Equivariant Reidemeister torsion on symmetric spaces

Kai Köhler Mathematisches Institut der Universität Bonn Beringstr. 1 D-53115 Bonn

e-mail: Koehler@Rhein.IAM.Uni-Bonn.De

January 5, 1996

Abstract

We calculate explicitly the equivariant Ray-Singer torsion for all symmetric spaces G/K of compact type with respect to the action of G. We show that it equals zero except for the odd-dimensional Graßmannians and the space SU(3)/SO(3). As a corollary, we classify up to diffeomorphism all isometries of these spaces which are homotopic to the identity; also, we classify their quotients by finite group actions up to homeomorphism.

Contents

1 Introduction 3 2 Equivariant Ray-Singer metrics 5 3 Homogeneous and symmetric spaces 6 4 The zeta function for symmetric spaces 8 5 The torsion for symmetric spaces 11 6 The classification 15

1 Introduction

In 1935, Franz and Reidemeister [9] established the following classification of lens spaces :

Theorem 1 (Franz, Reidemeister) Let Γ_{1}, Γ_{2} be cyclic groups acting isometrically and freely on the spheres S^{2n−1}, n > 1. Then the quotients Γ_{1}\S^{2n−1}, Γ_{2}\S^{2n−1} are diffeomorphic iff they are isometric, i.e. if Γ_{1} and Γ_{2} are conjugate in O(2n).

To prove this theorem they invented a real-valued combinatorial invariant of CW-complexes which is fine enough to distinguish the lens spaces, the Reidemeister torsion. Their result was generalized by de Rham [8] in 1964:

Theorem 2 (de Rham) Two isometries of S^{n} are diffeomorphic iff they are isometric.

Here, two transformations g_{1}, g_{2} of a manifold M are called diffeomorphic (resp. isometric) iff there exists a diffeomorphism (resp. an isometry) ϕ of M with ϕg_{1} = g_{2}ϕ. In 1973, Chapman [5] proved that the Reidemeister torsion is in fact a homeomorphism invariant; thus theorem 1 holds with “diffeomorphic” replaced by “homeomorphic”. In contrast, there exist counterexamples showing that this is not true for theorem 2 [4]. The first purpose of this article is to prove the following result:

Theorem 3 Two isometries homotopic to the identity of an odd-dimensional Graßmannian G_{2m,2p−1}(R) = SO(2m)/SO(2p − 1) × SO(2m − 2p + 1) or of SU(3)/SO(3) are diffeomorphic iff they are isometric.

Note that the connected component of the identity in the isometry group of G_{2m,2p−1} and SU(3)/SO(3) is given by SO(2m) and SU(3), resp. The isometry group consists of two or four copies of these groups [18].

The proof is given by calculating explicitly the equivariant Ray-Singer torsion for all compact symmetric spaces. The Ray-Singer torsion is defined as the derivative at zero of a certain zeta function associated to the spectrum of the Laplace operator on differential forms on a compact Riemannian manifold [17]. It has been determined by Ray for the lens spaces in an extensive calculation by determining first the eigenvalues and eigenspaces of the Laplacian on spheres. He found that the Reidemeister and Ray-Singer torsions are equal for these spaces. Using this result, Cheeger [6] and Müller [15] proved independently in 1978 the equality of the Reidemeister torsion and the Ray-Singer torsion. The second aim of our paper is to give a new, shorter proof of Ray‘s result.

Using the equality of Reidemeister torsion and Ray-Singer torsion and the topological invariance of the Reidemeister torsion, we show in the last section

Theorem 4 Let Γ_{1}, Γ_{2} ⊂ SO(2m) be finite groups acting isometrically and freely on an odd-dimensional Graßmannian G_{2m,2p−1}(R) (m > 1). Then the quotients Γ_{1}\G_{2m,2p−1}(R) and Γ_{2}\G_{2m,2p−1}(R) are homeomorphic iff they are isometric (more precisely: iff Γ_{1}, Γ_{2} are conjugate in O(2m)).

Let Γ_{1}, Γ_{2} ⊂ SU(3) be finite groups acting isometrically and freely on SU(3)/SO(3). Then the quotients are homeomorphic iff they are isometric (i.e. iff Γ_{1} is conjugate in SU(3) to Γ_{2} or \overline{Γ_{2}}).

Note that fixed-point free isometries of the above spaces are necessarily homotopic to the identity, except for G_{4p−2,2p−1}. Also, there are only a few sporadic fixed-point free isometries on even-dimensional Graßmannians, none of them homotopic to the identity [18, Th. 9.3.1]. Thus, the above theorem gives a rather complete topological classification of quotients of Graßmannians.

