A fixed point formula of Lefschetz type in Arakelov geometry III: representations of

Chevalley schemes and heights of flag varieties

Christian Kaiser^{1} Kai Köhler^{2}

October 9, 2001

Abstract

We give a new proof of the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z, except for three exceptional cases. This is done by applying the fixed point formula of Lefschetz type in Arakelov geometry to generalized flag varieties. Our proof involves the computation of the equivariant Ray-Singer torsion for all equivariant bundles over complex homogeneous spaces. Furthermore, we find several explicit formulae for the global height of any generalized flag variety.

2000 Mathematics Subject Classification: 14G40, 58J52, 20G05, 20G10, 14M17

1Mathematisches Institut/ Wegelerstr. 10/D-53115 Bonn/Germany/e-mail: ck@math.unibonn.de

2Centre de Mathématiques de Jussieu/C.P. 7012/2, place Jussieu/F-75251 Paris Cedex 05/France/e-mail : koehler@math.jussieu.fr/URL: http://www.math.jussieu.fr/˜ koehler

Contents

1 Introduction 2 2 Flag Varieties 4 3 Equivariant Ray-Singer torsion 10 4 A submersion formula for the analytic torsion 12 5 The torsion of generalized flag manifolds 14 6 Description of the L^{2}-metric 21 7 The main result 23 8 The height of generalized flag varieties 27

1 Introduction

One nice application of the classical Lefschetz fixed point formula is a geometric interpretation of the Weyl character formula for irreducible representations of a compact Lie group G_{c}, found by Atiyah and Bott [AB]. Namely, choose a maximal torus T_{c} of G_{c}, fix a weight λ and consider the line bundle associated to the T_{c}-representation of weight λ over the flag variety G_{c}/T_{c}. The action of G_{c} on G_{c}/T_{c} induces an action on the cohomology vector spaces of this line bundle. The Lefschetz fixed point formula provides a formula for the character of this G_{c}-representation as a sum over the fixed point set. The fixed point set can be identified with the Weyl group of G_{c}, and the resulting formula is the classical Weyl character formula.

The purpose of this article is to investigate the analogous problem in the context of Arakelov geometry. This way we recover the Jantzen sum formula for integral representations of Chevalley schemes over Spec Z. Then we use the Jantzen sum formula to compute the global height of ample line bundles over any generalized flag variety.

Thus instead of compact Lie groups, we consider Chevalley schemes G over Spec Z. The generalized flag varieties associated to G are acted upon by a maximal split torus of G. We apply to them the analogue of the fixed point formula in Arakelov geometry [KR2] due to Roessler and one of the authors. If V_{μ} denotes the weight space to the weight μ in the cohomology representation, then the classical Lefschetz formula computes the character ∑_{μ weight} μ dim V_{μ} . In the Arakelov situation, the cohomology is a Z-module whose tensor product with C carries an Hermitian metric, and the fixed point formula computes the arithmetic character

∑_{μ weight} μ log \frac{covol V_{free,μ}}{# V_{tor,μ}}

where V_{free}, V_{tor} indicate the free and torsion part, respectively. Having chosen a maximal split torus T ⊆ G we have a canonical maximal compact subgroup G_{c} in G(C). Our metric on the cohomology is G_{c}-invariant and thus determined up to a factor. The arithmetic character of the alternating sum of the cohomology modules is then expressed in terms of roots and the Weyl group of G_{c}. We have to exclude factors of type G_{2}, F_{4} and E_{8} for the technical reason that our proof uses Hermitian symmetric spaces, and these do not exist for these three cases.

The resulting formula (up to the determination of the L^{2}-metric) turns out to be the Jantzen sum formula [J2, p. 311] which has been proven originally using solely methods from representation theory and algebraic geometry. In contrast, in the proof presented here we mostly have to deal with the differential geometry of Laplace operators and we use only the representation theory of compact Lie groups. The arithmetic geometry is reduced to that of the fixed point scheme which consists of copies of Spec Z.

We have to compute the equivariant holomorphic Ray-Singer torsion associated to vector bundles on complex homogeneous spaces. This is based on the formula for the Ray-Singer torsion for Hermitian symmetric spaces which has been determined by explicitly describing the zeta function defining the torsion by one of the authors [K2]. Unfortunatly the Laplace operator on general complex homogeneous spaces is a far more difficult object than for Hermitian symmetric spaces. For example, its value on a given irreducible subrepresentation of the space of differential forms shall in general depend on the embedding of that subrepresentation, in contrast to the symmetric case. For that reason we use a geometric construction of the complex homogeneous spaces via the Hermitian symmetric ones instead of the direct approach of [K2]. Using the arithmetic Lefschetz formula or a formula due to Ma [Ma2], we can compare the analytic torsion of the base, of the total space and of the fiber of a fibration. Now we consider a tower of fibrations of complex homogeneous spaces whose fibers are Hermitian symmetric. The lowest base is a point and the total space is the full flag space. By an induction procedure, we get a formula for the torsion on any full flag space. Using again one fibration over an arbitrary complex homogeneous base and with total space and fiber being full flag spaces, we get an expression for the analytic torsion for that base.

In the last chapter, we use the Jantzen sum formula for the arithmetic character to derive formulae for the global height h(X, \overline{L}) of ample line bundles \overline{L} over any (generalized) flag variety X (now including the types G_{2}, F_{4}, E_{8}). The global height is defined using characteristic classes in Arakelov geometry.

The global height of generalized flag varieties has already been investigated in numerous cases using Arakelov intersection theory (compare [BoGS]). There are explicit formulae for projective spaces due to Gillet and Soulé [GS3, p. 212] and for even-dimensional quadrics by Cassaigne and Maillot [CM, Cor. 2.2.10]. Maillot [M] and Tamvakis [T1] found formulae for the height of Grassmannians. Tamvakis also found combinatorial algorithms giving the height of generalized flag varieties of type A_{n} [T2] and of Lagrangian Grassmannians [T3].

Let Ht denote the additive topological characteristic class associated to the power series

Ht(x) := ∑_{k=0}^{∞} \frac{(−x)^{k}}{2(k + 1)(k + 1)!} .

This is the Taylor expansion of the function x ↦ \frac{1}{2x} (log |x| − Γ′(1) − Ei(−x)) at x = 0 with Ei being the exponential-integral function. Let \overline{L} be an equivariant line bundle on G/P, ample over C. We construct in a simple way a virtual holomorphic vector bundle y on G_{C}/P_{C} such that the height with respect to \overline{L} is given by

h(G/P, \overline{L}) = (n + 1)! ∫_{G_{C}/P_{C}} Ht(y)e^{c_{1}(L)}

(Theorem 8.1). Also we give explicit formulae for the height in terms of roots and the Weyl group of G_{c}. From these formulae, we derive some simple properties of heights on generalized flag varieties. Tamvakis used Theorem 8.1 to find Schubert calculus expressions for the heights of some families of homogeneous spaces [T4].

Our method to compute the height is similar to the approach given in [KR3] by Roessler and one of the authors. Still, it is only an application of the classical Jantzen sum formula combined with a relation between the height and the asymptotics of the covolume of the cohomology of \overline{L}^{m} for large m ∈ N which is described in [SABK, VIII.2.3]. The proof does not make direct use of the Lefschetz fixed point formula in Arakelov geometry.

If one considers just the special case of Hermitian symmetric spaces, chapters 4-6 are not necessary for the proofs in chapters 7 and 8. In particular the computation of the analytic torsion for generalized flag varieties in these chapters is already known in the case of Hermitian symmetric spaces ([K2]).

Furthermore, the full main result of [K2] is not needed in this article. We apply it only in the very special case of actions with isolated fixed points. In this case, the proof in [K2] might be replaced by a considerably shorter one similar to the proof of [K3, Lemma 13].

