈ E(P) by

E_{ϵ}(P; x, λ) = ∑_{y∈x·Γ∩Ω_{P,ϵ}/Γ∩P} μ(y) |f(y)|_{ϵ}^{−(λ+δ)} ,

where Γ = GL(16, ℤ) and μ(y) is the normalized volume of P_{y,ℝ}/Γ ∩ P_{y} (P_{y} is the isotropy subgroup of P at y).

Theorem 2.3 (1) Let 𝔞_{P}^{∗+} be the open cone in 𝔞_{P,ℝ}^{∗} = X (P) ⊗ ℝ generated by characters corresponding to everywhere regular relative invariants and put

𝔞_{P,ℂ}^{∗+} = 𝔞_{P}^{∗+} + \sqrt{−1}𝔞_{P,ℝ}^{∗}.

Then E_{ϵ}(P; x, λ) are absolutely convergent if λ ∈ δ + 𝔞_{P,ℂ}^{∗+}.

(2) The series E_{ϵ}(P; x, λ) have analytic continuations to meromorphic functions of λ in 𝔞_{P,ℂ}^{∗}.

To describe the functional equations satisfied by E_{ϵ}(P; x, λ), we need some notational preliminaries. Let S_{4} be the symmetric group in 4 letters. For P = P_{e_{1},e_{2},e_{3},e_{4}} and σ ∈ S_{4}, we define

σP = P_{e_{σ^{−1}(1)},e_{σ^{−1}(2)},e_{σ^{−1}(3)},e_{σ^{−1}(4)}}.

Moreover σ defines an isomomorphism between the standard Levi subgroups of P and ^{σ}P by permutation of the diagonal entries and induces a linear isomorphism

σ : 𝔞_{P,ℂ}^{∗} → 𝔞_{^{σ}P,ℂ}^{∗}.

For p ∈ P, we write

p = ( \begin{matrix}p_{1} & & & ∗ \\ & p_{2} \\ & & p_{3} \\ 0 & & & p_{4} \end{matrix} ) .

We define the normalized zeta functions by

Ê_{ϵ}(P; x, λ) = ∏_{1≤μ<ν≤4} ζ̂(z_{μ} − z_{ν} + \frac{e_{μ} + e_{ν}}{2} ) E_{ϵ}(P; x, λ),

where λ = ∑_{μ=1}^{4} z_{μ} det p_{μ} ∈ 𝔞_{P,ℂ}^{∗} and ζ̂(z) = π^{−z/2}Γ(z/2)ζ(z) (ζ(z) = the Riemann zeta function).

Theorem 2.4 For σ ∈ S_{4} and ϵ ∈ E(^{σ}P), the following functional equation holds :

Ê_{ϵ}(^{σ}P; x, σλ) = ∑_{η∈E(P)} C_{ϵ,η}(P; σ, λ)Ê_{η}(P; x, λ),

where C_{ϵ,η}(P; σ, λ) are meromorphic functions of λ independent of x ∈ X_{ℚ} and have elementary expressions in terms of the gamma fucntion and the exponential function.

3. Let us collect here some data, which are necessary to make the functional equations explicit.

For simplicity, we put

P_{1} = P_{1,1,1,13}, P_{2} = P_{1,1,13,1}, P_{3} = P_{1,13,1,1}, P_{4} = P_{13,1,1,1}.

We write

δ(p) = δ_{P} (p) = δ_{1} det p_{1} + δ_{2} det p_{2} + δ_{3} det p_{3} + δ_{4} det p_{4}.

