Equivariant analytic torsion on ℙ^{n}ℂ

Kai KÖ OHLER

Université e de Paris-Sud Bâ at. 425 91405 ORSAY

Abstract for the Zentralblatt der Mathematik: The subject of the paper is to calculate an equivariant version of the complex Ray-Singer torsion for all bundles on the ℙ^{1}ℂ and for the trivial line bundle on ℙ^{n}ℂ, for isometries which have isolated fixed points. The result can for all n be expressed with a special function, which is very similar to the series defining the Gillet-Soul� R-genus.

1991 Mathematics Subject Classification: 58G26, 14J20, 53C30.

Equivariant analytic torsion on ℙ^{n}ℂ

Kai KÖ OHLER

Abstract: We calculate an equivariant version of the complex Ray-Singer torsion for all bundles on the ℙ^{1}ℂ and for the trivial line bundle on ℙ^{n}ℂ, for isometries with isolated fixed points. The result gives for all n a part of the Gillet-Soul� R-function.

Keywords: Determinants and determinant line bundles, Arakelov geometry, Homogeneous manifolds. 1990 AMS-Subject classification: 58G26, 14J20, 53C30.

Running title: Equivariant analytic torsion on ℙ^{n}ℂ Adress: Kai K�hler Math�matiques B�t. 425 F-91405 ORSAY France e-mail: Koehler@matups.matups.fr Introduction

The analytic torsion was constructed by Ray and Singer [RS] as an analytic analogue to the Reidemeister torsion. Bismut, Gillet and Soul� [BGS] proved as an extension of a result of Quillen important properties of the torsion in connection with vector bundles on fibrations:

Let π : M → B be a proper holomorphic map of compact complex manifolds and let ξ be a hermitian holomorphic vector bundle on M. Let Rπ_{∗}ξ be the right-derived direct image of ξ. Then the analytic torsion of the fibres of π induces a metric on the Knudsen-Mumford determinant λ^{KM} := (det Rπ_{∗}ξ)^{−1} which is a holomorphic line bundle on B. The curvature of this Quillen metric as well as its behaviour under changes of the metrics on M and ξ was expressed in [BGS] explicitly by means of secondary Bott-Chern classes. In particular this gives a refinement of the Riemann-Roch theorem for families.

On the other hand let i : Y ↪ X be an embedding of compact complex manifolds. Let η be a hermitian holomorphic vector bundle on Y and let ξ be a resolution of η by a complex of vector bundles on X. Bismut and Lebeau [BL] calculated the relation between the Quillen metrics of η and ξ. With the help of this result, Gillet and Soul� [GS2] were able to prove a Riemann-Roch theorem in Arakelov geometry for the first Chern class of the direct image (see [S] for the theorem and some background information). This theorem was later proved by Faltings [F] for higher degrees.

The proof of the Riemann-Roch theorem uses a calculation of Gillet, Soul� and Zagier [GS1] of the torsion for the trivial line bundle on the complex projective spaces ℙ^{n}ℂ. This led Gillet and Soul� to conjecture this theorem, which was the initial motivation for [BL]. In particular this rather difficult calculation gives in particular the Gillet-Soul� R-genus, which appears explicitly in the theorem. This is the additive genus associated to the series

R(x) = ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} (2ζ′(−ℓ) + ζ(−ℓ) ∑_{j=1}^{ℓ} \frac{1}{j})\frac{x^{ℓ}}{ℓ!} ,

where ζ is the Riemann zeta function. To obtain this series, one has to caculate the torsion of ℙ^{n}ℂ for every n.

Let us consider now a holomorphic isometry g of a hermitian vector bundle E over a compact K�hler manifold M. One can define in a natural way an equivariant version of the torsion. This equivariant torsion appeared already in Ray’s [R] calculation of the real analytic torsion for lens spaces.

In this paper we present the calculation of the equivariant analytic torsion for all holomorphic bundles on ℙ^{1}ℂ and for the trivial line bundle on ℙ^{n}ℂ, where the projective spaces are equipped with the Fubini-Study metric. We consider only rotations with isolated fixpoints. For a rotation by angles ∈ π . ℚ, we obtain a closed expression involving the gamma function. For arbitrary angles a function R^{rot} , which is similar to the Gillet-Soul� R-function, appears as an infinite series. This is relatively easy to calculate because the defining ζ -function Z has no singularities in contrast to the situation in [GS1].