This article is a part of the habilitation thesis of the second author.

Acknowledgements The authors wish to thank Gerd Faltings, Günter Harder, Jens-Carsten Jantzen, Damian Roessler and Matthias Weber for many valuable discussions and comments. We are also very grateful to the referees for their detailed comments.

2 Flag Varieties

In this chapter we recall the definition and some properties of the flag variety of a Chevalley group over Z. Further on we describe the fixed point scheme under the action of some multiplicative subgroup scheme.

Notations: We denote the category of schemes resp. sets by \underline{Scheme} resp. \underline{Set}. For any scheme S we denote by \underline{Scheme_{S}} the category of S-schemes. We identify a S-scheme by the functor from \underline{Scheme_{S}} to \underline{Set} which it represents. If X and S are schemes we set X_{S} := X×S and consider it as an S-scheme. If S = Spec A is affine we also write X_{A}. For A = C we also denote by X_{C} the complex analytic variety X(C). For a Z-module M and ring A we set M_{A} := M ⊗A. Lie algebras associated to smooth group schemes shall be denoted by the corresponding german letter.

Let G be a semisimple Chevalley group of rank r. This is a smooth affine group scheme over Spec Z with connected semisimple groups as geometric fibres. A closed subgroup scheme of G_{S} is called a Borel subgroup, resp. a parabolic subgroup, if it is smooth and of finite presentation over S and a Borel subgroup, resp. a parabolic subgroup, in every geometric fibre.

Consider the functors from \underline{Scheme} to \underline{Set} which map S to the set of all Borel subgroups, resp. parabolic subgroups, of G_{S}. These functors are representable by smooth projective schemes Bor(G), resp. Par(G), over Spec Z [SGA3, ch. XXVI, 3.3]. The group G acts by conjugation on these schemes.

We fix now a maximal split torus T ⊆ G with group of characters X^{∗}(T) and cocharacters X_{∗}(T), set of roots Σ and Weyl group (of the root system) W_{G}. The adjoint action of T on 𝔤 gives the weight decomposition in free Z-modules:

𝔤 = 𝔱 ⊕ ⨁_{α∈Σ} 𝔤_{α}.

Consider a parabolic subgroup P containing T. Since the root spaces 𝔤_{α} are of rank one there is a subset R ⊆ Σ with

𝔭 = 𝔱 ⊕ ⨁_{α∈R} 𝔤_{α}.

The subset R ⊂ Σ is closed (i.e. if α, β ∈ R and α + β ∈ Σ then α + β ∈ R) and R ∪ −R = Σ . We call such a subset of roots a parabolic subset.

Lemma 2.1 For each parabolic subset of roots R ⊂ Σ there exists a unique parabolic subgroup P (R) in G containing T such that 𝔭(R) = 𝔱 ⊕ ⨁_{α∈R} 𝔤_{α}.

Proof: The same proof as in [SGA3, XXII, 5.5.1]. Q.E.D.

For a S-group scheme H acting on a S-scheme Y we denote by Y^{H} the functor of fixed points, i.e. for a S-scheme S′ we have

Y^{H}(S′) := {y ∈ Y (S′) | ∀ϕ : S′′ → S′, ∀h ∈ H(S′′) : hϕ^{∗}(y) = ϕ^{∗}(y)}.

In the braces ϕ is an arbitrary S-morphism and ϕ^{∗} the induced map on points.

Proposition 2.2 The functor morphism Par(G)^{T} → Par(G) is represented by the closed embedding of copies of Spec Z given by P (R) ∈ Par(G)(Z):

⨆_{R⊆Σ parabolic} Spec Z → Par(G).

Proof: Let P be a parabolic subgroup over S. Then P is its own normalizer ([SGA3, XXII, 5.8.5]), i.e. for any S-scheme S′ and any g ∈ G(S′) with int(g)(P_{S′}) = P_{S′} we have g ∈ P (S′). Hence we have P ∈ Par(G)^{T} (S) iff T_{S} ⊂ P. The parabolic subgroups containing T_{S} are determined by their Lie algebra ([SGA3, XXII, 5.3.5]). For S connected we have 𝔭 = 𝔱 ⊗ O_{S} ⊕ ⨁_{α∈R} 𝔤_{α} ⊗ O_{S} for some parabolic subset of roots ([SGA3, XXVI, 1.4]), and hence P = P (R)_{S}. Since clearly Par(G)^{T} (S_{1} ⊔S_{2}) = Par(G)^{T} (S_{1})×Par(G)^{T} (S_{2}) the claim follows. Q.E.D.

Let us fix an ordering Σ = Σ^{+}∪Σ^{−} with base Π. A parabolic subset R containing Σ^{−}, resp. its associated parabolic subgroup P (R), is called ”standard”. For a standard parabolic subset R we set type(R) = {α ∈ Π | α ∈ R ∩ −R}. For k a field every parabolic subgroup P of G_{k} is conjugate to an unique standard parabolic P (R). We set type(P) := type(R). For general S it is a locally constant function s → type(P_{s}).

We have a type decomposition in open and closed subsets:

Par(G) = ⨆_{Θ⊆Π} Par(G)_{Θ}

with Par(G)_{Θ} classifying parabolic subgroups of type Θ. Two parabolic groups have the same type if and only if they are locally conjugate w.r.t. the fpqc or étale topology ([SGA3, XXVI, 3.3]). Since parabolic subgroups are their own normalizer we see that Par(G)_{Θ} represent also the fpqc- or étale-sheafification of the functor S → G(S)/P (S) for any P ⊆ G of type Θ. We write therefore Par(G)_{Θ} =: G/P. It is a connected scheme. We have Par(G)_{∅} = Bor(G) and Par(G)_{Π} = Spec Z. Denoting the standard parabolic of type Θ by P_{Θ}, Proposition 2.2 implies

Corollary 2.3 For Θ ⊆ Π we have

(G/P_{Θ})^{T} = ⨆_{w∈W_{G}/W_{Θ}} Spec Z.

Here W_{Θ} is the subgroup generated by the reflections at roots in Θ.

Remark : By [SGA3, XXVI, 3.16 (iii)] the fixed point scheme (G/P)^{T} is finite étale over Spec Z, from that Corollary 2.3 also follows.

We consider now a finite diagonalisable closed subgroup scheme μ ⊆ T. This means we fix a finite abelian group M and a surjective map ψ : X^{∗}(T) → M. Then μ ≃ Spec Z[M] with comultiplication given by Z[M] → Z[M] ⊗ Z[M], m ↦ m ⊗ m. We assume now the following property for μ:

Definition 2.4 A finite diagonalisable closed subgroup scheme μ ⊆ T has the property (Reg) iff the restriction of ψ to the set of roots Σ is injective.

Proposition 2.5 If μ ⊆ T has property (Reg) then Par(G)^{μ} = Par(G)^{T}.

Proof: Take P ∈ Par(G)(S). We have to show that P is fixed by μ_{S} iff it is fixed by T_{S}. For this we can assume G to be adjoint. From [SGA3, XXII, 5.3.3] it follows that P is invariant under the action of T_{S} resp. μ_{S} iff the same is true for 𝔭. It is a general property of a diagonalisable group scheme acting linearly on a module M with weight decomposition M = ⨁_{α} M_{α}, that a submodule N ⊆ M is invariant iff N = ⨁_{α}(N ∩ M_{α}). Since by property (Reg) the weight decompositions of 𝔤 with respect to μ and T coincide the claim follows. Q.E.D.