Then

(δ_{1}, δ_{2}, δ_{3}, δ_{4}) = \{ \begin{matrix}(\frac{15}{2}, \frac{13}{2}, \frac{11}{2}, −\frac{3}{2}) & ⋯ & P = P_{1}, \\ (\frac{15}{2}, \frac{13}{2}, −\frac{1}{2}, −\frac{15}{2}) & ⋯ & P = P_{2}, \\ (\frac{15}{2}, \frac{1}{2}, −\frac{13}{2}, −\frac{15}{2}) & ⋯ & P = P_{3}, \\ (\frac{3}{2}, −\frac{11}{2}, −\frac{13}{2}, −\frac{15}{2}) & ⋯ & P = P_{4}. \end{matrix}

We can choose 4 characters χ_{0}, χ_{1}, χ_{2}, χ_{3} such that χ_{0}^{±1}, χ_{1}, χ_{2}, χ_{3} generate the semigroup X_{H}^{+}(P) consiting of characters corresponding to everywhere regular relative P -invariants. These characters are given by the following and form a basis of the free abelian group X_{H}(P): Case P = P_{1}:

χ_{0}(p) = p_{1}p_{2}p_{3} · det p_{4}, χ_{1}(p) = p_{1}^{2}p_{2}^{2}, χ_{2}(p) = p_{1}^{4}p_{2}^{2}p_{3}^{2}, χ_{3}(p) = p_{1}^{4}p_{2}^{4}p_{3}^{4}.

Case P = P_{2}:

χ_{0}(p) = p_{1}p_{2} · det p_{3}p_{4}, χ_{1}(p) = p_{1}^{2}p_{2}^{2}, χ_{2}(p) = p_{1}^{4}p_{2}^{2} det p_{3}^{2}, χ_{3}(p) = p_{1}^{6}p_{2}^{6} · det p_{3}^{4}.

Case P = P_{3}: χ_{0}(p) = p_{1} ·det p_{2} ·p_{3}p_{4}, χ_{1}(p) = p_{4}^{−2}p_{3}^{−2}, χ_{2}(p) = p_{4}^{−4}p_{3}^{−2} ·det p_{2}^{−2}, χ_{3}(p) = p_{4}^{−6}p_{3}^{−6} ·det p_{2}^{−4}. Case P = P_{4}:

χ_{0}(p) = det p_{1} · p_{2}p_{3}p_{4}, χ_{1}(p) = p_{3}^{−2}p_{4}^{−2}, χ_{2}(p) = p_{2}^{−2}p_{3}^{−2}p_{4}^{−4}, χ_{3}(p) = p_{2}^{−4}p_{3}^{−4}p_{4}^{−4}. We write

λ = z_{1} det p_{1} + z_{2} det p_{2} + z_{3} det p_{3} + z_{4} det p_{4} ∈ 𝔞_{P,ℂ}^{∗}

and put

α_{i} = z_{i} − z_{i+1} (i = 1, 2, 3).

Then, by the fact that χ_{0}^{±1}, χ_{1}, χ_{2}, χ_{3} generate the semigroup X_{H}^{+}(P), we have

𝔞_{P,ℂ}^{∗+} = {λ ∈ 𝔞_{P,ℂ}^{∗} | (∗)} ,

where

(∗) : \{ \begin{matrix}Re(α_{3}) > Re(α_{1}) > 0, Re(α_{2}) > 0 & ⋯ & P = P_{1}, \\ 2Re(α_{2}) + Re(α_{1}) > Re(α_{3}) > Re(α_{1}) > 0 & ⋯ & P = P_{2}, \\ 2Re(α_{2}) + Re(α_{3}) > Re(α_{1}) > Re(α_{3}) > 0 & ⋯ & P = P_{3}, \\ Re(α_{1}) > Re(α_{3}) > 0, Re(α_{2}) > 0 & ⋯ & P = P_{4}. \end{matrix}

The group S_{4} acts on E(P) naturally. The following is the simplest case of the functional equation given in Theorem 2.4: Case σ = (i, i + 1) (i = 1, 2, 3) and ^{σ}P = P: If σϵ = ϵ, then we have

Ê_{ϵ}(P; x, σλ) = Ê_{ϵ}(P; x, λ).