The equivariant Ray-Singer torsion associated to an isometry g acting on M has been investigated by Lott and Rothenberg. They compared it with an equivariant Reidemeister torsion for finite group actions. Using Ray‘s calculation, they found that the equivariant torsion for spheres is mainly given by sums of the digamma function; this enabled them to give a new proof of theorem 2 for orientation-preserving actions on odd-dimensional spheres. We shall apply their method to deduce theorem 3 from our result for the torsion.

Also, we shall show that the equivariant torsion equals zero for all symmetric spaces G/K with respect to the action of any g ∈ G, except for products of G_{2m,2p−1}(R) or SU(3)/SO(3) with some G′/K′ so that G′ and K′ have the same rank. A similar result has been shown by Moscowici and Stanton for locally symmetric spaces of the noncompact type [14].

Our method to obtain the value of the torsion is similar to the one used in a previous paper on holomorphic Ray-Singer torsion on Hermitian symmetric spaces [12]. For a symmetric space G/K an eigenvalue of the Laplacian is determined by its eigenspace as a G-representation. This reduces the problem of determining the zeta function to a problem in finite-dimensional representation theory. Nevertheless, there are big differences between the real and the holomorphic situation: In the complex case, the equivariant torsion is always non-trivial and depends in a rather subtle way on the fixed-point set of the isometry g ∈ G, in sharp contrast to our result theorem 11.

2 Equivariant Ray-Singer metrics

Let F be a complex flat hermitian vector bundle over a compact oriented Riemannian manifold M. Let

d : Γ(Λ^{q}T^{∗}M ⊗ F) → Γ(Λ^{q+1}T^{∗}M ⊗ F)

denote the de Rham operator with coefficients in F and let d^{∗} denote its formal adjoint with respect to the L^{2}-metric. Consider the Hodge-Laplacian Δ_{q} := (d + d^{∗})^{2} acting on q-forms with coefficients in F. We denote the eigenspace of Δ_{q} corresponding to an eigenvalue λ ∈ Spec Δ_{q} by Eig_{λ}(Δ_{q}). Let g be an isometry of M preserving the hermitian bundle F. Consider the zeta function

Z_{g}(s) := ∑_{q>0}(−1)^{q}q ∑_{{λ∈SpecΔq \atop λ≠0}} λ^{−s} Tr g_{|Eig_{λ}(Δ_{q})}^{∗}

for ℜe s > dim M/2. Classically, this zeta function has a meromorphic continuation to the complex plane which is holomorphic at zero.

Definition 1 The equivariant analytic torsion is defined as

τ_{g}(M, F) := e^{− \frac{1}{2} Z′_{g}(0)}.

This object has been defined by Ray [16]. We shall denote the torsion with coefficients in the trivial line bundle by τ_{g}(M). Ray showed the following property of τ_{g}: Consider a free action of a finite group Γ on M. Let ρ : Γ → U(1) be an unitary representation, thus defining a flat hermitian line bundle F on Γ\M. Then the usual non-equivariant Ray-Singer torsion with coefficients in F is given by

log τ(Γ\M, F) = \frac{1}{#Γ} ∑_{g∈Γ} ρ̄(g) log τ_{g}(M).

This is known to be a topological invariant if the cohomology H^{∗}(M, F) with coefficients in F vanishes. The equivariant torsion has been investigated by Lott and Rothenberg [13] for flat metrics on F. They showed that it equals zero on even dimensional manifolds or for orientation reversing actions on odd dimensional manifolds. They proved the result

Theorem 5 (Lott, Rothenberg) Assume that g is homotopic to the trivial action. Choose a sequence (f_{ν}) ∈ ∐_{ν∈Z} R, ∑ f_{ν} = 0. Then the torsion of powers of g weighted with (f_{ν})

∑ f_{ν}τ_{g^{ν}}(M)

is a diffeomorphism invariant.

This result has been generalized strongly by Bismut and Zhang [2].

3 Homogeneous and symmetric spaces

Let G be a connected compact Lie group and let K be a connected compact subgroup. Let T_{G} ⊃ T_{K} denote the maximal tori of G and K. We denote the Lie algebras of G, K, T_{G} and T_{K} by 𝔤, 𝔨, 𝔱_{G} and 𝔱_{K}, respectively. We fix compatible orderings on 𝔱_{G}^{∗} and 𝔱_{K}^{∗}. The action of K on the homogeneous space G/K induces a representation Ad_{G/K} on the tangent space of G/K at the class of [1] ∈ G, i.e. on 𝔤/𝔨. Let Ψ denote the set of weights of this representation, the isotropy representation, and let Δ_{G} and Δ_{K} be the sets of roots of G resp. K. Then the weights of the adjoint representation of G on 𝔤 are given by Δ_{G} and the weight {0} with multiplicity the rank of G. The weights of the action of K on 𝔤 are given by Δ_{K}, Ψ and rk K-times the {0}, thus