Let ⟨·, ·⟩ : X^{∗}(T) × X_{∗}(T) → Z denote the natural pairing. For a coroot α^{∨} we set H_{α} := dα^{∨}(1) ∈ 𝔱. For each α ∈ Σ there is a homomorphism x_{α} : G_{a} → G such that dx_{α} : Lie G_{a} → 𝔤_{α} induces an isomorphism. This map is unique up to a sign change. We can choose the maps x_{α} in such a way that with X_{α} := dx_{α}(1) we get a Chevalley base {X_{α}, H_{μ} | α ∈ Σ, μ ∈ Π} of 𝔤 ⊗ C. For G simply connected it is actually a base of 𝔤. There is a unique involution τ acting on G ([J2, Corollary II.1.16 and proof]) with:

1) τ_{|T} : t ↦ t^{−1} 2) τ ∘ x_{α}(·) = x_{−α}(−·).

It is independent from the consistent choice above.

We can now define a maximal compact subgroup G_{c} of G_{C}:

G_{c} := (G_{C})^{τ∘ι}.

Here ι denotes the complex conjugation.

Lemma 2.6 The group N_{G}(T)(Z) ∩ G_{c} acts on T by conjugation through the full Weyl group.

Proof: For each root α set n_{α} = x_{α}(1)x_{−α}(−1)x_{α}(1). By [J2, p.176] we have n_{α} ∈ N_{G}(T)(Z) and it acts on T by the reflection at α. Since τ(x_{α}(1)x_{−α}(−1)x_{α}(1)) = x_{−α}(−1)x_{α}(1)x_{−α}(−1) it is only an easy calculation in SL_{2} to check τ(n_{α}) = n_{α} [J2, p.176]. Q.E.D.

We can use the maximal compact group G_{c} to endow the complex flag variety (G/P)_{C} with a canonical hermitian metric which fits nicely with the integral structure. Fix a standard parabolic subgroup P ⊆ G. Setting K := P_{C} ∩ G_{c} the inclusion induces an isomorphism (𝔤/𝔭) ⊗ C ≃ 𝔤_{c}/𝔨. For α a weight in 𝔤/𝔭 it maps the integral generator of the α-weight space X_{α} to X_{α} − X_{−α}. The negative of the Killing form κ(·, ·) induces a positive definite real bilinear form (·, ·) on 𝔪 := 𝔤_{c}/𝔨. Denote the complex structure on 𝔪 by J. Set 𝔪^{1,0} := {X ∈ 𝔪 ⊗ C | JX = iX} and define X^{1,0} := (X − iJX)/2 for X ∈ 𝔪. The metric on 𝔪 induces a G_{c}-invariant Hermitian metric on 𝔪^{1,0} and thus an Hermitian metric (·, ·)_{B} on (G/P)_{C}. Using the Killing form we get a norm on (𝔱 ⊗ R)^{∨} ≅ X^{∗}(T) ⊗ R. We get (using [Hu, p.37])

(X_{α}, X_{α}) = −κ(X_{α} − X_{−α}, X_{α} − X_{−α}) = 2κ(X_{α}, X_{−α}) = \frac{4}{∥α∥^{2}}

and thus ∥X_{α}^{1,0}∥_{B}^{2} = \frac{2}{∥α∥^{2}} . By corollary 2.3 the normal bundle of (G/P)^{T} is canonically isomorphic to the restriction of the relative tangent bundle T f to (G/P)^{T}, where f : G/P → Spec Z is the structure morphism. Using Lemma 2.6 and denoting by Ψ ⊆ Σ^{+} the set of weights in 𝔤/𝔭, the above calculation gives Lemma 2.7 For α ∈ Ψ the generators of the α-weight spaces of the normal bundle of (G/P)^{T} have squarelength \frac{2}{∥α∥^{2}} on every component.

Let P be a standard parabolic subgroup. For λ ∈ X^{∗}(P) we define an invertible sheaf L_{λ} on G/P by

L_{λ}(U) = {f ∈ O(π^{−1}U) | f(xp) = λ(p)^{−1}f(x) ∀x ∈ (π^{−1}U)(S), p ∈ P (S) and ∀S}.

Here π : G → G/P is the canonical map and U ⊆ G/P an open subset. Considering X^{∗}(P) as a subgroup of X^{∗}(B) we have a canonical isomorphism H^{∗}(G/P, L_{λ}) ≃ H^{∗}(G/B, L_{λ}). Hence in the following we consider only the ”full flag variety”. The groups H^{∗}(λ) := H^{∗}(G/B, L_{λ}) are finitely generated Z-modules with an algebraic action of G. We use the ”dotted action” of the

Weyl group: w. μ = w(μ + ρ) − ρ. Here ρ = \frac{1}{2} ∑_{α∈Σ^{+}} α. Let I_{+} denote the set

of dominant weights w.r.t. Σ^{+}. For the action of G_{Q} on H^{∗}(w. λ)_{Q} we have, denoting the irreducible representation of highest weight λ by V_{ρ+λ} and by l(w) the length of a w ∈ W_{G}:

Theorem 2.8 (Borel-Weil-Bott): a) If λ + ρ ∈ I_{+} but λ ∉ I_{+}, then

H^{∗}(w. λ)_{Q} = 0 for all w ∈ W_{G}.

b) If λ ∈ I_{+} then we have for all w ∈ W_{G} and all i ∈ N_{0}:

H^{i}(w. λ)_{Q} ≃ \{ \begin{matrix}V_{ρ+λ} & if i = l(w) \\ 0 & otherwise. \end{matrix}

For an abelian group M we denote by M_{tor} its torsion and set M_{free} = M/M_{tor}. Then H^{i}(w. λ)_{free} is a lattice in H^{i}(w. λ)_{Q}. The free part of the following Proposition is due to Andersen [A1, 2.10], the torsion part is a remark of Faltings. We set X = G/B and n + 1 = dim X.

Proposition 2.9 Let λ be an arbitrary weight. For all i ∈ Z there are perfect pairings of G-modules

H^{i}(λ)_{free} × H^{n−i}(−λ − 2ρ)_{free} → Z
H^{i}(λ)_{tor} × H^{n+1−i}(−λ − 2ρ)_{tor} → Q/Z.

Proof: Applying Grothendieck duality [Ha2] to the smooth and proper structure morphism f : X → Spec Z gives an isomorphism in the derived category of bounded complexes of abelian groups

RΓ(L_{−λ−2ρ})[n] →^{∼} RHom_{Z}(RΓ(L_{λ}), Z).

The right hand side is a composition of derived functors and hence the limit of a (in this case: second quadrant) spectral sequence (E_{r}^{i,j}), with E_{2}^{i,j} = Ext_{Z}^{j}(H^{−i}(λ), Z). Since Ext_{Z}^{j}(·, ·) = 0 ∀j ≥ 2 the spectral sequence degenerates at E_{2}, and we get exact sequences

0 → Ext_{Z}^{1}(H^{j}(λ), Z) → H^{n+1−j}(−λ − 2ρ) → Hom(H^{j−1}(λ), Z) → 0.

The group Ext_{Z}^{1}(H^{j}(λ), Z) = Ext_{Z}^{1}(H^{j}(λ)_{tor}, Z) ≃ Hom(H^{j}(λ)_{tor}, Q/Z) is torsion and Hom(H^{j−1}(λ), Z) is torsionfree. The claim follows. Q.E.D.

For a T -module A, which is a finite abelian group, we set char A := − ∑_{μ} μ log #A_{μ}.

Corollary 2.10 char H^{i}(λ)_{tor} = char H^{n+1−i}(w_{0}.λ)_{tor}.