If σϵ ≠ ϵ, then we have

( \begin{matrix}Ê_{ϵ}(P; x, σλ) \\ Ê_{σϵ}(P; x, σλ) \end{matrix} ) = ( \begin{matrix}sec \frac{π(z_{i+1}−z_{i})}{2} & tan \frac{π(z_{i+1}−z_{i})}{2} \\ tan \frac{π(z_{i+1}−z_{i})}{2} & sec \frac{π(z_{i+1}−z_{i})}{2} \end{matrix} ) ( \begin{matrix}Ê_{ϵ}(P; x, λ) \\ Ê_{σϵ}(P; x, λ) \end{matrix} ) .

Put

Γ_{P} (λ) = ∏_{1≤μ<ν≤4} Γ_{ℝ} (z_{μ} − z_{ν} + \frac{e_{μ} + e_{ν}}{2} ) , Γ_{ℝ}(z) = π^{−z/2}Γ (\frac{z}{2}) ζ(z).

In the following, for ϵ ∈ E(P), we put ϵ_{i} = ϵ(χ_{i}) (i = 0, 1, 2, 3).

Case σ = (2, 3), P = P_{1,1,13,1}, ^{σ}P = P_{1,13,1,1}: In this case, unless ϵ_{0} = η_{0} and ϵ_{2} = η_{2}, we have

C_{ϵ,η}(P; σ, λ) = 0.

If ϵ_{0} = η_{0} and ϵ_{2} = η_{2}, we have

C_{ϵ,η}(P; σ, λ) = const × exp(linear form of λ) × \frac{Γ_{^{σ}P} (σλ)}{Γ_{P} (λ)}

×A_{η_{1},η_{2}ϵ_{1}}^{(4,4)} (\frac{α_{1} + 2α_{2} − α_{3}}{4} − 2) × \{ \begin{matrix}A_{η_{3},η_{2}ϵ_{3}}^{(2,4)} (\frac{α_{3}−α_{1}−6}{4}) & if η_{2} = +1, \\ A_{η_{3},η_{2}ϵ_{3}}^{(3,3)} (\frac{α_{3}−α_{1}−6}{4}) & if η_{2} = −1, \end{matrix}

where

( \begin{matrix}A_{++}^{(p,q)}(z) & A_{+−}^{(p,q)}(z) \\ A_{−+}^{(p,q)}(z) & A_{−−}^{(p,q)}(z) \end{matrix} ) = Γ_{ℝ}(2z + 2)Γ_{ℝ}(2z + p + q) ( \begin{matrix}cos π (x + \frac{q+1}{2}) & sin \frac{pπ}{2} \\ sin \frac{qπ}{2} & cos π (x + \frac{p+1}{2}) \end{matrix} ) .

Note that this is the gamma matrix of the functional equation satisfied by

|x_{1}^{2} + ⋯ + x_{p}^{2} − x_{p+1}^{2} − ⋯ − x_{p+q}^{2}|^{z} .

Case σ = (3, 4), P = P_{1,1,1,13}, ^{σ}P = P_{1,1,13,1}: In this case, unless ϵ_{0} = η_{0} and ϵ_{1} = η_{1}, we have

C_{ϵ,η}(P; σ, λ) = 0.

If ϵ_{0} = η_{0} and ϵ_{1} = η_{1}, we have

C_{ϵ,η}(P; σ, λ) = const × exp(linear form of λ) × \frac{Γ_{^{σ}P} (σλ)}{Γ_{P} (λ)}

×B_{(η_{1}η_{2},η_{1}η_{3}),(η_{1}ϵ_{2}ϵ_{3},η_{1}ϵ_{2})}^{(3,4)} (\frac{α_{1} − 1}{2}, \frac{α_{3} − α_{1} − 6}{4}) ,

where

B_{ϵ,η}^{(3,4)}(z_{1}, z_{2}) = Γ_{ℝ}(2z_{1} + 2z_{2} + 3)Γ_{ℝ}(2z_{1} + 2z_{2} + 8)Γ_{ℝ}(2z_{2} + 2)Γ_{ℝ}(2z_{2} + 7) × b_{ϵ,η}^{(3,4)}(z_{1}, z_{2})

and the functions b_{ϵ,η}^{(3,4)}(z_{1}, z_{2}) are given by the following table:

