Maass zeta functions attached to positive definite quadratic forms and prehomogeneous vector spaces

Fumihiro Sato (Rikkyo University, Tokyo)

December 4, 1998

1 Results of Maass

Let m, n be positive integers with m ≥ n. Put Γ = SL_{n}(ℤ) and let L be a Γ-invariant lattice in M_{m,n}(ℝ). We write simply SL(n) for SL_{n}(ℝ) and SO(n) for the standard maximal compact subgroup of SL(n). Take an automorphic form ϕ ∈ A(SO(n)\SL(n)/Γ; χ) and a homogeneous polynomial Q ∈ ℂ[M_{m,n}]^{SL^{n}}. In [M1], [M2] and [M3], Maass introduced a zeta function which is a generalization of the Epstein and the Koecher zeta functions, which is defined by

ζ(Q, ϕ, L; s) = ∑_{{x∈L/Γ \atop rank x=n}} \frac{Q(x)ϕ(x̄)}{(det^{t}xx)^{s+deg Q/2n}} ,  (1)

where x̄ = (det^{t}xx)^{−1/n} ^{t}xx and ϕ is regarded as a function on the space of positive symmetric matrices of size n with determinant 1.

In [M3], Maass proved that ζ(Q, ϕ, L; s) has a meromorphic continuation as a function of s and satisfies certain functional equation. However his functional equation is not explicit at all unless Q is harmonic (the harmonic case was studied in [M1]). To describe the functional equation Maass introduced a modification of the zeta function (1). Put

∂_{ij} = ∑_{l=1}^{n} x_{il} \frac{∂}{∂x_{jl}} (i, j = 1, …, m)

and define a differential operator L_{Q}(∂), which is a polynomial of ∂_{ij} (1 ≤ i, j ≤ m), by

L_{Q}(∂)e^{−πtr^{t}xx} = Q(x)e^{−πtr^{t}xx}.

Maass’ modified zeta function is then given by

ζ̃(Q, ϕ, L; s) = ∑_{{x∈L/Γ \atop rank x=n}} L_{Q}(∂) (\frac{ϕ(x̄)}{(det^{t}xx)^{s}} ) .  (2)

Put

ξ(Q, ϕ, L; s) = π^{−ns} ∏_{i=1}^{n} Γ(s − α_{i}) · ζ̃(Q, ϕ, L; s),

where α_{1}, …, α_{n} are constants depending on the eigenvalues of ϕ. Then the functional equation reads

ξ(Q, ϕ, L; \frac{m}{2} − s) = ξ(Q̌, ϕ̌, L^{∗}; s),

where ϕ̌(x̄) = ϕ(x̄^{−1}) and

Q̌(y) = ∫_{M_{m,n}(ℝ)} Q(x + iy)e^{−πtr^{t}xx} dx.

If Q is SL_{n}-invariant, then ζ̃(Q, ϕ, L; s) coincides with ζ(Q, ϕ, L; s) up to some polynomial of s. However Q̌ is not SL_{n}-invariant, unless Q is harmonic; and Maass considered it hopeless to calculate (2) explicitly. Here is a quotation from Maass [M2]:

“Eine Darstellung der Reihe (2) als Linearkombination von gewöhnlichen Dirichletreihen mit Potenzen von s als Koefficienten scheint abwegig zu sein. Man nehme den Standpunkt ein, daß der Reihentypus (2) keiner Reduktion mehr bedarf. Die explizite Ausführung der Differentiationen im (2) ist im allgemeinen gar nicht möglich, da schon die Bildung von L_{Q}(∂) bei gegebenen Q(x) im allgemeinen schwierig zu übersehen ist.”

However, contrary to Maass’ opinion, we have good reasons to expect a more explicit functional equation. Indeed, applying the method developed in [S3] to the prehomogeneous vector space (SO(m) × GL_{n}, M_{m.n}), we show that the Maass zeta functions satisfy a functional equation of usual form.