Proof: For n_{0} ∈ N_{G}(T)(Z) acting on T by w_{0} we consider the automorphism τ̃ := Int(n_{0}) ∘ τ. This automorphism respects B, acts therefore on G/B, and puts L_{λ} to L_{−w_{0}λ}. Hence it induces an isomorphism H^{i}(λ) →^{∼} H^{i}(−w_{0}λ), giving the equality char Hom(H^{i}(λ)_{tor}, Q/Z) = char H^{i}(−w_{0}λ)_{tor}. The claim follows from the Proposition. Q.E.D.

We assume in the following λ to be dominant. We choose a generator v_{w.λ} of the λ-weight space of H^{l(w)}(w. λ)_{free}. By irreducibility there is a unique G_{c}-invariant hermitian metric (·, ·)_{τ} on H^{l(w)}(w. λ)_{C} with (v_{w.λ}, v_{w.λ})_{τ} = 1. Its restriction to H^{l(w)}(w. λ)_{R} is a τ-contravariant form ([J1, p. 35]), i.e. (gv, v′)_{τ} = (v, τ(g)^{−1}v′)_{τ}, and was studied by Jantzen [J1] and Andersen [A1, p. 515]. By theorem 2.8 above we have for any w ∈ W and for w_{0} ∈ W the longest element an unique isomorphism:

T_{w_{0}} : H^{l(w)}(w. λ)_{Q} → H^{l(w_{0}w)}(w_{0}w. λ)_{Q}

with T_{w_{0}}(v_{w.λ}) = v_{w_{0}w.λ}.

Proposition 2.11 The lattices T_{w_{0}}(H^{l(w)}(w. λ)_{free}) and H^{l(w_{0}w)}(w_{0}w. λ)_{free} are dual w.r.t. the pairing (·, ·)_{τ}.

Proof: This follows immediately from the explicit presentation of a contravariant form in [A1, p.515]. For convenience of the reader we recall it. For n_{0} ∈ N_{G}(T)(Z) acting on T by w_{0} consider as above τ̃ := Int(n_{0}) ∘ τ, inducing an isomorphism ϕ : H^{l(w)}(w. λ) →^{∼} H^{l(w)}(−w_{0}(w. λ)). We set ψ := n_{0}^{−1} ∘ ϕ. It is independent of the choice of n_{0} and ψ(gv) = τ(g)ψ(v). Using Serre duality pairing, for v, v′ ∈ H^{l(w)}(w. λ)_{Q} we set :

β(v, v′) = (T_{w_{0}}(v), ψ(v′)).

Note that β(v_{w.λ}, v_{w.λ}) = ±1. We have β(gv, v′) = β(v, τ(g)^{−1}v′). Hence it defines a non trivial τ-contravariant form on H^{l(w)}(w. λ)_{Q}. We denote by T_{w_{0}∗}(β) the transported form on H^{l(w_{0}w)}(w_{0}w. λ)_{Q}. Then w.r.t. this form H^{l(w_{0}w)}(w_{0}w. λ)_{free} and T_{w_{0}}(H^{l(w)}(w. λ)_{free}) are dual lattices, as Serre duality induces a perfect pairing on H_{free}^{•} (Proposition 2.9). On the other hand by uniqueness of normalized τ-contravariant forms we have T_{w_{0}∗}(β) = ±(·, ·)_{τ}. Q.E.D.

Take w_{λ} in the λ-weight space of V_{ρ+λ} and w_{−λ} in the −λ-weight space of V_{ρ+λ}^{∨} with ⟨w_{λ}, w_{−λ}⟩ = 1, and consider the associated matrix coefficient c_{w_{λ},w_{−λ}}(g) = ⟨w_{λ}, gw_{−λ}⟩. The following result shall be used to refine our results in the case of positive line bundles:

Proposition 2.12 The function c_{w_{λ},w_{−λ}} is a generator of the λ-weight space of H^{0}(λ).

Proof: It is clear that c_{w_{λ},w_{−λ}} lies in the λ-weight space of H^{0}(λ)_{Q}. Since c_{w_{λ},w_{−λ}}(1) = 1 it is enough to show that it defines a regular function on G. We can do this in two ways:

1) We can realize V_{ρ+λ}, V_{ρ+λ}^{∨} and the natural pairing between them by H^{0}(λ)_{Q}, H^{l(w_{0})}(−λ − 2ρ)_{Q} and the Serre duality pairing. By Proposition 2.9 this defines already a perfect pairing over Z. Hence we can take appropriate elements in the integral structure for w_{λ} and w_{−λ}. Therefore c_{w_{λ},w_{−λ}} is a regular function on G.

2) Using [J2, Proposition I.10.12] we have to show that μ(c_{w_{λ},w_{−λ}}) ∈ Z for all distributions μ ∈ Dist(G). Almost by definition we have μ(c_{w_{λ},w_{−λ}}) = ⟨μw_{λ}, w_{−λ}⟩. By [J2, p. 185] Dist(G) is the subalgebra of U(𝔤) generated by X_{α}^{n}/n! with α ∈ Σ and n ∈ N, and by all \binom{H}{m} with H ∈ 𝔱 and m ∈ N. There is an integral PBW-decomposition U(𝔤) = U^{−}U^{0}U^{+} with U^{−} = ⟨X_{α}^{n}/n! | α ∈ Σ^{−}, n ∈ N⟩, U^{+} = ⟨X_{α}^{n}/n! | α ∈ Σ^{+}, n ∈ N⟩ and U^{0} = ⟨\binom{H}{m} | H ∈ 𝔱, m ∈ N⟩. Since the vectors w_{λ} and w_{−λ} are highest weight vectors w.r.t. B resp. the opposite Borel subgroup B^{−} one has only to consider μ ∈ U^{0}. Considering U^{0} as a polynomial algebra on X^{∗}(T) we get μ(c_{w_{λ},w_{−λ}}) = μ(λ). Since 𝔱 = \{dϕ(1) | ϕ ∈ X_{∗}(T)\} we see that the polynomials \binom{H}{m} take integral values on X_{∗}(T). The claim follows. Q.E.D.

3 Equivariant Ray-Singer torsion

In this section we repeat the definition of the equivariant Ray-Singer analytic torsion introduced in [K1]. Let M be a compact n-dimensional hermitian manifold with associated Kähler form ω. Let \overline{E} denote an hermitian holomorphic vector bundle on M and let

¯∂: Γ^{∞}(Λ^{q}T^{∗0,1}M ⊗ E) → Γ^{∞}(Λ^{q+1}T^{∗0,1}M ⊗ E)

be the Dolbeault operator. As in [GS5], we equip 𝔄^{0,q}(M, E) := Γ^{∞}(Λ^{q}T^{∗0,1}M⊗ E) with the hermitian L^{2}-metric

(η, η′) := ∫_{M}⟨η(x), η′(x)⟩\frac{ω^{∧n}}{(2π)^{n}n!}.  (1)

Let ¯∂^{∗} be the adjoint of ¯∂ relative to this metric and let □_{q} := (¯∂ + ¯∂^{∗})^{2} be the Kodaira-Laplacian acting on Γ^{∞}(Λ^{q}T^{∗0,1}M ⊗ E) with spectrum σ(□_{q}). We denote by Eig_{λ}(□_{q}) the eigenspace of □_{q} corresponding to an eigenvalue λ. Consider a holomorphic isometry g of M and assume given a holomorphic isometry g^{E} : g^{∗}\overline{E} → \overline{E}. This induces an action on section s of E as (g · s)_{|x} := g^{E}(s_{|gx}) for x ∈ M. With this convention, the action g^{T M} on T M induced by g is given by

g^{T M} = (T g)^{−1} : T_{gp}M → T_{p}M

(p ∈ M) with T g being the differential map of g. The fixed point set of g shall be denoted by M^{g}. This notation is chosen different from [KR2], [KR3] to avoid other notational problems. The Dolbeault cohomology H^{0,q}(M, E) := ker □_{q} shall be equipped with the restriction of the L^{2}-metric. The element g induces an isometry g^{∗} of H^{0,q}(M, E) via pullback. Then the equivariant Quillen metric is defined via the zeta function

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

for Re s ≫ 0. Classically, this zeta function has a meromorphic continuation to the complex plane which is holomorphic at zero ([Do]).