(Δ_{G} ∪ {0} · rk G)_{|𝔱_{K}} = Δ_{K} ∪ Ψ ∪ {0} · rk K  (1)

(counted with multiplicity). In particular, the dimension #Ψ of G/K is odd-dimensional iff rk G − rk K is so. The space of forms Γ(Λ^{q}T^{∗}G/K) is an infinite dimensional G-representation which contains the space of its irreducible subrepresentations (V_{π}, π) as a L^{2}-dense subspace. Thus,

Γ(Λ^{q}T^{∗}G/K) ⊃^{dense} ⨁_{π} Hom_{G} (V_{π}, Γ(Λ^{q}T^{∗}G/K)) ⊗ V_{π}.  (2)

In this imbedding, the tensor product Hom_{G} (V_{π}, Γ(Λ^{q}T^{∗}G/K)) ⊗ V_{π} is the direct sum of dim Hom_{G} (V_{π}, Γ(Λ^{q}T^{∗}G/K)) copies of the representations (V_{π}, π). By a Frobenious law due to Bott [3], there is a canonical isomorphism

Hom_{G} (V_{π}, Γ(Λ^{q}T^{∗}G/K)) ≅ Hom_{K} (V_{π}, Λ^{q}Ad_{G/K})  (3)

(Note that (𝔤/𝔨 ⊗ C)^{∗} ≅ 𝔤/𝔨 ⊗ C via the metric). In particular, the representations (V_{π}, π) which occur are finite dimensional.

Let (X_{1}, …, X_{N}) be an orthonormal basis of 𝔤 with respect to the negative Killing form. The Casimir operator of 𝔤 is defined as the following element of the universal enveloping algebra of 𝔤

Cas := − ∑ X_{j}.X_{j} .  (4)

Ikeda and Taniguchi proved the following beautiful result [10]:

Theorem 6 (Ikeda, Taniguchi) Assume that G/K is a symmetric space equipped with the metric induced by the Killing form. Then the Laplacian Δ acts on the V_{π}’s as Cas with respect to the imbedding (2).

The Casimir is known to act by multiplication with a constant on irreducible representations. Thus, the eigenspaces of the Laplacian correspond to the irreducible representations π with multiplicity dim Hom_{K}(V_{π}, Λ^{q}Ad_{G/K}) and its eigenvalue there depends only on π.

Let ρ_{G} := \frac{1}{2} ∑_{α∈Δ^{+}} α be half the sum of the positive roots of G and let W_{G} be the Weyl group of G. Let ⟨·, ·⟩ and ∥ · ∥ denote the metric and the norm on 𝔱_{G}^{∗} induced by the Killing form. We denote the sign of an element w ∈ W_{G} by (−1)^{w}. As usual, we define

(α, ρ_{G}) := \frac{2⟨α, ρ_{G}⟩}{∥α∥^{2}}  (5)

for any weight α. For an irreducible representation π we denote by b_{π} the sum of its highest weight and ρ_{G}. Then, classically, the action of the Casimir is given by

π(Cas) = ∥b_{π}∥^{2} − ∥ρ_{G}∥^{2}.

To abbreviate we set

Alt_{G}{b} := ∑_{w∈W_{G}}(−1)^{w}e^{2πiwb}.

Then the Weyl character formula for the character χ_{b_{π}} of the representation evaluated at t ∈ T_{G} may be written as

χ_{b_{π}}(t) = \frac{Alt_{G}\{b_{π}\}(t)}{Alt_{G}\{ρ_{G}\}(t)}.

This formula provides the definition of the so-called virtual (or formal) character χ_{b} for any b equal to ρ_{G}+some weight. This extends to all of G by setting χ_{b} to be invariant under the adjoint action. The corresponding virtual representation shall be denoted by V_{b}. Occasionally we shall use the notation χ_{b}^{G}, χ_{b}^{K}, V_{b}^{G}, V_{b}^{K} to distinguish G- and K-representations. From now on we shall consider the irreducible symmetric space G/K as being equipped with any G-invariant metric ⟨·, ·⟩_{⋄}. All these metrics are proportional to the metric induced by the Killing form [1, Th. 7.44] on the irreducible factors of G/K. We shall denote the dual metric and norm on 𝔱_{G}^{∗} by ⟨·, ·⟩_{⋄}, ∥ · ∥_{⋄}, too.