2 Zeta integrals of the Maass zeta functions

We decompose the ring ℂ[M_{m,n}]^{SL^{n}} into a direct sum of irreducible SO(m)-module:

ℂ[M_{m,n}]^{SL^{n}} = ⨁_{π} R_{π}.  (3)

Then an irreducible representation π of SO(m) appears in the decomposition if and only if there exists a non-zero SO(n) × SO(m − n)-invariant vector in W_{π}, the representation space of π. The multiplicity

of π is then equal to 1. Fix a w_{0} ∈ W_{π}^{SO(n)×SO(m−n)} with ∥w_{0}∥ = 1. Put x_{0} = ( \begin{matrix}I_{n} \\ 0_{m−n,n} \end{matrix} ). There

exists an equivalence

W_{π} ⟶ R_{π}, w ⟼ Q_{w}  (4)

satisfying

⟨w, π(k)w_{0}⟩ = Q_{w}(kx_{0}) (k ∈ SO(m)).

Fix π ∈ ŜO(m) appearing in the decomposition (3) and an automorphic form ϕ ∈ A(SL(n)/Γ; π, χ). We define the End(W_{π})-valued zeta integral by

Z_{ϕ,π}(f, L; s) = ∫_{0}^{∞} t^{2ns} d^{×}t ∫_{SO(m)} π(k) dk ∫_{SL(n)/Γ} ϕ(g) ∑_{{x∈L \atop rank x=n}} f(tkxg^{−1}) dg.  (5)

Lemma When Re s is sufficiently large, the zeta integral Z_{ϕ,π}(f, L; s) is absolutely convergent.

Set V′ = {x ∈ M_{m,n}(ℝ) | rank x = n} and write an element x ∈ V′ as

x = t_{x}k_{x}x_{0}g_{x}^{−1} (t_{x} ∈ ℝ_{+}^{×}, k_{x} ∈ SO(m), g_{x} ∈ SL(n)).

By a routine manipulation, we have

Z_{ϕ,π}(f, L; s) = ∑_{{x∈L/Γ \atop rank x=n}} \frac{1}{(det^{t}xx)^{s}} ∫_{V′}(det^{t}yy)^{s−m/2}f(y)M_{x}ϕ(ȳ) dy,  (6)

where

M_{x}ϕ(ȳ) = ∫_{SO(n)×SO(m−n)} π(k_{y}hk_{x}^{−1})ϕ(g_{y}h_{1}g_{x}^{−1}) dh_{1} dh_{2}, h = ( \begin{matrix}h_{1} & 0 \\ 0 & h_{2} \end{matrix} ) .

Let pr be the orthogonal projection of W_{π} onto W_{π}^{SO(m−n)} and π_{1} : SO(n) → GL(W_{π}^{SO(m−n)}) the subrepresentation of SO(n) of π|_{SO(n)}. Then

M_{x}ϕ(ȳ) = ∫_{SO(n)} ϕ(g_{y}h_{1}g_{x}^{−1})(π(k_{y}) ∘ pr ∘ π_{1}(h_{1}) ∘ pr ∘ π(k_{x}^{−1})) dh_{1}.

For a fixed g′ ∈ SL(n), consider the integral

I(g) = ∫_{SO(n)} ϕ(gh_{1}g′)π_{1}(h_{1}) dh_{1}

as a function of g on SL(n). Then I(g) is a eigenfunction of Z(SL(n)) corresponding to χ = χ_{λ} and satisfies

I(k_{2}gk_{1}^{−1}) = π_{1}(k_{1})I(g).

Hence, by [H, Corollary 7.4] there exists some T ∈ Hom(W_{π_{1}}, W_{π_{1}}^{M}) depending on g′ such that

I(g) = Ψ_{π_{1}}(g; λ)T, Ψ_{π_{1}}(g; λ) = ∫_{SO(n)} d(b_{gk})^{−(λ+ρ)}π_{1}(k) dk,

where g = k_{g}b_{g} (k_{g} ∈ SO(n), b_{g} ∈ B(n)), ρ = ∏_{i=1}^{n−1} Λ_{i} and d(b)^{−(λ+ρ)} = ∏_{i=1}^{n−1} d_{i}(b)^{−(λ_{i}+1)}. Therefore we can find a c_{ϕ}(x) ∈ Hom(W_{π_{1}}, W_{π_{1}}^{M}) such that

M_{x}ϕ(ȳ) = π(k_{y}) ∘ pr ∘ Ψ_{π_{1}}(g_{y}; λ)c_{ϕ}(x).