Definition 3.1 Set λ_{g}(M, E) := det_{g}H^{0,∗}(M, E) := ⨂_{q}(det_{g}H^{0,2q}(M, E) ⊗ det_{g}H^{0,2q+1}(M, E)^{∨}). The equivariant analytic torsion [K1] is defined as

T_{g}(M, \overline{E}) := Z′_{g}(0) .

For group actions with isolated fixed points on M, we consider the equivariant characteristic classes defined on M^{g} which equal on every fixed point p

Td_{g}(E)_{|p} = det(1 − (g_{|p}^{E})^{−1})^{−1}  (2)

and

ch_{g}(E)_{|p} = Tr g_{|p}^{E} .

In particular, Td_{g}(T M)_{|p} = det(1 − T g_{|p})^{−1} for the infinitesimal action T g_{|p} at a fixed point p ∈ M. These classes are defined in a more general context in [KR2, Th. 3.5].

4 A submersion formula for the analytic torsion

In this section we state a special case of a result due to Xiaonan Ma [Ma2], namely for isolated fixed points. In this special case, it is also a direct consequence of the arithmetic Lefschetz formula in its formulation in [KR3, end of section 2, (b)]. We refer to [KR2, sections 3, 4 and 6.2] for notation and definitions.

Consider a holomorphic Kähler fibration f : M → B of Kähler manifolds with fibre Z. Assume given an isometric automorphism g of f such that all fixed points are isolated. Take an equivariant holomorphic Hermitian vector bundle \overline{E} on M and assume that the dimension of the cohomology groups H^{∗}(Z, E_{|Z}) does not vary on B. For ζ ∈ S^{1} the short exact sequence

0 → T Z → T M → f^{∗}T B → 0

decomposes over the fixed point set M^{g} into ζ-eigenspaces with respect to the action of g

0 → T Z_{ζ} → T M_{ζ} → f^{∗}T B_{ζ} → 0 .

Let s_{ζ} denote a non-zero element in det T M_{ζ} and let s′_{ζ} ∈ det T Z_{ζ} ⊗ det f^{∗}T B_{ζ} be induced from s_{ζ} by the sequence above. Consider the characteristic class Td_{g} as defined in [KR2, section 3.3]. We define the map T̃d_{g}(\overline{T M}, \overline{T B}) : M_{g} → C as

T̃d_{g}(\overline{T M}, \overline{T B}) := Td_{g}(T M) ∑_{ζ≠1} \frac{1}{ζ − 1} log \frac{∥s_{ζ}∥^{2}}{∥s′_{ζ}∥^{2}} .  (3)

See [Ma2] for the general definition which corresponds to the definition in [KR2, Th. 3.4] up to a sign. T̃d_{g} is a Bott-Chern secondary class which transgresses in that general case a difference of Chern-Weil differential forms

\frac{¯∂∂}{2πi}T̃d_{g}(\overline{T M}, \overline{T B}) = Td_{g}(\overline{T M}) − π^{∗}Td_{g}(\overline{T B})Td_{g}(\overline{T Z}) .

Of course in our case, for isolated fixed points, we have \frac{¯∂∂}{2πi} T̃d_{g}(\overline{T M}, \overline{T B}) = 0. For our proof, we need to assume that f extends over Spec Z to a flat projective map of arithmetic varieties f : X → Y. We assume furthermore that g corresponds to the action of a diagonalisable group scheme μ_{N} as in [KR2, section 4] and that E extends to an equivariant locally free sheaf over X.

Consider the structure morphisms f_{X} : X → Spec Z, f_{Y} : Y → Spec Z and endow the direct image sheafs R^{·}f_{X∗}E, R^{·}f_{∗}E and R^{·}f_{Y ∗}(R^{·}f_{∗}E) over C with the L^{2}-metrics induced by Hodge theory. The direct image sheaf equipped with these metrics shall be denoted by R^{·}f_{X∗}\overline{E} etc. The Leray spectral sequence provides an isomorphism R^{·}f_{X∗}E → R^{·}f_{Y ∗}(R^{·}f_{∗}E). Consider over C the equivariant Knudsen-Mumford determinant lines λ_{M,g} := det_{g}(H^{·}(M, \overline{E}))^{−1} and λ_{B,g} := det_{g}(∑_{j,k}(−1)^{j+k}H^{j}(B, H^{k}(Z, \overline{E}_{|Z})))^{−1} and let σ : λ_{B,g} → λ_{M,g} denote the isomorphism between them induced by the Leray spectral sequence.

Thus, log ∥σ∥_{L^{2},g}^{2} = log \frac{∥σs∥_{λ_{M} ,g}^{2}}{∥s∥_{λ_{B} ,g}^{2}} for non-vanishing s ∈ λ_{M,g}.

Theorem 4.1 The equivariant analytic torsions of M, B and Z are related by the equation

log ∥σ∥_{L^{2},g}^{2} − T_{g}(M, \overline{E})

= −T_{g}(B, R^{·}f_{∗}\overline{E}) − ∫_{B^{g}} Td_{g}(T B)T_{g}(Z, \overline{E}) + ∫_{M^{g}} T̃d_{g}(\overline{T M}, \overline{T B})ch_{g}(E) .

As all fixed points are isolated in our case, it does not make any difference in this formula whether one considers classes Td_{g}(T B) or characteristic differential forms like Td_{g}(\overline{T B}) as in [Ma2]. In [Ma2], Ma does not need the existence of models over Spec Z.

Proof: We use the notations of [KR2, Th. 6.14]. Then the arithmetic Lefschetz formula applied subsequently to f_{X}, f (in its generalization in [KR3, end of section 2, (b)]) and to f_{Y} yields

2d̂eg ĉh_{μ_{N}} (R^{·}f_{X∗}\overline{E}) − T_{g}(M, \overline{E}) = 2d̂eg ĉh_{μ_{N}} (f_{X!}\overline{E})

= 2d̂eg f_{X∗}^{μ_{N}} (T̂d_{μ_{N}} (\overline{T f_{X}})ĉh_{μ_{N}} (\overline{E})) − ∫_{M^{g}} Td_{g}(T M)R_{g}(T M)ch_{g}(E)

= 2d̂eg f_{Y ∗}^{μ_{N}} [T̂d_{μ_{N}} (\overline{T f_{Y}})f_{∗}^{μ_{N}} (T̂d_{μ_{N}} (\overline{T f})ĉh_{μ_{N}} (\overline{E})) ]

+ ∫_{M^{g}} T̃d_{g}(\overline{T M}, \overline{T B})ch_{g}(E) − ∫_{M^{g}} Td_{g}(T M)R_{g}(T M)ch_{g}(E)

= 2d̂eg f_{Y ∗}^{μ_{N}} [T̂d_{μ_{N}} (\overline{T f_{Y}})(ĉh_{μ_{N}} (R^{·}f_{∗}\overline{E}) − T_{g}(Z, \overline{E})

+ ∫_{Z^{g}} Td_{g}(T Z)R_{g}(T Z)ch_{g}(E))]

+ ∫_{M^{g}} T̃d_{g}(\overline{T M}, \overline{T B})ch_{g}(E) − ∫_{M^{g}} Td_{g}(T M)R_{g}(T M)ch_{g}(E)
= 2d̂eg ĉh_{μ_{N}} (R^{·}f_{Y ∗}(R^{·}f_{∗}\overline{E})) − T_{g}(B, R^{·}f_{∗}\overline{E})
+ ∫_{B^{g}} Td_{g}(T B)R_{g}(T B)ch_{g}(R^{·}f_{∗}E)