The similarity of R^{rot} and R might help to find an equivariant Riemann-Roch formula in Arakelov geometry, where the two functions correspond to the extremal cases: isolated fixed points or identity map. In fact, Bismut [B3] found further evidence for such a formula: He constructed analytic torsion forms associated to a short exact sequence of hermitian holomorphic vector bundles equipped with a holomorphic unitary endomorphism g. In his result, a series R(φ, x) appears with the properties

R(0, x) = R(x), R(φ, 0) = R^{rot}(φ) .

As the appearance of the R-genus in [B2] gave evidence for the existence of the Riemann-Roch theorem, he now conjectures an equivariant Riemann-Roch formula.

The function R^{rot} can be obtained as follows: Let for 0 < φ < 2π and s > 0, ζ^{rot}(φ, s) be the Dirichlet series

ζ^{rot}(φ, s) := ∑_{k≥1} \frac{sin kφ}{k^{s}} .

Then ζ^{rot} can be seen as the imaginary part of a Lerch zeta function. We set R^{rot}(φ) := \frac{∂}{∂s}ζ^{rot}(φ, 0). The following is obtained by classical results:

Proposition 1. R^{rot} is equal to

R^{rot}(φ) = \frac{C + log φ}{φ} − ∑_{{ℓ≥1 \atop ℓ odd}} ζ′(−ℓ)(−1)^{\frac{ℓ+1}{2}} \frac{φ^{ℓ}}{ℓ!} .

If φ = 2π \frac{p}{q} with p, q ∈ ℕ, 0 < p < q, then

R^{rot}(φ) = −\frac{1}{2} log q. cot \frac{φ}{2} + ∑_{ℓ=1}^{q−1} log Γ (\frac{j}{q} ). sin jφ .

In the last chapter we give some other functional properties of R^{rot} . Let E := O(k_{1}) ⊕ … ⊕ O(k_{n}) be a holomorphic vector bundle on ℙ^{1}ℂ, equipped with the standard metric (i.e. the curvature of O(1) is the Fubini-Study K�hler form). By a theorem of Grothendieck, each holomorphic vector bundle on ℙ^{1}ℂ is of this form. Then we find

Theorem 2. The equivariant analytic torsion τ(E, φ) with respect to a rotation by an angle φ ∈]0, 2π[ is given by

−2 log τ(E, φ) = \frac{2R^{rot}(φ)}{sin \frac{φ}{2}}.∑_{j=1}^{n} cos(k_{j}+1)\frac{φ}{2}+∑_{j=1}^{n} ∑_{m=1}^{|k_{j}+1|} \frac{sin(2m − |k_{j} + 1|)\frac{φ}{2}}{sin \frac{φ}{2}} log j .

We see in particular that the equivariant torsion τ gives already for the trivial line bundle O on ℙ^{1}ℂ the function

log τ(O, φ) = cot \frac{φ}{2} . (i ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} ζ′(−ℓ)\frac{(iφ)^{ℓ}}{ℓ!} − \frac{C + log φ}{φ}) .

Let now Φ := ( ..._{0  iφ_{n+1}}^{iφ_{1}  0} ) be an element of the (canonical) maximal

Cartan subalgebras of 𝔰𝔲(n+1), hence an infinitesimal rotation on ℙ^{n}ℂ ≅ SU(n + 1)/S(U(1) × U(n)). Assume that all the φ_{j} are distinct. Then we have Theorem 3. The equivariant torsion τ(O, e^{Φ}) for the trivial line bundle O on ℙ^{n}ℂ is given by

−2 log τ(O, e^{Φ}) = (−1)^{n} ∑_{{j,k=1 \atop j≠k}}^{n+1} 2iR^{rot}(φ_{j}−φ_{k}) ∏_{{ℓ=1 \atop ℓ≠k}}^{n+1}(e^{i(φ_{k}−φ_{ℓ})}−1)^{−1}− log n! .

I) Definition of the torsion

Let M be a K�hler manifold of complex dimension n with holomorphic tangent bundle T M and K�hler form ω_{M}, ξ a hermitian vector bundle on M and \overline{∂} the Dolbeault operator acting on sections of Λ^{q}T^{∗(0,1)}M ⊗ ξ. We define a hermitian product on the vector space of smooth sections of Λ^{q}T^{∗(0,1)}M ⊗ ξ by

(η, η′) := ∫_{M}(η(x), η′(x))\frac{ω^{n}}{(2π)^{n}n!}

as in [GS1]. Consider the adjoint operator \overline{∂}^{∗} relative to this product and the Kodaira-Laplace operator

□_{q} := (\overline{∂} + \overline{∂}^{∗})^{2} : Γ(Λ^{q}T^{∗(0,1)}M ⊗ ξ) → Γ(Λ^{q}T^{∗(0,1)}M ⊗ ξ) . Let g be a holomorphic isometry of M. Assume that the bundle and its hermitian metric are holomorphically invariant under the induced action of g. Let Eig_{λ}(□_{q}) be the eigenspace of □_{q} corresponding to the eigenvalue λ and g^{∗} the of g induced action on Γ(Λ^{q}T^{∗(0,1)}M ⊗ ξ).