4 The zeta function for symmetric spaces

By theorem 6, the equivariant zeta function defining the torsion is given by

Z(s) = ∑_{q=1}^{n}(−1)^{q}q ∑_{π} \frac{dim Hom_{K}(V_{π}, Λ^{q}Ad_{G/K})}{(∥b_{π}∥_{⋄}^{2} − ∥ρ_{G}∥_{⋄}^{2})^{s}}χ_{b_{π}}.  (6)

In the case K = {1}, dim G = n > 1, we observe that

Z(s) = (∑_{q=1}^{n}(−1)^{q}q dim Λ^{q}𝔤) ∑_{π} \frac{dim V_{π} · χ_{b_{π}}}{(∥b_{π}∥_{⋄}^{2} − ∥ρ_{G}∥_{⋄}^{2})^{s}}
= −n(1 − 1)^{n−1} ∑_{π} \frac{dim V_{π} · χ_{b_{π}}}{(∥b_{π}∥_{⋄}^{2} − ∥ρ_{G}∥_{⋄}^{2})^{s}} .

Thus the torsion τ_{g}(G) equals zero for all compact Lie groups except the circle. Our key result is the following

Lemma 7 Let G/K be a n-dimensional homogeneous space.

• If rk G > rk K + 1 then the virtual representation

∑_{q=1}^{n}(−1)^{q}qΛ^{q}Ad_{G/K}

is trivial.

• Assume that rk G = rk K + 1 and let L denote the line of those weights of G which restrict to zero on 𝔱_{K}. If (V_{π}, π) is an irreducible Grepresentation then the sum

∑_{q=1}^{n}(−1)^{q}q dim Hom_{K}(V_{π}, Λ^{q}Ad_{G/K}) · χ_{b_{π}}

equals the sum of −χ_{ρ_{G}+wα} over those [w] ∈ W_{G}/W_{K} and α ∈ L such

that b_{π} lies in the W_{G}-orbit of ρ_{G} + wα.

By this lemma and theorem 6 we get the following expression for the equivariant zeta function Z:

Lemma 8 For any odd-dimensional symmetric space G/K, the zeta function Z(s) is given by

Z(s) = ∑_{[w]∈W_{G}/W_{K} ; {α∈L \atop ⟨wα,wα+2ρ_{G}⟩>0}} \frac{−χ_{ρ_{G}+wα}}{⟨wα, wα + 2ρ_{G}⟩_{⋄}^{s}} = \frac{1}{#W_{K}} ∑_{w∈W_{G} ; {α∈L \atop ⟨wα,wα+2ρ_{G}⟩>0}} \frac{−χ_{ρ_{G}+wα}}{⟨wα, wα + 2ρ_{G}⟩_{⋄}^{s}}

if rk G = rk K + 1 and zero otherwise.

In particular, one observes the following consequence:

Corollary 9 Let G/K_{1}, G/K_{2} be two symmetric spaces with G-conjugate tori T_{K_{1}} and T_{K_{2}}. Then the associated zeta functions Z_{1} and Z_{2} are proportional by the factor #W_{K_{1}}/#W_{K_{2}}.

This shows already the desired classification of SO(2m)-actions on odddimensional Graßmannians G_{2m,2p−1}(R).

Theorem 10 Two isometries g_{1}, g_{2} ∈ SO(2m) of the Graßmannian G_{2m,2p−1} are conjugate by a diffeomorphism iff they are conjugate in O(2m). If g_{1}, g_{2} act fixed-point free, than they are conjugate by a homeomorphism iff they are conjugate in O(2m).

Proof The maximal tori of SO(2p − 1) × SO(2m − 2p + 1) are conjugate in SO(2m) for all p. By corollary 9, the equivariant torsion of G_{2m,2p−1} at g ∈ SO(2m) equals the torsion of S^{2m−1} at g up to a non-zero constant. In [13], Lott and Rothenberg proved that an element of SO(2m) is determined up to conjugacy in O(2m) by certain linear combinations of this torsion (see the next section). Thus, the result for the spheres extends to all Graßmannians.

□

Proof of lemma 7 Let χ^{K} denote the virtual character in the representation ring of K given by

χ^{K} := ∑_{q=1}^{n}(−1)^{q}qχ (Λ^{q}Ad_{G/K})

where χ (Λ^{q}Ad_{G/K}) denotes the character of the K-representation Λ^{q}Ad_{G/K}. By classical representation theory, one knows

∑_{q=1}^{n}(−1)^{q}q dim Hom_{K}(V_{π}, Λ^{q}Ad_{G/K}) = ∫_{K} \overline{χ^{K}} · χ_{π}dvol_{K}.  (7)

Using the Weyl integral formula this transforms to

\frac{1}{#W_{K}} ∫_{T_{K}} \overline{Alt_{K}\{ρ_{K}\}}Alt_{K}\{ρ_{K}\}\overline{χ^{K}} · χ_{π}dvol_{T_{K}}

(where T_{K} is identified with the quotient of 𝔱_{K} by the integral lattice). Classically the restriction of Alt_{G}{ρ_{G}} and Alt_{K}{ρ_{K}} to 𝔱_{K} is given by

Alt_{G}{ρ_{G}}_{|𝔱_{K}} = ∏_{α∈Δ_{G}^{+}} 2i sin πα_{|𝔱_{K}}

and

Alt_{K}{ρ_{K}} = ∏_{α∈Δ_{K}^{+}} 2i sin πα.