− ∫_{B^{g}} Td_{g}(\overline{T B})T_{g}(Z, \overline{E}) + ∫_{B^{g}} Td_{g}(T B) ∫_{Z^{g}} Td_{g}(T Z)R_{g}(T Z)ch_{g}(E)

+ ∫_{M^{g}} T̃d_{g}(\overline{T M}, \overline{T B})ch_{g}(\overline{E}) − ∫_{M^{g}} Td_{g}(T M)R_{g}(T M)ch_{g}(E)
= 2d̂eg ĉh_{μ_{N}} (R^{·}f_{Y ∗}(R^{·}f_{∗}\overline{E})) − T_{g}(B, R^{·}f_{∗}\overline{E})
− ∫_{B^{g}} Td_{g}(\overline{T B})T_{g}(Z, \overline{E}) + ∫_{M^{g}} T̃d_{g}(\overline{T M}, \overline{T B})ch_{g}(\overline{E}) .

Furthermore

2d̂eg ĉh_{μ_{N}} (R^{·}f_{X∗}\overline{E}) − 2d̂eg ĉh_{μ_{N}} (R^{·}f_{Y ∗}(R^{·}f_{∗}\overline{E})) = log ∥σ∥_{L^{2},g}^{2}

by the Leray spectral sequence, which in particular identifies the torsion subgroups of the direct images. Thus we get the statement of the theorem. Q.E.D.

Remark. By K-theory, one could in fact reduce oneself to the case where the direct images are locally free and non-zero only in degree 0. Then one can consider the equivariant determinant lines over Spec Z and identify them directly as graded direct sums of Z-modules of rank one (up to a sign). Thus one could avoid the use of the spectral sequence entirely.

An arithmetic Lefschetz formula for more general fibrations (combined with a general arithmetic Grothendieck-Riemann-Roch theorem) would provide this formula not only for non-isolated fixed points, but for the torsion forms associated to a double fibration M → B → S with a Kähler manifold S as treated in [Ma1] in the non-equivariant case.

5 The torsion of generalized flag manifolds

Set T_{c} := G_{c} ∩T_{C}. We denote the roots of K by Σ_{K}; in general, we shall denote the objects corresponding to subgroups of G by writing this subgroup as an index. Then M := G_{C}/P_{C} is canonically isomorphic to G_{c}/K and Σ^{+} = Ψ∪Σ_{K}^{+}. Recall that we identify 𝔪^{1,0} with the tangent space T_{e}M^{1,0} and that the negative of the Killing form induces an Hermitian metric on T M. Fix X_{0} ∈ 𝔱_{c} in the closure of the positive Weyl chamber such that its stabilizer with respect to the adjoint action of G_{c} equals K. Then the metric (·, ·)_{X_{0}} on 𝔪 which equals for X ∈ 𝔪_{α}, Y ∈ 𝔪_{β}

(X, Y)_{X_{0}} = \{ \begin{matrix}0 & α ≠ β \\ (X, Y)_{B}α(X_{0}) & α = β \end{matrix}

induces a Kähler metric on M [Be, Ch. 8.D].

Set 𝔱_{reg} := {X ∈ 𝔱_{c}|α(X) ∉ Z ∀α ∈ Σ}. For X ∈ 𝔱_{c} let e^{X} ∈ T_{c} denote the associated group element. We denote the G_{c}-representation with highest weight λ by V_{ρ+λ}^{G} and its character is denoted by χ_{ρ+λ}. In general, for a weight λ and X ∈ 𝔱_{reg} we define χ_{ρ+λ} by the Weyl character formula

χ_{ρ+λ}(e^{X}) := \frac{∑_{w∈W_{G}}(−1)^{l(w)}e^{2πi(ρ+λ)(wX)}}{∏_{α∈Σ^{+}} 2i sin πα(X)} .

An irreducible K-representation V_{ρ_{K}+λ}^{K} induces a G_{c}-invariant holomorphic vector bundle E_{ρ_{K}+λ}^{K} on M. As V_{ρ_{K}+λ}^{K} carries a K-invariant Hermitian metric which is unique up to a factor, we get corresponding G_{c}-invariant metrics on E_{ρ_{K}+λ}^{K}. Set Ψ^{+} := {α ∈ Ψ|⟨α^{∨}, ρ + λ⟩ ≥ 0} and Ψ^{−} := {α ∈ Ψ|⟨α^{∨}, ρ + λ⟩ < 0} with α^{∨} = 2α/∥α∥^{2}.

Define for ϕ ∈ R and Re s > 1

ζ_{L}(s, ϕ) = ∑_{k=1}^{∞} \frac{e^{ikϕ}}{k^{s}} .  (4)

The function ζ_{L} has a meromorphic continuation to the complex plane in s which is holomorphic for s ≠ 1. Set ζ′_{L}(s, ϕ) := ∂/∂s(ζ_{L}(s, ϕ)). Let P : Z → C be a function of the form

P (k) = ∑_{j=0}^{m} c_{j}k^{n_{j}}e^{ikϕ_{j}}  (5)

with m ∈ N_{0}, n_{j} ∈ N_{0}, c_{j} ∈ C, ϕ_{j} ∈ R for all j. We define P^{odd}(k) := (P (k) − P (−k))/2. Also we define as in [K2]

ζP := ∑_{j=0}^{m} c_{j}ζ_{L}(−n_{j}, ϕ_{j}),  (6)

ζ′P := ∑_{j=0}^{m} c_{j}ζ′_{L}(−n_{j}, ϕ_{j}),  (7)

\overline{ζ}P := ∑_{j=0}^{m} c_{j}ζ_{L}(−n_{j}, ϕ_{j}) ∑_{ℓ=1}^{n_{j}} \frac{1}{ℓ}.  (8)

Res P (p) := ∑_{{j=0 \atop ϕ_{j} ≡0 mod2π}}^{m} c_{j} \frac{p^{n_{j}+1}}{2(n_{j} + 1)}  (9)

and  P^{∗}(p) := − ∑_{{j=0 \atop ϕ_{j} ≡0 mod2π}}^{m} c_{j} \frac{p^{n_{j}+1}}{4(n_{j} + 1)} ∑_{ℓ=1}^{n_{j}} \frac{1}{ℓ}  (10)

for p ∈ R. In particular, for X ∈ 𝔱_{reg} we get (compare [K2, Th. 10, top of p. 108])

ζχ_{ρ+λ−kα}(e^{X}) = \frac{∑_{w∈W_{G}}(−1)^{l(w)}e^{2πi(ρ+λ)(wX)}(e^{2πiα(wX)} − 1)^{−1}}{∏_{β∈Σ^{+}} 2i sin πβ(X)} . (11)

Assume now that M = G_{c}/K is an Hermitian symmetric space. Then the isotropy representation is irreducible, hence all G_{c}-invariant metrics coincide with (·, ·)_{B} up to a factor. Fix one metric associated to X_{0} with stabilizer K. In [K2, section 11], the analytic torsion on G_{c}/K has been calculated for vector bundles \overline{E}_{ρ_{K}+λ}^{K} with λ ∈ I_{+}. We shall now extend this result to arbitrary λ.