Consider the ζ -function

Z(g, s) := ∑_{\begin{matrix}q>O \\ λ∈ Spec⊓ ⊔_{q} \\ λ≠0 \end{matrix}}(−1)^{q+1}qλ^{−s} Trg_{| Eig_{λ}(□_{q})}^{∗}

for s ≫ 0. The equivariant torsion of M relative to the action of g is then defined as an exponential of the derivative at zero Z′(g, 0) of the holomorphic continuation of Z(g,.),

τ(g) := e^{− \frac{1}{2} Z′(g,0)} .

The eigenvalues and eigenspaces for the Kodaira Laplacian for the trivial line bundle on ℙ^{n}ℂ were determined explicitly by Ikeda and Taniguchi [IT]. If one regards ℙ^{n}ℂ as SU(n + 1)/S(U(1) × U(n)), the eigenspaces can be described by sums of irreducible representations of SU(n + 1). We are using their method and results in our proof; see also Malliavin and Malliavin [MM].

II) The Laplacian on O(k)-bundles over ℙ^{1}ℂ

Let ℙ^{1}ℂ be the one-dimensional complex projective space equipped with the usual Fubini-Study metric. That means, ℙ^{1}ℂ is isometric to the 2-sphere with radius 1/2. Take G := SU(2) and K := S(U(1) × U(1)) with the corresponding Lie algebras 𝔤 and 𝔨. We equip G with the metric

𝔤^{2} → ℝ
(X, Y) ↦ −2trXY ,

which is minus one half of the Killing form. Then we may represent ℙ^{1}ℂ as the homogeneous space G/K with the induced metric.

Let Λ be the weight of 𝔤 which acts on the Cartan subalgebras 𝔨 by diag(iφ, −iφ) ↦ \frac{φ}{2π} and let

ρ_{k}^{K} : 𝔨 → ℂ
( \begin{matrix}iφ & 0 \\ 0 & −iφ \end{matrix} ) ↦ e^{ikφ}

be the of kΛ, k ∈ ℤ, induced representation of K. This gives an action of K on the right of G × ℂ as follows:

(g, x) . h = (gh, ρ_{k}^{K}(h^{−1})x)

for g ∈ G, x ∈ ℂ and h ∈ K. Then the holomorphic line bundle O(k) is the homogeneous vector bundle

O(k) = G ×_{ρ_{−k}^{K}} ℂ := (G × ℂ)/K .

It is well known that O(2) ≅ T ℙ^{1}ℂ ≅ T^{∗(0,1)}ℙ^{1}ℂ. By a theorem of Grothendieck [G], each holomorphic vector bundle E on ℙ^{1}ℂ is a direct sum

E = O(k_{1}) ⊕ … ⊕ O(k_{n}) ,

k_{1}, …, k_{n} ∈ ℤ, so it suffices to calculate the torsion for O(k). Obviously, Z′(., 0) behaves additively under direct sum of vector bundles.

We equip O(k) with the induced metric. If ∇ is the unique holomorphic hermitian connection on the bundle of forms with coefficients in O(k), ΛT^{∗(0,1)}ℙ^{1}ℂ ⊗ O(k), and (e_{1}, e_{2}) a real orthonormal frame in the real tangent bundle T_{ℝ}ℙ^{1}ℂ, we define the horizontal (or Bochner) Laplacian as

Δ := ∑_{1}^{2}(∇_{e_{n}})^{2} − ∑_{1}^{2} ∇_{∇_{en} e_{n}} .

We know that the curvature tensor of O(1) is simply −2i times the K�hler form of ℙ^{1}ℂ. By applying Licherowicz’s formula (cf. Bismut [B1, Prop. 1.2]), we find that the Kodaira Laplacian acting on T^{∗(0,1)}ℙ^{1}ℂ ⊗ O(k) is given by

\overline{□}^{0,1} = −\frac{1}{2}Δ + \frac{k}{2} + 1 .

To find a better expression for Δ, we consider the Casimir Operators of G and K. For a given compact Lie algebra with Killing form B and orthonormal basis {X_{1}, …, X_{n}} with respect to B, its Casimir operator is defined as

Cas := − ∑_{i} X_{i} . X_{i} .