Equation (1) shows that the restriction of χ^{K} is given by

χ^{K} = \frac{∂}{∂s}_{|s=1} det(1 − sAd_{G/K})_{|𝔱_{K}}  (8)
= \frac{∂}{∂s}_{|s=1}(1 − s)^{rk G−rk K} ∏_{α∈Δ_{G}−Δ_{K}} (1 − se^{2πiα})  (9)

= {− ∏_{α∈Δ_{G}−Δ_{K}} (1 − e^{2πiα}) ; 0 if rk G = rk K + 1 rk G > rk K + 1  (10)

This shows the first part of lemma 7. Assume now that rk G = rk K + 1. Then

χ^{K}_{|𝔱_{K}} = −\frac{∏_{α∈Δ_{G}} 2i sin πα_{|𝔱_{K}}}{∏_{α∈Δ_{K}} 2i sin πα} = −\frac{\overline{Alt_{G}\{ρ_{G}\}}Alt_{G}\{ρ_{G}\}}{\overline{Alt_{K}\{ρ_{K}\}}Alt_{K}\{ρ_{K}\}},

hence equation (7) yields

∑_{q=1}^{n}(−1)^{q}q dim Hom_{K}(V_{π}, Λ^{q}Ad_{G/K}) = −\frac{1}{#W_{K}} ∫_{T_{K}} \overline{Alt_{G}\{ρ_{G}\}}Alt_{G}\{b_{π}\}dvol_{T_{K}}

which finishes the proof of lemma 7. □

5 The torsion for symmetric spaces

Classically, a compact symmetric space decomposes as a Riemannian manifold into a product of irreducible symmetric spaces G_{ν}/K_{ν}, where the metric on G_{ν}/K_{ν} is induced by a negative real number times the Killing form of G_{ν}. By lemma 8 the equivariant torsion is non-zero only if rk G = rk K+1. Also it is zero for all Lie groups except the circle. By the classification of irreducible symmetric spaces, among them only the odd-dimensional Graßmannians

G_{2m,2p−1}(R) = SO(2m)/SO(2p−1)×SO(2m−2p+1) (m, p ∈ N, m ≥ p).

and the 5-dimensional space SU(3)/SO(3) can have non-zero torsion. Thus, the torsion is non-zero only for the spaces

G_{2m,2p−1} × G′/K′ and SU(3)/SO(3) × G′/K′, (T)

where G′/K′ is an arbitrary symmetric space with rk G′ = rk K′. We imbed SO(2p−1)×SO(2m−2p+1) in SO(2m) as K = \{\binom{A 0}{0 B}|A ∈ SO(2p−1), B ∈ SO(2m − 2p + 1)}. To diagonalize the standard maximal torus, we imbed

SO(3) in SU(3) after conjugation by the matrix ( {−i \atop i} ; 0 1 1 0 0 0 ; \sqrt{2}) /\sqrt{2}. As maximal tori of G we choose

𝔱_{SO(2m)} := \{2π ( ..._{{0 −λ_{m} \atop λ_{m} 0}}^{{0 −λ_{1} \atop λ_{1} 0}} ) |λ_{1}, …, λ_{m} ∈ R\}

and

𝔱_{SU(3)} := \{2πi ( \begin{matrix}λ_{1} \\ & λ_{2} \\ & & λ_{3} \end{matrix} ) |λ_{1} + λ_{2} + λ_{3} = 0\}.

Let e_{ν} ∈ 𝔱_{SO(2m)}^{∗}, 1 ≤ ν ≤ m resp. f_{ν} ∈ 𝔱_{SU(3)}^{∗}, 1 ≤ ν ≤ 3 denote the weight mapping one of the above matrices to λ_{ν}, ordered according to their index. We set 𝔱_{K} as the kernel of e_{p} resp. f_{3}; thus, these weights generate L ≅ Z. They shall be denoted by α_{0}. The orbit of α_{0} under W_{G}/W_{K} is then given by {±e_{ν}}_{ν=1}^{m} and {f_{ν}}_{ν=1}^{3}, respectively.