Proposition 5.1 Assume that G_{c}/K is Hermitian symmetric. Then the zeta function defining the analytic torsion of \overline{E}_{ρ_{K}+λ}^{K} is given by

Z_{t}(s) = − ∑_{α∈Ψ^{+}} α^{∨}(X_{0})^{s} ∑_{k>⟨α^{∨},ρ+λ⟩} \frac{χ_{ρ+λ−kα}(t)}{k^{s}(k − ⟨α^{∨}, ρ + λ⟩)^{s}}

+ ∑_{α∈Ψ^{−}} α^{∨}(X_{0})^{s} ∑_{k>−⟨α^{∨},ρ+λ⟩} \frac{χ_{ρ+λ+kα}(t)}{k^{s}(k + ⟨α^{∨}, ρ + λ⟩)^{s}} .

Proof: According to [K2, (114)], the irreducible representations occurring as eigenspaces of the Laplacian acting on 𝔄^{0,q}(M, E) are given by the infinitesimal characters ρ + λ + kα with k > 0. Consider now the case ⟨α^{∨}, ρ + λ⟩ < 0 and assume 0 < k < ⟨−α^{∨}, ρ + λ⟩/2. Let S_{α} denote the reflection of the weights by the hyperplane orthogonal to α. Then the multiplicity of V_{ρ+λ+kα} cancels with that of V_{S_{α}(ρ+λ+kα)} = V_{ρ+λ+(−k−⟨α^{∨},ρ+λ⟩)α}. In case ⟨α^{∨}, ρ + λ⟩ is even, the representation V_{ρ+λ−⟨α^{∨},ρ+λ⟩α/2} vanishes. Also, the representation V_{ρ+λ−⟨α^{∨},ρ+λ⟩α} corresponds to the eigenvalue 0, thus it does not contribute to the zeta function. Thus, Z_{t}(s) is given by

Z_{t}(s) = ∑_{α∈Ψ} α^{∨}(X_{0})^{s} ∑_{k>max\{0,⟨−α^{∨},ρ+λ⟩\}} \frac{χ_{ρ+λ+kα}}{k^{s}(k + ⟨α^{∨}, ρ + λ⟩)^{s}}

which equals the above formula after another application of S_{α} for α ∈ Ψ^{+}. Q.E.D.

Theorem 5.2 Assume that G_{c}/K is Hermitian symmetric. Then the equivariant analytic torsion of \overline{E}_{ρ_{K}+λ}^{K} is given by

T_{t}(G_{c}/K, \overline{E}_{ρ_{K}+λ}^{K}) = −2 ∑_{α∈Ψ} ζ′χ_{ρ+λ−kα}^{odd} − 2 ∑_{α∈Ψ} χ_{ρ+λ−kα}^{∗}(⟨α^{∨}, ρ + λ⟩)

− ∑_{α∈Ψ} ζχ_{ρ+λ−kα} · log α^{∨}(X_{0}) − χ_{ρ+λ} ∑_{α∈Ψ^{+}} log α^{∨}(X_{0})

− ∑_{α∈Ψ^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα} · log k + ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩} χ_{ρ+λ+kα} · log k .

Proof: According to [K2, Lemma 8] and [K2, (61),(63)], we find

Z′_{t}(0) = ∑_{α∈Ψ^{+}} [ − 2ζ′χ_{ρ+λ−kα}^{odd} − 2χ_{ρ+λ−kα}^{∗}(⟨α^{∨}, ρ + λ⟩)

− ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα} · log k]

− ∑_{α∈Ψ^{+}} [ζχ_{ρ+λ−kα} − ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα}] log α^{∨}(X_{0})

+ ∑_{α∈Ψ^{−}} [2ζ′χ_{ρ+λ+kα}^{odd} + 2χ_{ρ+λ+kα}^{∗}(⟨−α^{∨}, ρ + λ⟩)

+ ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩} χ_{ρ+λ+kα} · log k] − ∑_{α∈Ψ^{−}} ζχ_{ρ+λ−kα} log α^{∨}(X_{0})

As

− ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα} = −χ_{ρ+λ−⟨α^{∨},ρ+λ⟩α} = χ_{ρ+λ}

for α ∈ Ψ^{+} and

ζ′χ_{ρ+λ+kα}^{odd} = −ζ′χ_{ρ+λ−kα}^{odd} ,
χ_{ρ+λ+kα}^{∗}(⟨−α^{∨}, ρ + λ⟩) = −χ_{ρ+λ−kα}^{∗}(⟨α^{∨}, ρ + λ⟩)

we get the statement of the theorem. Q.E.D.

According to [K2, (103),(116)], for any generalized flag manifold the equation

∫_{M^{t}} Td_{t}(T M)R_{t}(T M)ch_{t}(E_{ρ_{K}+λ}^{K}) = − ∑_{α∈Ψ} [2ζ′χ_{ρ+λ−kα}^{odd} + \overline{ζ}χ_{ρ+λ−kα}^{odd}]

holds. For fixed X ∈ 𝔱_{reg}, according to [K2, eq. (66)] the characters χ_{ρ+λ−kα}(e^{iX}) can be written as a linear combination of exponential functions in k. Polynomial functions in k do not occur in this case. Thus, by definition χ_{ρ+λ−kα}^{∗} and \overline{ζ}χ_{ρ+λ−kα} vanish. Hence, we get the following simpler formula for generic X:

Corollary 5.3 Assume that M = G_{c}/K is Hermitian symmetric and choose X ∈ 𝔱_{reg}. Then

T_{t}((M, g_{X_{0}}), \overline{E}_{ρ_{K}+λ}^{K}) − ∫_{M^{t}} Td_{t}(T M)R_{t}(T M)ch_{t}(E_{ρ_{K}+λ}^{K}) =

− ∑_{α∈Ψ} ζχ_{ρ+λ−kα} · log α^{∨}(X_{0}) − χ_{ρ+λ} ∑_{α∈Ψ^{+}} log α^{∨}(X_{0})

− ∑_{α∈Ψ^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα} · log k + ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩} χ_{ρ+λ+kα} · log k .

We say that a semisimple Lie group has tiny weights if none of its simple components is of type G_{2}, F_{4} or E_{8}.

Theorem 5.4 Let G_{C} be a semisimple Lie group having tiny weights and let M = G_{c}/K be an associated (generalized) flag manifold. Let \overline{E}_{ρ_{K}+λ}^{K} be a G_{c}- invariant holomorphic Hermitian vector bundle on M and fix a Kähler metric g_{X_{0}} associated to X_{0} ∈ 𝔱_{c} on M. Choose X ∈ 𝔱_{reg} and set t := e^{X}. Then the equivariant analytic torsion associated to the action of t is given by

T_{t}((M, g_{X_{0}}), \overline{E}_{ρ_{K}+λ}^{K}) − ∫_{M^{t}} Td_{t}(T M)R_{t}(T M)ch_{t}(E_{ρ_{K}+λ}^{K}) =

− ∑_{α∈Ψ} ζχ_{ρ+λ−kα} · log α^{∨}(X_{0}) + Cχ_{ρ+λ}

− ∑_{α∈Ψ^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα} · log k + ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩} χ_{ρ+λ+kα} · log k

where the constant C ∈ R does not depend on t.

The formula for the torsion of Hermitian symmetric spaces (Theorem 5.2) is based on the particularly nice behaviour of the Laplacian on them. In particular, the value of the Laplacian on an irreducible subrepresentation V_{ρ+λ} of 𝔄^{0,q}(M, E) does not depend on the embedding of that representation. Also, the Killing metric provides a canonical Kähler metric in that case. All this is not true for general complex homogeneous spaces. Thus we have to find an alternative approach for them. Our proof proceeds by constructing first a tower of fibrations of flag manifolds with total space a full flag manifold G_{C}/B_{C}, such that all fibers are Hermitian symmetric spaces. As the torsion is known for the fibers, we can deduce the value of the torsion for the total space of every fibration from the value on the base. Thus we finally get the value for the full flag manifold. Then we use the fibration G_{C}/B_{C} → G_{C}/P_{C} with both total space and fiber being full flag manifolds to compute the torsion for any flag manifold. We need first a Lemma and a Proposition:

Lemma 5.5 Let G_{C} be a reductive group having tiny weights. Then there is an ordering (α_{1}, …, α_{m}) of the base Π such that with Θ_{j} := {α_{1}, …, α_{j}}, 0 ≤ j ≤ m, all quotients of subsequent parabolic subgroups P_{Θ_{j+1}}/P_{Θ_{j}}, 0 ≤ j < m, are Hermitian symmetric spaces.