Cas is independent of the choice of the basis. Let Cas_{G} be the Casimir operator of G, acting on C^{∞}(G) by derivation, and Cas_{K} the Casimir operator of K, acting on ℂ via the representation ρ_{−k−2}^{K}. Then it is easily verified (cf. for example [BGV, Prop. 5.6]) that

2Δ = Cas_{G} + Cas_{K}

on sections of T^{∗(0,1)}ℙ^{1}ℂ ⊗ O(k) ≅ G ×_{ρ_{−k−2}^{K}} ℂ. The factor 2 appears because we take half of the negative Killing form as metric on G. For X ∈ 𝔨 we have ρ_{−k−2}^{K}(X) = −i(k + 2), so

ρ_{−k−2}^{K}(Cas_{K}) = (k + 2)^{2} ,

hence

Lemma 4.

\overline{□}^{0,1} = −\frac{1}{4} Cas_{G} − \frac{k}{2} (\frac{k}{2} + 1) .

III) Construction of the defining ζ -function

Let (ρ_{ℓ}^{G}, E_{ℓ}^{G}) be the irreducible representation G → End(E_{ℓ}^{G}) with highest weight ℓΛ, ℓ ∈ ℕ. Then we have ρ_{ℓ}^{G}(Cas_{G}) = −ℓ(ℓ + 2) . Id_{E_{ℓ}^{G}}.

To determine the eigenspaces of \overline{□}^{0,1} , we use as Ikeda and Taniguchi the following Frobenius law of Bott [Bo]:

Proposition 5. For finite dimensional representations (ρ^{K}, E^{K}) and (ρ^{G}, E^{G}) of K and G, we have the canonical isomorphism of vector spaces

Hom_{G}(E^{G}, Γ(G ×_{ρ^{K}} E^{K})) ≅ Hom_{K}(E^{G}, E^{K}) .

Now we know that the characters χ_{ℓ}^{G} of ρ_{ℓ}^{G} and χ_{k}^{K} of ρ_{k}^{K} are given by

χ_{ℓ}^{G} ( \begin{matrix}e^{iφ} & 0 \\ 0 & e^{−iφ} \end{matrix} ) = \frac{sin(ℓ + 1)φ}{sin φ}

(cf. Br�cker, tom Dieck [BD, Ch. 5, p. 267]), and

χ_{k}^{K} ( \begin{matrix}e^{iφ} & 0 \\ 0 & e^{−iφ} \end{matrix} ) = e^{ikφ} ,

hence we find the decomposition

χ_{ℓ}^{G} = \{ \begin{matrix}∑_{\begin{matrix}|n|≤ℓ \\ n even \end{matrix}} χ_{n}^{K} & when ℓ even \\ ∑_{\begin{matrix}|n|≤ℓ \\ n odd \end{matrix}} χ_{n}^{K} & when ℓ odd . \end{matrix}

Now we can see by Proposition 5 that (ρ_{ℓ}^{G}, E_{ℓ}^{G}) occurs as irreducible subspace of Γ(G ×_{ρ_{n}^{K}} ℂ) iff |n| ≤ ℓ and n ≡ ℓ(mod 2):

Lemma 6. Γ(T^{∗(0,1)}ℙ^{1}ℂ ⊗ O(k)) contains the L^{2} -dense subspace

⨁_{ℓ≥0} E_{|k+2|+2ℓ}^{G} .

The density of this subspace follows from the Peter-Weyl theorem (cf. [Bo]). By Lemma 4, the eigenvalues of \overline{□}^{0,1} for O(k) are given by

\{ \begin{matrix}ℓ(ℓ + k + 1) on E_{k+2ℓ}^{G} for ℓ ≥ 1 & when k ≥ −1 \\ ℓ(ℓ − k − 1) on E_{−k−2+2ℓ}^{G} for ℓ ≥ 0 & when k < −1 . \end{matrix}

So we finally obtain the Lemma 7. Let g := ( \begin{matrix}e^{iφ} & 0 \\ 0 & e^{−iφ} \end{matrix} ) ∈ G, φ ∈]0, π[, be an element of the maximal torus K (which corresponds to the rotation of S^{2} by the angle 2φ). Then the ζ -function Z_{k}(g,.) of the O(k)-bundle on ℙ^{1}ℂ is for s > \frac{1}{2} given by

Z_{k}(g, s) = ∑_{\begin{matrix}ℓ≥0 \\ E_{|k+2|+2ℓ}^{G}⊄ker \overline{□}^{0,1} \end{matrix}} χ_{|k+2|+2ℓ}^{G} . (\overline{□}^{0,1}|_{E_{|k+2|+2ℓ}^{G}})^{−s}

= ∑_{ℓ≥1} \frac{sin(2ℓ + |k + 1|)φ}{sin φ} . ℓ^{−s}(ℓ + |k + 1|)^{−s} .