Let ψ_{even} : R/Z → R denote the map

[x] ↦ C + \frac{1}{2}(ψ_{0}([x]) + ψ_{0}([−x]))

where C denotes Euler’s constant and ψ_{0} is the digamma function evaluated on the fundamental domain ]0, 1], i.e. ψ_{0}([x]) := ψ(x + ⌈1 − x⌉) with ⌈x⌉ the integer part of x. We show now the following formula for the equivariant Ray-Singer torsion:

Theorem 11 Let G/K be a space of type (T) which has no circle as irreducible component. Let α_{0} be a generator of L. Then the logarithm of the equivariant torsion for t ∈ T_{G} is given by

−\frac{1}{2}Z′_{t}(0) = log ∏_{[w]∈W_{G}/W_{K}} \frac{⟨wα_{0}, ρ_{G}⟩_{⋄}}{π} − ∑_{[w]∈W_{G}/W_{K}} ψ_{even}(wα_{0}(t))

and for general g ∈ G it is obtained by conjugating g into T_{G}.

Using the formula for the values of ψ at rational numbers this reproves Ray’s result for the lens spaces [16]. Before proving the theorem we show two auxiliary lemmas. Let S_{β} denote the reflection of 𝔱_{G}^{∗} in the hyperplane orthogonal to β ∈ 𝔱_{G}^{∗}. We need the following symmetry:

Lemma 12 Let G/K be a space of the type (T). Then, for all [w] ∈ W_{G}/W_{K} there exists w̃ ∈ W_{G} with (−1)w̃ = 1 such that

S_{wα_{0}}ρ_{G} = w̃ρ_{G}.

In particular, (wα_{0}, ρ_{G}) = −(w̃^{−1}wα_{o}, ρ_{g}) is an integer and the map [w] ↦ [w̃^{−1}w] ∈ W_{G}/W_{K} is bijective.

Proof The proof reduces to that for the cases G_{2m,2p−1} and SU(3)/SO(3). Classically, ρ_{SO(2m)} = ∑_{ν=1}^{m}(m − ν)e_{ν} and ρ_{SU(3)} = f_{1} − f_{3}. Hence, S_{wα_{0}}ρ_{G} is a weight and thus given by w̃ρ_{G} for some unique w̃ ∈ W_{G}. One observes immediately that (−1)w̃ = 1. By the invariance of ⟨·, ·⟩ under reflections, one shows

⟨wα_{0}, ρ_{G}⟩ = ⟨S_{wα_{0}}wα_{0}, S_{wα_{0}}ρ_{G}⟩ = −⟨wα_{0}, w̃ρ_{G}⟩ = −⟨w̃^{−1}wα_{0}, ρ_{G}⟩;

as S_{wα_{0}}ρ_{G} = ρ_{G} − (wα_{0}, ρ_{G})wα_{0}, one notices that (wα_{0}, ρ_{G}) is an integer. As S_{w̃^{−1}wα_{0}}ρ_{G} = w̃^{−1}S_{wα_{0}}(w̃ρ_{G}) = w̃^{−1}ρ_{G}, the map [w] ↦ [w̃^{−1}w] is an involution. □

Also, we need a lemma about values of zeta functions at zero. For p, n ∈ Z and h, ϕ ∈ R, h > 0, let ζ̃_{n,p,h}(s, ϕ) denote the zeta function

ζ̃_{n,p,h}(s, ϕ) := ∑_{{k∈Z \atop k(k+p)>0}} \frac{k^{n}e^{ikϕ} log(hk)^{2}}{(k(k + p))^{s}} + ∑_{{k∈Z\\{0\} \atop k(k+p)≤0}} \frac{k^{n}e^{ikϕ} log(hk)^{2}}{|k|^{2s}}

for ℜe s > \frac{n+1}{2}. This zeta function has a meromorphic extension to the complex plane.

Lemma 13 The value at zero of ζ̃ is independent of p. For n = 0 it takes the value

\frac{1}{2}ζ̃_{0,p,h}(0, ϕ) = log \frac{2π}{h} + ψ_{even}([\frac{ϕ}{2π}]).

Proof The proof may be given easily by applying the more general result of [12, Lemma 7]; instead, for the case ϕ ≢ 0 mod 2π we shall give an alternative proof which is better adapted to this particular situation. Choose ϕ ∈]0, 2π[. First, observe that in this case

ζ̃_{n,p,h}(s, ϕ) = i^{−n−1} (\frac{∂}{∂ϕ})^{n+1} ζ̃_{−1,p,h}(s, ϕ)

for n ≥ 0. For n < 0, the series defining ζ̃_{n,p,h}(s, ϕ) converges at s = 0. Clearly its value there does not depend on p. Using Kummer’s Fourier series for the logarithm of the Gamma function [11, sect. 5] one finds

ζ̃_{−1,p,h}(0, ϕ) = ∑_{k>0} \frac{4i sin kϕ log hk}{k}
= 2i(C + log \frac{2π}{h})(ϕ − π) + 2πi log Γ(\frac{ϕ}{2π}) − 2πi log Γ(1 − \frac{ϕ}{2π}).