Substituting this expression in (6), we obtain

Z_{ϕ,π}(f, L; s) = ζ^{(∞)}(f; π, λ, s) · ζ_{ϕ,π}(L; s),  (7)

ζ_{ϕ,π}(L; s) = ∑_{{x∈L/Γ \atop rank x=n}} \frac{c_{ϕ}(x)}{(det^{t}xx)^{s}} ,

ζ^{(∞)}(f; π, λ, s) = ∫_{V′}(det^{t}yy)^{s−m/2}f(y)π(k_{y}) ∘ pr ∘ Ψ_{π_{1}}(g_{y}; λ) dy.

Note that ζ^{(∞)}(f; π, λ, s) depends on the eigenvalues of ϕ, not on ϕ itself and can be rewritten as follows:

ζ^{(∞)}(f; π, λ, s) = ∫_{V′}(det^{t}yy)^{s−m/2}d(\overline{^{t}yy})^{(λ+ρ)/2}(π(k_{y}) ∘ pr) ⊗ f^{SO(n)}(y) dy,  (8)

where f^{SO(n)}(y) = ∫_{SO(n)} f(yk) dk.

This expression of ζ^{(∞)} is more tractable than in (7), since the right hand side is of the form of the integral considered in [S4].

3 Relation to the original Maass zeta functions

Fix a vector w_{0} ∈ W_{π}^{SO(n)×SO(m−n)} with ∥w_{0}∥ = 1 as before. Let f_{1}(x) ∈ S(M_{m,n}(ℝ)) be a rapidly decreasing function satisfying f_{1}(kx) = f_{1}(x) (∀k ∈ SO(m)) and put f = Q_{w_{0}}f_{1}.

Proposition For a Q ∈ R_{π}, let w be an element in W_{π} such that Q = Q_{w} (cf. (4)). Then

⟨ζ_{ϕ,π}(L; s)w, w_{0}⟩ = ζ(Q, ϕ, L; s).

We can consider general matrix coefficients of c_{ϕ}(x) as automorphic forms on (SO(n) × SO(m − n))\SO(m) × SL(n)/Γ. Namely, for v, w ∈ W_{π}, there exists an automorphic form Φ such that

⟨c_{ϕ}(x)w, v⟩ = Φ(k_{x}^{−1}, g_{x}^{−1}).

The inner product ⟨c_{ϕ}(x)w, v⟩ vanishes unless v ∈ W_{π}^{SO(m−n)}.

If Φ is (SO(n) × SO(m − n)) × SO(n)-invariant, then we obtain the original Maass zeta function (1). This is the case where w = w_{0} and is given in the proposition above.

4 Functional equation satisfied by ζ^{(∞)}(f; π, λ, s)

To prove the functional equation, it is necessary to consider ζ^{(∞)}(f; π, λ, s) as a local zeta function for the prehomogeneous vector space (SO(m) × B(n), ρ, M_{m,n}), where B(n) is the subgroup of GL(n) consisting of all upper triangular matrices and the representation is the restriction of ρ to SO(m)×B(n). Then we can use the results in [S4] to derive the functional equation of ζ^{(∞)}(f; π, λ, s).

First we examine the structure of (SO(m) × B(n), ρ, M_{m,n}). The fundamental relative invariants of this space are given by

d_{i}(^{t}xx) (i = 1, …, n)

and the character χ_{i} corresponding to d_{i} is χ_{i}(k, b) = d_{i}(b)^{2}. The open orbit is given by

Ω = {x ∈ V | d_{i}(^{t}xx) ≠ 0 (1 ≤ i ≤ n)} .

Denote by T_{n} the group of diagonal matrices in B(n) and by U_{n} the unipotent radical of B(n). Let ℂ[Ω] be the ring of regular functions on Ω. Then

ℂ[Ω] = ℂ[M_{m,n}] [d_{1}^{−1}, …, d_{n}^{−1}] .