In particular, Z_{k}(g, s) = Z_{−k−2}(g, s). This is in fact an immediate consequence of the Poincar� duality.

IV) The derivative at zero of the Lerch zeta function Define for 0 < φ < 2π, Re s > 0 the zeta function ζ^{rot}(φ, s) by

ζ^{rot}(φ, s) := ∑_{ℓ=1}^{∞} \frac{sin ℓφ}{ℓ^{s}} .

ζ^{rot} continuous holomorphically to the whole complex plane. Let φ = 2π \frac{p}{q}, p, q ∈ ℕ, 0 < p < q be a rational angle and ζ(.,.) the Hurwitz zeta function. We obtain

ζ^{rot}(φ, s) = ∑_{j=1}^{q} ∑_{ℓ=0}^{∞} \frac{sin(ℓq + j)φ}{(ℓq + j)^{s}} = ∑_{j=1}^{q} \frac{sin jφ}{q^{s}} ∑_{ℓ=0}^{∞} (ℓ + \frac{j}{q} )^{−s}

= ∑_{j=1}^{q} \frac{sin jφ}{q^{s}}ζ(s, \frac{j}{q}) .

By using the equations (see for example [WW, Chap. XIII])

ζ(0, x) = \frac{1}{2} − x and \frac{∂}{∂s}_{| s=0}ζ(s, x) = log \frac{Γ(x)}{\sqrt{2π}}

we find

\frac{∂}{∂s}ζ^{rot}(φ, 0) = ∑_{j=1}^{q} sin jφ. (log \frac{Γ(\frac{j}{q})}{\sqrt{2π}} − log q. (\frac{1}{2} − \frac{j}{q} )) .

Because of ∑_{j=1}^{q} sin jφ = 0 and ∑_{j=1}^{q} \frac{j}{q} sin jφ = −\frac{1}{2} cot \frac{φ}{2} this is equal to

\frac{∂}{∂s}ζ^{rot}(φ, 0) = −\frac{1}{2} log q. cot \frac{φ}{2} + ∑_{j=1}^{q} sin jφ. log Γ(\frac{j}{q}) .

V) The derivative at zero for arbitrary angles

We are using Kummer’s Fourier series for the logarithm of the Γfunction

log Γ(x) = \frac{1}{2} log 2π+∑_{n≥1} (\frac{cos 2πnx}{2n}+\frac{C + log 2πn}{nπ} sin 2πnx) (0 < x < 1) .

With the orthogonal relations

∑_{j=1}^{q} sin \frac{2πjp}{q} cos \frac{2πjn}{q} = 0 ,
∑_{j=1}^{q} sin \frac{2πjp}{q} sin \frac{2πjn}{q} = \frac{q}{2} . (δ_{p≡n( mod q)} − δ_{p≡−n( mod q)})

and the Fourier series of the identity function

x log q = \frac{log q}{2} − ∑_{n≥1} \frac{log q}{nπ} sin 2πnx (0 < x < 1) ,

it follows that

\frac{∂}{∂s}ζ^{rot}(φ, 0) = \frac{q}{2} . [\frac{C + log 2π \frac{p}{q}}{pπ} + ∑_{n≥1} (\frac{C + log (2π \frac{nq+p}{q})}{(nq + p)π} − \frac{C + log (2π \frac{nq−p}{q})}{(nq − p)π})]

= \frac{C + log φ}{φ} + ∑_{n≥1} (\frac{C + log(2πn + φ)}{2πn + φ} − \frac{C + log(2πn − φ)}{2πn − φ}) .

We have the identities (see [WW] or Bismut and Soul� [B2, Appendix])

∑_{n≥1} (\frac{1}{n + x} − \frac{1}{n − x}) = π cot πx − \frac{1}{x} = −2 ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} ζ(ℓ + 1)x^{ℓ} ,
∑_{n≥1} (\frac{log n}{n + x} − \frac{log n}{n − x}) = 2x ∑_{n≥1} \frac{−log n}{n^{2}} ∑_{ℓ≥0} (\frac{x}{n})^{2ℓ} = 2 ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} ζ′(ℓ + 1)x^{ℓ}

and

∑_{n≥1} ( \frac{log(1 + \frac{x}{n})}{n + x} − \frac{log(1 − \frac{x}{n})}{n − x}) = ∑_{n≥1} \frac{2}{n} ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} \frac{x}{n}^{ℓ} ∑_{j=1}^{ℓ} \frac{1}{j}

= 2 ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} ζ(ℓ + 1) ∑_{j=1}^{ℓ} \frac{1}{j} . x^{ℓ} ,

so we obtain

\frac{∂}{∂s}ζ^{rot}(φ, 0) = \frac{C + log φ}{φ} + \frac{1}{π} ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} (\frac{ζ′(ℓ + 1)}{ζ(ℓ + 1)} + ∑_{j=1}^{ℓ} \frac{1}{j} − C − log 2π). ζ(ℓ + 1) . (\frac{φ}{2π} )^{ℓ}

= \frac{C + log φ}{φ} − ∑_{\begin{matrix}ℓ≥1 \\ odd \end{matrix}} ζ′(−ℓ)(−1)^{\frac{ℓ+1}{2}} \frac{φ^{ℓ}}{ℓ!} .