To prove the lemma for the case ϕ ≡ 0 one may use the Taylor expansion for |k + p|^{−s} = |k|^{−s}|1 + p/k|^{−s} as in [11, sect. 6] or [12]. □

Proof of theorem 11 The derivative of Z(s) is given by

Z′(s) = ∑_{[w]∈W_{G}/W_{K} ; {k∈Z \atop k(k+(wα_{0},ρ_{G}))>0}} \frac{χ_{ρ_{G}+kwα_{0}} (log |k| + log |k + (wα_{0}, ρ_{G})| + log ∥α_{0}∥_{⋄}^{2})}{(k(k + (wα_{0}, ρ_{G}))∥α_{0}∥_{⋄}^{2})^{s}}

= ∑_{[w]∈W_{G}/W_{K} ; {k∈Z \atop k(k+(wα_{0},ρ_{G}))>0}} \frac{χ_{ρ_{G}+kwα_{0}} log |k|}{(k(k + (wα_{0}, ρ_{G}))∥α_{0}∥_{⋄}^{2})^{s}}

+ ∑_{[w]∈W_{G}/W_{K} ; {k∈Z \atop k(k−(wα_{0},ρ_{G}))>0}} \frac{χ_{ρ_{G}+(k−(wα_{0},ρ_{G}))wα_{0}} log |k|}{(k(k − (wα_{0}, ρ_{G}))∥α_{0}∥_{⋄}^{2})^{s}} + Z(s) log ∥α_{0}∥_{⋄}^{2}. (11)

By lemma 12 one obtains

ρ_{G} + (k − (wα_{0}, ρ_{G}))wα_{0} = S_{wα_{0}}(ρ_{G} − kwα_{0}) = w̃(ρ_{G} + kw̃^{−1}wα_{0})

with (−1)w̃ = 1. Hence the second sum in line (11) equals

∑_{[w]∈W_{G}/W_{K} ; {k∈Z \atop k(k+(w̃^{−1}wα_{0},ρ_{G}))>0}} \frac{χ_{ρ_{G}+kw̃^{−1}wα_{0}} log |k|}{(k(k + (w̃^{−1}wα_{0}, ρ_{G}))∥α_{0}∥_{⋄}^{2})^{s}} .

By the bijectivity of [w] ↦ [w̃^{−1}w] one finds

Z′(s) = ∑_{[w]∈W_{G}/W_{K} ; {k∈Z \atop k(k+(wα_{0},ρ_{G}))>0}} \frac{χ_{ρ_{G}+kwα_{0}} log ∥kα_{0}∥_{⋄}^{2}}{(k(k + (wα_{0}, ρ_{G}))∥α_{0}∥_{⋄}^{2})^{s}} .

Fix g ∈ G and consider the character χ_{ρ_{G}+kwα_{0}}(g) as a function in k ∈ Z. One shows easily that χ_{ρ_{G}+kwα_{0}}(g) is a linear combination of functions of the type k ↦ k^{n}e^{iϕk} (n ∈ Z, ϕ ∈ R) [12, eq. 81]. The characters χ_{ρ_{G}+kwα_{0}} are zero for 0 > k(k + (wα_{0}, ρ_{G})) = (∥ρ_{G} + kα_{0}∥_{⋄}^{2} − ∥ρ_{G}∥_{⋄}^{2})/∥α_{0}∥_{⋄}^{2}. For k = −(wα_{0}, ρ_{G}) we have by lemma 12 the result

χ_{ρ_{G}−(wα_{0},ρ_{G})wα_{0}} = χ_{S_{wα_{0}}ρ_{G}} = (−1)w̃ = 1.

Thus, we may write Z′(s) as

Z′(s) = ∑_{[w]∈W_{G}/W_{K}} ( ∑_{{k∈Z \atop k(k+(wα_{0},ρ_{G}))>0}} \frac{χ_{ρ_{G}+kwα_{0}} log ∥kα_{0}∥_{⋄}^{2}}{(k(k + (wα_{0}, ρ_{G}))∥α_{0}∥_{⋄}^{2})^{s}}

+ ∑_{{k∈Z\\{0\} \atop k(k+(wα_{0},ρ_{G}))≤0}} \frac{χ_{ρ_{G}+kwα_{0}} log ∥kα_{0}∥_{⋄}^{2}}{∥kα_{0}∥_{⋄}^{2s}} − \frac{log ∥(wα_{0}, ρ_{G})α_{0}∥_{⋄}^{2}}{∥(wα_{0}, ρ_{G})α_{0}∥_{⋄}^{2s}} ).

Lemma 13 states that the value at zero of Z′(s) equals the value at zero of the zeta function

Z̃(s) = ∑_{[w]∈W_{G}/W_{K}} ( ∑_{k∈Z\\{0\}} \frac{χ_{ρ_{G}+kwα_{0}} log ∥kα_{0}∥_{⋄}^{2}}{∥kα_{0}∥_{⋄}^{2s}} − \frac{log ∥(wα_{0}, ρ_{G})α_{0}∥_{⋄}^{2}}{∥(wα_{0}, ρ_{G})α_{0}∥_{⋄}^{2s}} ).