ϵ\η  ++  +−  −+  −−
++ cos π(s_{1} + s_{2}) sin πs_{2}  0  0  0
+−  0  − sin π(s_{1} + s_{2}) sin πs_{2}  0  − sin πs_{2}
−+  0  1  cos π(s_{1} + s_{2}) cos πs_{2}  sin π(s_{1} + s_{2})
−−  0  − cos πs_{2}  − cos π(s_{1} + s_{2})  − sin π(s_{1} + s_{2}) cos πs_{2}

Note that the matrix (B_{ϵ,η}^{(3,4)}) is the gamma matrix of the local functional equation of the prehomogeneous vector space (SO(3, 4) × P_{1,1}, M(7, 2)).

4. Now we apply the result of the preceding section to the prehomogeneous vector space (Spin(10)×P_{1,1,1}, half-spin⊗□). First we recall the relation between prehomogeneous vector spaces and weakly spherical homogeneous spaces given in Lemma 1.1. The weakly spherical space Spin(10)\GL(16) we have just studied is closely related to the prehomogeneous vector space (Spin(10) × P_{1,1,1}, half-spin ⊗ □). One can express the zeta functions associated with the latter space in terms of the Riemann zeta function and E_{ϵ}(P_{1,1,1,13}; x, λ). Moreover there exists a similar relation between E_{ϵ}(P_{13,1,1,1}; x, λ) and the zeta functions associated to the prehomogeneous vector space dual to (Spin(10) × P_{1,1,1}, half-spin ⊗ □). Then it can be shown that the functional equation obtained from the general theory of prehomogeneous vector spaces is nothing but the functional equation corresponding to the cyclic permutation \binom{1 2 3 4}{2 3 4 1}, which connects E_{ϵ}(P_{13,1,1,1}; x, λ) to E_{η}(P_{1,1,1,13}; x, λ). It is rather hard to calculate the functional equation explicitly; however, if we decompose the functional equation as

E(P_{1,1,1,13}; ∗, ∗) ⟶^{(3,4)} E(P_{1,1,13,1}; ∗, ∗) ⟶^{(2,3)} E(P_{1,13,1,1}; ∗, ∗) ⟶^{(1,2)} E(P_{13,1,1,1}; ∗, ∗),

then, as the formulas in §3 shows, the functional equations corresponding to the transpositions (1, 2), (2, 3), (3, 4) are simple enough for explicit calculation. These simple functional equations are not visible if we stick to the view point of prehomogeneous vector spaces.

5. The detail of the proof of the above results will apear in [S5]. The result in §2 can be extended to arbitrary weakly spherical homogeneous spaces of GL(n) (see [S2] and [S3]). This generalization is based on Lemma 1.1 and the theory of zeta functions associated with prehomogeneous vector spaces ([S1]). To extend the result further to reductive groups other than GL(n), we can no longer appeal to the theory of prehomogeneous vector spaces. We are sure that the key of further generalization is the study of the regularization of period integrals of Eisenstein series ([S4]).

References

[KKO] S.Kasai, T.Kimura and S.Otani: A classification of simple weakly spherical homogeneous spaces (I), to appear in J. of Algebra.

[S1] F.Sato: Zeta functions in several variables associated with prehomogeneous vector spaces I: Functional equations, Tôhoku Math. J. 34(1982), 437–483.

[S2] F.Sato: Eisenstein series on weakly spherical homogeneous spaces and zeta functions of prehomogeneous vector spaces, Comment. Math. Univ. St. Pauli, 44(1995), 129– 150.

[S3] F.Sato: Eisenstein series on weakly spherical homogeneous spaces of GL(n), Preprint, 1995.

[S4] F.Sato: Regularization of Eisenstein periods, in preparation.

[S5] F.Sato: Eisenstein series on Spin(10)\GL(16), in preparation.