This gives the Proposition 1 by continuity.

VI) The torsion on ℙ^{1}ℂ

Recall now the zeta function Z_{k} of Lemma 7 with φ≠0. By a Taylor expansion of the denominator with respect to \frac{|k+1|}{ℓ}, we find for s ↘ 0

\frac{∂}{∂s}Z_{k}(g, s) = − ∑_{ℓ≥1} \frac{sin(2ℓ + |k + 1|)φ}{sin φ} . (\frac{log ℓ}{ℓ^{s}(ℓ + |k + 1|)^{s}} + \frac{log(ℓ + |k + 1|)}{ℓ^{s}(ℓ + |k + 1|)^{s}})

= − ∑_{ℓ≥1} \frac{sin(2ℓ + |k + 1|)φ}{sin φ} . \frac{log ℓ}{ℓ^{2s}} . (1 + \frac{|k + 1|}{ℓ})^{−s}

− ∑_{ℓ>|k+1|} \frac{sin(2ℓ − |k + 1|)φ}{sin φ} . \frac{log ℓ}{ℓ^{2s}} . (1 − \frac{|k + 1|}{ℓ})^{−s}

= − ∑_{ℓ≥1} \frac{2 cos |k + 1| φ sin 2ℓφ}{sin φ} . \frac{log ℓ}{ℓ^{2s}} + ∑_{ℓ=1}^{|k+1|} \frac{sin(2ℓ − |k + 1|)φ}{sin φ} . \frac{log ℓ}{ℓ^{2s}} + O(s)

= \frac{2 cos |k + 1| φ}{sin φ} \frac{∂}{∂s}ζ^{rot}(2φ, 2s) + ∑_{ℓ=1}^{|k+1|} \frac{sin(2ℓ − |k + 1|)φ}{sin φ} . \frac{log ℓ}{ℓ^{2s}} + O(s) ,

hence for s = 0

\frac{∂}{∂s}Z_{k}(g, 0) = \frac{2 cos |k + 1| φ}{sin φ}R^{rot}(2φ) + ∑_{ℓ=1}^{|k+1|} \frac{sin(2ℓ − |k + 1|)φ}{sin φ} log ℓ .

Remark that this computation breaks down for φ = 0 because of the singularity of the Riemann ζ -function. The isomorphism g corresponds to a rotation of the sphere by an angle 2φ, so we obtain Theorem 2.

VII) The zeta function on ℙ^{n}ℂ

Now we regard as in [IT] the complex projective space ℙ^{n}ℂ as the homogeneous space SU(n + 1)/S(U(1) × U(n)). Let

𝔥 := {( ..._{0  iφ_{n+1}}^{iφ_{1}  0} )| ∑_{1}^{n+1} φ_{j} = 0}

be the canonical maximal Cartan subalgebra of the Lie algebra 𝔰𝔲(n + 1). Let Λ_{j}, 1 ≤ j ≤ n, be the fundamental weight

Λ_{j} : diag(iφ_{1}, …, iφ_{n+1}) ↦ ∑_{1}^{j} \frac{φ_{k}}{2π} .

In the following, Λ(k, 0, q) denotes the irreducible SU(n+1) -representation with highest weight given by (k−q)Λ_{1}+Λ_{q}+kΛ_{n} for all k ≥ q, n ≥ q ≥ 0. Ikeda and Taniguchi found that the spaces

⨁_{k≥0} Λ(k, 0, 0) (q = 0)

⨁_{k≥q} Λ(k, 0, q) ⊕ ⨁_{k≥q+1} Λ(k, 0, q + 1)  (0 < q < n)

⨁_{k≥n} Λ(k, 0, n) (q = n)

can be regarded as L^{2} -dense subspaces of Γ(Λ^{q}T^{∗(0,1)}ℙ^{n}ℂ), where the Laplacian acts on Λ(k, 0, q) by multiplication with k(k + n + 1 − q). We denote by χ(k, 0, q) the character to the representation Λ(k, 0, q). Hence we find for our zeta function

Z(., s) = ∑_{q=1}^{n−1}(−1)^{q+1}q( ∑_{k≥q} \frac{χ(k, 0, q)}{k^{s}(k + n + 1 − q)^{s}} + ∑_{k≥q+1} \frac{χ(k, 0, q + 1)}{k^{s}(k + n − q)^{s}} )

+(−1)^{n+1}n ∑_{k≥n} \frac{χ(k, 0, n)}{k^{s}(k + 1)^{s}}

= ∑_{q=1}^{n}(−1)^{q+1} ∑_{k≥q} \frac{χ(k, 0, q)}{k^{s}(k + n + 1 − q)^{s}} .