The sum of the characters over W_{G}/W_{K} equals

∑_{[w]∈W_{G}/W_{K}} χ_{ρ_{G}+kwα_{0}} = \frac{1}{#W_{K}} ∑_{w∈W_{G}} \frac{∑_{w′∈W_{G}}(−1)^{w′}e^{2πi(w′ρ_{G}+kw′wα_{0})}}{Alt_{G}\{ρ_{G}\}}

= \frac{1}{#W_{K}} \frac{∑_{w∈W_{G}} e^{2πikwα_{0}} ∑_{w′∈W_{G}}(−1)^{w′}e^{2πiw′ρ_{G}}}{Alt_{G}\{ρ_{G}\}}
= ∑_{[w]∈W_{G}/W_{K}} e^{2πikwα_{0}}.

Hence we may apply the formula in lemma 13 to Z̃(s). This proves the theorem. □

6 The classification

To proof theorem 3 for the case SU(3)/SO(3) we apply the method used by Lott and Rothenberg for the spheres. In [13, Prop. 32] they proved

Lemma 14 (Franz, Lott, Rothenberg) Let x_{ν} ∈ R/Z, 1 ≤ ν ≤ N be elements of the circle. Then the x_{ν} are determined up to order and sign by the sequence

(∑_{ν=1}^{N} ψ_{even}(nx_{ν}))_{n∈Z} .

This lemma has been proven by Franz for rational x_{ν}. For the action of a torus element t := 2πidiag(λ_{1}, λ_{2}, λ_{3}) this means that λ_{1}, λ_{2} and λ_{3} are determined up to order and sign by the torsion τ_{t^{n}} of powers of t. As λ_{1} + λ_{2} + λ_{3} = 0, all λ_{ν} have to change sign if one of them does so. Now a change of the order is an element of W_{SU(3)}, thus given by conjugation with an element of SU(3). A change of the sign of all λ_{ν} is obtained by the symmetry around [1] ∈ SU(3)/SO(3) composed with interchanging λ_{1} and λ_{2} (recall our nonstandard imbedding of SO(3)). Thus, all isometries which have the same torsion are actually isometric.

Theorem 4 follows by the same method which has been applied by de Rham for the spheres [7, sect. 8]: First let Γ_{1}, Γ_{2} ⊂ SO(2m) act freely on G_{2m,2p−1} and assume that they are conjugate by a homeomorphism. Thus we may consider them as two euclidean representations ρ_{1}, ρ_{2} : Γ → SO(2m) of one single group Γ. Now theorem 10 implies that for each γ ∈ Γ there is an element g ∈ O(2m) such that

gρ_{1}(γ)g^{−1} = ρ_{2}(γ).

In particular, Tr ρ_{1}(γ) = Tr ρ_{2}(γ). Hence ρ_{1} and ρ_{2} have the same character, thus they are isomorphic as euclidean representations and conjugate in O(2m).

Now consider two finite groups Γ_{1}, Γ_{2} ⊂ SU(3) acting freely on SU(3)/SO(3). Let Γ_{1}, Γ_{2} be conjugate by a homeomorphism. Consider them as unitary representations ρ_{1}, ρ_{2} : Γ → SU(3). For each γ ∈ Γ there exists g̃ ∈ SU(3) ⋊ Z/2Z = Aut(SU(3)) such that

g̃ρ_{1}(γ)g̃^{−1} = ρ_{2}(γ).

In other terms, there is some g ∈ SU(3) such that

gρ_{1}(γ)g^{−1} = ρ_{2}(γ) or \overline{ρ_{2}(γ)}.

Consider the (normal) subgroups of Γ

Γ^{+} := {γ ∈ Γ | Tr ρ_{1}(γ) = Tr ρ_{2}(γ)},  (12)
Γ^{−} := \{γ ∈ Γ | Tr ρ_{1}(γ) = \overline{Tr ρ_{2}(γ)}\}.  (13)

Then Γ = Γ^{+} ∪ Γ^{−}, and for reasons of cardinality Γ has to equal at least one of them. Thus, ρ_{1} is conjugate to ρ_{2} or \overline{ρ_{2}} via some h ∈ U(3). Hence they are conjugate by

\frac{1}{\sqrt[3]{det h}}h ∈ SU(3).

Acknowledgement During the preparation of this paper I have been supported by the Max-Planck-Institut in Bonn and by the Bennigsen-Foerder-Preis. I want to thank Wolfgang Lück and Christian Kaiser for helpful discussions.

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