Denote by R the subring of ℂ[Ω] of all U_{n}-invariants:

R = ℂ[Ω]^{U_{n}} = ℂ[M_{m,n}]^{U_{n}} [d_{1}^{−1}, …, d_{n}^{−1}] .

We consider R as an SO(m) × T_{n}-module. Then we have the decomposition

R = ⨁_{ψ∈X(T)} ⨁_{π:class n} R_{π,ψ} = ⨁_{e_{1},...,e_{n}=−∞}^{∞} ∏_{i=1}^{n} d_{i}(^{t}xx)^{e_{i}} ( ⨁_{ψ∈X(T)/X(T)^{2}} ⨁_{π:class n} R_{π,ψ}) ,

where π is said to be class n if W_{π}^{SO(m−n)} ≠ {0}. Fix a complete set of representatives of X(T)/X(T)^{2} so that R_{π,ψ} is contained in ℂ[M_{m,n}] and put

H = ⨁_{ψ∈X(T)/X(T)^{2}} ⨁_{π:class n} R_{π,ψ}.

We put M = {t ∈ T | t^{2} = 1}. Then the mapping

H ∋ Q(x) ⟼ [k ↦ Q(k · x_{0})] ∈ C(SO(m)/SO(m − n))

induces an isomorphism as SO(m) × M-modules. Here the action of M on C(SO(m)/SO(m − n)) is given by

(m · f)(k · SO(m − n)) = f(km̃ · SO(m − n)), m̃ = ( \begin{matrix}m & 0 \\ 0 & h_{2} \end{matrix} ) , h_{2} ∈ O(m − n), det m = det h_{2}.

Lemma We denote by C(SO(m)/SO(m − n))_{π,ψ} the image of R_{π,ψ} in C(SO(m)/SO(m − n)). Then the multiplicity of π in C(SO(m)/SO(m − n))_{π,ψ} is equal to the dimension of

W_{π,ψ} = \{w ∈ W_{π} | π(( \begin{matrix}m & 0 \\ 0 & h \end{matrix} ))w = ψ(m)w (m ∈ M, h ∈ O(m − n), det m · det h = 1)\} .

Moreover there exists an ℝ-linear isomorphism

W_{π} ⊗ W_{π,ψ} ∋ v ⊗ w ⟼ Q_{v⊗w} ∈ R_{π,ψ}, Q_{v⊗w}(kx_{0}) = ⟨v, π(k)w⟩ .

Put

ψ^{(1)} = ψ|_{T∩SL(n)} = ∏_{i=1}^{n−1} Λ_{i}^{ψ_{i}} and d(ψ) = \frac{deg Q_{v⊗w}}{n} .

Note that d(ψ) is independent of the choice of v ⊗ w. Since

Q_{v⊗w}(x)d(\overline{^{t}xx})^{−ψ^{(1)}/2}(det^{t}xx)^{−d(ψ)/2} = ⟨v, π(k_{x})w⟩,

we obtain

⟨ζ^{(∞)}(f; π, λ, s)w, v⟩ = ∫_{V′}(det^{t}yy)^{s−(m+d(ψ))/2}d(\overline{^{t}yy})^{(λ−ψ^{(1)}+ρ)/2}\overline{Q_{v⊗w}(y)}f^{SO(n)}(y) dy.  (9)

Put

γ(λ, s) = π^{−2(s_{1}+2s_{2}+···+ns_{n})+n(m−2)/2} · \{∏_{i=1}^{n} sin π (s_{i} + ⋯ + s_{n} − \frac{m − i − 1}{2})

× Γ (s_{i} + ⋯ + s_{n} − \frac{m − i − 1}{2}) Γ (s_{i} + ⋯ + s_{n} − \frac{n − i}{2})\} ,

where

s_{i} = \frac{λ_{i} + 1}{2} (1 ≤ i ≤ n − 1),  s_{n} = s − \frac{1}{2n} ∑_{i=1}^{n−1} iλ_{i} − \frac{n − 1}{4}.  (10)

Let pr_{π,ψ} be the orthogonal projection of W_{π}^{SO(m−n)} onto W_{π,ψ}. We define a character ψ̌ by

ψ̌(t) = ψ(Jt^{−1}J^{−1}), J = ( \begin{matrix}& & & & & ±1 \\ & & & & 1 \\ & · & · & · \\ 1 \end{matrix} ) ∈ SL(n).