The “telescope” effect in the summation is not caused by accident, but by the natural splitting of each eigenspace Eig_{λ}(□) into Eig_{λ}(□) ∩ ker \overline{∂} and Eig_{λ}(□) ∩ ker \overline{∂}^{∗}, which are isomorphic. The character χ_{Λ} of an irreducible SU(n + 1)-module with highest weight Λ = m_{1}Λ_{1} + m_{2}(Λ_{2} − Λ_{1}) + … + m_{n}(Λ_{n} − Λ_{n−1}), m_{1} ≥ … ≥ m_{n} ≥ m_{n+1} = 0, can classically be calculated by Weyl’s character formula. One finds with e_{j} := e^{iφ_{j}}

χ_{Λ}( ..._{0  iφ_{n+1}}^{iφ_{1}  0} ) = \frac{det(e_{j}^{m_{ℓ}+n+1−ℓ})_{j,ℓ=1}^{n+1}}{det(e_{j}^{n+1−ℓ})_{j,ℓ=1}^{n+1}} .

In our case one gets after a rotation of the first q rows

χ(k, 0, q) = exceptional → q -th row | \begin{matrix}e_{1}^{n} & … & e_{n+1}^{n} \\ ⋮ & & ⋮ \\ e_{1}^{n+1−(q−1)} & & e_{n+1}^{n+1−(q−1)} \\ e_{1}^{n+1−q+k} & & e_{n+1}^{n+1−q+k} \\ e_{1}^{n+1−(q+1)} & & e_{n1+}^{n+1−(q+1)} \\ ⋮ & & ⋮ \\ e_{1} & & e_{n+1} \\ e_{1}^{−k} & … & e_{n+1}^{−k} \end{matrix} | : | \begin{matrix}e_{1}^{n} & … & e_{n+1}^{n} \\ ⋮ & & ⋮ \\ 1 & … & 1 \end{matrix} |.(−1)^{q+1} .

We see immediately

χ(k − (n + 1 − q), 0, q) = −χ(−k, 0, q)

and χ(k, 0, q) = 0 for k ∈ {−n, …, q − 1} \ {0, q − n − 1}.

VIII) The torsion on ℙ^{n}ℂ

Remark that χ(k, 0, q) as a function in k can be regarded as a linear combination of exponentials exp ik(φ_{j} − φ_{ℓ}) with 1 ≤ j, ℓ ≤ n + 1. So the function

∑_{k≥1} \frac{log k}{k^{2s}} χ(k, 0, q)

is a linear combination of Lerch ζ -functions. Hence it follows, if all the φ_{j} are distinct, for s ↘ 0

Z′(., s) = ∑_{q=1}^{n}(−1)^{q+1}( ∑_{k≥q} \frac{χ(k, 0, q)logk}{k^{s}(k + n + 1 − q)^{s}} − ∑_{k≥n+1} \frac{χ(−k, 0, q)log k}{k^{s}(k − n − 1 + q)^{s}} )

= ∑_{q=1}^{n}(−1)^{q+1}( ∑_{k≥1} (χ(k, 0, q) − χ(−k, 0, q)) \frac{log k}{k^{2s}}

+ \frac{log(n + 1 − q)}{(n + 1 − q)^{2s}}χ(q − n − 1, 0, q)) + O(s)

= ∑_{q=1}^{n}(−1)^{q+1} ∑_{k≥1} (χ(k, 0, q) − χ(−k, 0, q)) \frac{log k}{k^{2s}} − log n! + O(s) ,

because of χ(q − n − 1, 0, q) = (−1)^{q}. The Laplace expansion theorem for determinants shows

∑_{q=1}^{n}(−1)^{q+1}χ(k, 0, q)= 1 − ∑_{j=1}^{n+1} e_{j}^{k} | \begin{matrix}e_{1}^{n} & … & e_{n+1}^{n} \\ ⋮ & & ⋮ \\ e_{1} & … & e_{n+1} \\ e_{1}^{−k} & … & e_{n+1}^{−k} \end{matrix} | : | \begin{matrix}e_{1}^{n} & … & e_{n+1}^{n} \\ ⋮ & & ⋮ \\ 1 & … & 1 \end{matrix} | .