Theorem There exists a linear mapping b_{π,ψ}(λ, s) ∈ Hom(W_{π,ψ}, W_{π,ψ̌}) ⊗ ℂ(λ_{1}, …, λ_{n−1}, s) such that the following functional equation holds for f ∈ S(V):

ζ^{(∞)}(f̂; π, λ, s) ∘ pr_{π,ψ}

= (−2πi)^{−d(ψ)n}γ(λ − ψ^{(1)}, s − d(ψ)/2)ζ^{(∞)}(f; π, λ̌, \frac{m}{2} − s) ∘ (pr_{π,ψ̌} ∘ b_{π,ψ}(λ, s) ∘ pr_{π,ψ}) .

The theorem is a special case of the main theorem in [S4]. The proof is based on a similar argument to that in [RS].

5 Functional equation satisfied by ζ_{ϕ,π}

Theorem (1) The zeta function ζ_{ϕ,π}(L, s) has an analytic continuation to a meromorphic function of s in ℂ. Moreover, there exists a linear mapping τ_{π,ψ}(λ, s) ∈ End(W_{π,ψ}) ⊗ ℂ[λ_{1}…. , λ_{n−1}, s] such that

τ_{π,ψ}(λ, s) ∘ pr_{π,ψ} ∘ ζ_{ϕ,π}(L; s)

is an entire function.

(2) For any v ∈ W_{π} and any w ∈ W_{π,ψ̌}, the following functional equation holds:

⟨ζ_{ϕ,π}(L^{∗}; \frac{m}{2} − s)v, w⟩ = v(L)^{−1}(−2πi)^{−d(ψ)n}γ(λ − ψ^{(1)}, s − d(ψ)/2) ⟨ζ_{ϕ,π}(L; s)v, b_{π,ψ}(λ, s)^{∗}(w)⟩ ,

where b_{π,ψ}(λ, s)^{∗} is the adjoint operator of b_{π,ψ}(λ, s).

As noticed earlier, the original Maass zeta functions appear when w ∈ W_{π}^{SO(n)×SO(m−n)}. The space W_{π}^{SO(n)×SO(m−n)} of spherical vectors is contained in W_{π,ψ_{0}} (ψ_{0} = 1) and W_{π,ψ_{0}} is not one-dimensional in general. In such a case, even if we start from w = w_{0}, the right hand side may contain several terms not of the form (1). This explains why it was difficult to obtain a reasonable functional equation of the Maass zeta functions. There exists an exceptionally happy case where W_{π}^{SO(m−n)} is one-dimensional and coincides with W_{π}^{SO(n)×SO(m−n)}. This is the case where R_{π} consists of harmonic polynomials (see [G]), which Maass treated in [M1] succesfully.

References

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[H] S. Helgason, A duality for symmetric spaces with application to group representations II. Adv. in Math. 22(1976), 187–219.

[M1] H. Maass, Spherical functions and quadratic forms. J. Indian Math. Soc. 20(1956), 117–162.

[M2] H. Maass, Zetafunktionen mit Größencharakteren und Kugelfunktionen. Math. Ann. 134(1957), 1–32.

[M3] H. Maass, Siegel’s modular forms and Dirichlet series. Lect. Notes in Math. No.216, Springer, 1971.

[RS] H.Rubenthaler and G.Schiffmann, Opérateurs différentiels de Shimura et espaces préhomogènes, Invent. Math. 90(1987), 409–442.

[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, The Maass zeta functions attached to positive definite quadratic forms. Adv. Studies in pure Math. 21(1992), 409–443.

[S3] F. Sato, Zeta functions of prehomogeneous vector spaces with coefficients related to periods of automorphic forms. Proc. Ind. Acad. (K.G.Ramanathan memorial issue) 104(1994), 99–135.

[S4] F. Sato, Zeta functions with polynomial coefficients associated with prehomogeneous vector spaces. Comment. Math. Univ. St. Pauli 45(1996), 177–211.