Hence we obtain some Vandermonde determinants:

∑_{q}(−1)^{q+1}(χ(k, 0, q) − χ(−k, 0, q)) =

− ∑_{\begin{matrix}j,ℓ=1 \\ j≠ℓ \end{matrix}}^{n+1} ((\frac{e_{j}}{e_{ℓ}} )^{k} − (\frac{e_{ℓ}}{e_{j}} )^{k})(−1)^{n+ℓ} | \begin{matrix}e_{1}^{n} & … & e_{ℓ}^{n} & … & e_{n+1}^{n} \\ ⋮ & & & & ⋮ \\ e_{1} & … & e_{ℓ} & … & e_{n+1} \end{matrix} | : | \begin{matrix}e_{1}^{n} & … & e_{n+1}^{n} \\ ⋮ & & ⋮ \\ 1 & … & 1 \end{matrix} |

= (−1)^{n} ∑_{j,ℓ=1}^{n+1} ((\frac{e_{j}}{e_{ℓ}} )^{k} − (\frac{e_{ℓ}}{e_{j}} )^{k}) ∏_{\begin{matrix}k=1 \\ k≠ℓ \end{matrix}}^{n+1} (\frac{e_{ℓ}}{e_{k}} − 1)^{−1}

(the indicates that the ℓ-th column is missing). By using

(\frac{e_{j}}{e_{ℓ}} )^{k} − (\frac{e_{ℓ}}{e_{j}} )^{k} = 2i sin k(φ_{j} − φ_{ℓ})

and the definition of R^{rot}(φ), we find Theorem 3.

IX) Remarks about the function R^{rot}

The function R^{rot} has a rather simple definition and hence a lot of special properties. Here we only give a few of them.

Theorem 8. The following identities hold

(1) R^{rot}(φ) = −R^{rot}(2π − φ)  (0 < φ < 2π) ,
(2) 2R^{rot}(2φ) = R^{rot}(φ) + R^{rot}(π + φ) + log 2 . cot φ (0 < φ < π) ,

(3) 3R^{rot}(3φ) = R^{rot}(φ) + R^{rot}(\frac{2π}{3} + φ)

− R^{rot}(\frac{2π}{3} − φ) + \frac{3}{2} log 3 . cot \frac{3φ}{2}  (0 < φ < \frac{2π}{3} ) ,
(4) R^{rot}(π + φ) = ∫_{0}^{∞} log x \frac{sinh φx}{sinh πx}dx  (−π < φ < π) .

Proof: 1) is trivial by the definition of R^{rot} . 2) follows from

2^{1−s}ζ^{rot}(2φ, s) = ζ^{rot}(φ, s) + ζ^{rot}(π + φ, s) .

We see by the formulas of § IV that ζ^{rot}(φ, 0) = \frac{1}{2} cot \frac{φ}{2} . The result follows then by derivation. In the same way, one gets 3) from

3^{1−s}ζ^{rot}(3φ, s) = ζ^{rot}(φ, s) + ζ^{rot}(\frac{2π}{3} + φ) − ζ^{rot}(\frac{2π}{3} − φ) .

To see the integral formula 4) we are using the Fourier series

−\frac{π}{2} \frac{sinh φx}{sinh πx} = ∑_{1}^{∞} \frac{(−1)^{ℓ}ℓ}{x^{2} + ℓ^{2}} sin ℓφ (|φ| < π)

and the definite integral

∫_{0}^{∞} \frac{x^{−s}dx}{x^{2} + ℓ^{2}} = \frac{π}{2ℓ^{1+s} cos \frac{sπ}{2}} (|s| < 1) .

We have for |s| < 1.

ζ^{rot}(π + φ, s) = ∑_{1}^{∞} \frac{(−1)^{ℓ} sin ℓφ}{ℓ^{s}} = \frac{2}{π} cos \frac{πs}{2} ∑_{1}^{∞} ∫_{0}^{∞} \frac{(−1)^{ℓ}ℓx^{−s}dx}{x^{2} + ℓ^{2}} sin ℓφ

= − cos \frac{πs}{2} ∫_{0}^{∞} x^{−s} \frac{sinh φx}{sinh πx}dx .

The desired result follows. □

Acknowledgement: I would like to thank Prof. J.-M. Bismut for sharing the idea of this problem. Also I would like to thank Prof. C. Deninger for helpful comments. This article is a part of the author’s thesis.

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