Proof: The only simple groups whose quotients by parabolic subgroups are never Hermitian symmetric spaces are the groups G_{2}, F_{4} and E_{8} [He, Ch. X §6.3]. If G_{C} has tiny weights, then the Dynkin diagram of 𝔤_{2}, 𝔣_{4} or 𝔢_{8} can never occur as a subdiagram of the Dynkin diagram of 𝔤_{C}. Thus, there is always a simple root α such that the quotient G_{C}/P_{Π\{α}} is Hermitian symmetric (namely, those α such that their coefficient in the highest root is 1, compare [He, p. 476ff]), and the Levi component of P_{Π\{α}} has again tiny weights. We get the statement by induction on the number of roots. Q.E.D.

Proposition 5.6 Theorem 5.4 holds for the spaces G_{C}/P_{Θ_{j}} with Θ_{j} as in Lemma 5.5. In particular, it holds for G_{C}/B_{C}.

Proof: The proof proceeds by induction on the number of elements of Π \ Θ_{j}. For Θ_{j} = Π, the analytic torsion vanishes and the statement is trivial. Now assume that the statement is true for all Θ_{k} ⊃ Θ_{j}. To shorten the notation, we shall denote K_{Θ_{j}} and K_{Θ_{j+1}} by H and K, respectively. Thus, the associated holomorphic fibration

π : G_{c}/H → G_{c}/K

has as fiber the Hermitian symmetric space K/H. The Kähler metric on G_{c}/H induces a Kähler metric on K/H. We fix a Kähler metric on G_{c}/K associated to X_{1} ∈ 𝔱_{c}. Notice that X_{1} and X_{0} have to be different as their stabilizers are different.

By theorem 2.8 applied to K/H and the base change formulae, we find that a certain shift of the direct image R^{•}π_{∗}E_{ρ_{H}+λ}^{H} is given by the vector bundle E_{ρ_{K}+λ}^{K} associated to the irreducible K-representation with highest weight λ. Also, the cohomologies H^{•}(G_{c}/H, E_{ρ_{H}+λ}^{H}) and H^{•}(G_{c}/K, E_{ρ_{K}+λ}^{K}) can both be identified with the irreducible G-representation E_{ρ+λ}^{G}, and all cohomology groups except one vanish. As a G-invariant metric on an irreducible representation is unique up to a constant, the isomorphism σ in section 4 simply corresponds to a multiplication of the metric with a constant, and the equivariant metrics considered in theorem 4.1 differ just by a constant C′ ∈ R times χ_{ρ+λ}(t).

Let T̃d_{t}(T G_{c}/H, T G_{c}/K, g_{X_{0}}, g_{X_{1}}) denote the equivariant Bott-Chern secondary class associated to the short exact sequence

0 → T K/H → T G_{c}/H → π^{∗}T G_{c}/K → 0

of vector bundles on G_{c}/H, equipped with the Hermitian metrics induced by X_{0}, X_{0} and X_{1}, respectively. By theorem 4.1 we get the formula

T_{t}(G_{c}/H, \overline{E}_{ρ_{H}+λ}^{H}) = log ∥σ∥_{L^{2},t}^{2} + T_{t}(G_{c}/K, \overline{E}_{ρ_{K}+λ}^{K})
+ ∫_{(G_{c}/K)^{t}} Td_{t}(T G_{c}/K)T_{t}(K/H, \overline{E}_{ρ_{H}+λ}^{H})
− ∫_{(G_{c}/H)^{t}} T̃d_{t}(T G_{c}/H, T G_{c}/K, g_{X_{0}}, g_{X_{1}})ch_{t}(E_{ρ_{H}+λ}^{H})

The fixed point set (G_{c}/K)^{t} is given by W_{G}/W_{K}. For simplicity, we assume Ψ^{−} = ∅; the proof remains the same in the general case.

By applying the induction hypothesis and [K2, Theorem 18] for the torsion of the fiber, we get

T_{t}(G_{c}/H, \overline{E}_{ρ_{H}+λ}^{H}) = C′χ_{ρ+λ}(t) + ∫_{(G_{c}/K)^{t}} Td_{t}(T G_{c}/K)R_{t}(T G_{c}/K)ch_{t}(E_{ρ_{K}+λ}^{K})

− ∑_{α∈Σ^{+}\Σ_{K}^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα}(t) log k − ∑_{α∈Σ^{+}\Σ_{K}^{+}} ζχ_{ρ+λ−kα}(t) log α^{∨}(X_{1})

+C′′χ_{ρ+λ}(t)

+ ∑_{w∈W_{G}/W_{K}} Td_{t}(T G_{c}/K)( (∫_{(K/H)^{t}} Td_{t}(T K/H)R_{t}(T K/H)ch_{t}(E_{ρ_{H}+λ}^{H}))_{|wK}

− ∑_{α∈Σ_{K}^{+}\Σ_{H}^{+}} ∑_{k=1}^{⟨α^{∨},ρ_{K}+λ⟩} χ_{ρ_{K}+λ−kα}^{K}(wt) log k − ∑_{α∈Σ_{K}^{+}\Σ_{H}^{+}} ζχ_{ρ_{K}+λ−kα}^{K}(wt) log α^{∨}(X_{0})

−χ_{ρ_{K}+λ}^{K}(t) ∑_{α∈Σ_{K}^{+}\Σ_{H}^{+}} log α^{∨}(X_{0}))

− ∫_{(G_{c}/H)^{t}} T̃d_{t}(T G_{c}/H, T G_{c}/K, g_{X_{0}}, g_{X_{1}})ch_{t}(\overline{E}_{ρ_{H}+λ}^{H}) .

Next we notice that

∫_{(G_{c}/K)^{t}} Td_{t}(T G_{c}/K)R_{t}(T G_{c}/K)ch_{t}(E_{ρ_{K}+λ}^{K})

+ ∫_{(G_{c}/K)^{t}} Td_{t}(T G_{c}/K) ∫_{(K/H)^{t}} Td_{t}(T K/H)R_{t}(T K/H)ch_{t}(E_{ρ_{H}+λ}^{H})

= ∫_{(G_{c}/K)^{t}} Td_{t}(T G_{c}/K)R_{t}(T G_{c}/K) ∫_{(K/H)^{t}} Td_{t}(T K/H)ch_{t}(E_{ρ_{H}+λ}^{H})
+ ∫_{(G_{c}/H)^{t}} Td_{t}(T G_{c}/H)R_{t}(T K/H)ch_{t}(E_{ρ_{H}+λ}^{H})
= ∫_{(G_{c}/H)^{t}} Td_{t}(T G_{c}/H)R_{t}(T G_{c}/K)ch_{t}(E_{ρ_{H}+λ}^{H})
+ ∫_{(G_{c}/H)^{t}} Td_{t}(T G_{c}/H)R_{t}(T K/H)ch_{t}(E_{ρ_{H}+λ}^{H})
= ∫_{(G_{c}/H)^{t}} Td_{t}(T G_{c}/H)R_{t}(T G_{c}/H)ch_{t}(E_{ρ_{H}+λ}^{H}) .

Also, by the classical Lefschetz fixed point formula (in this case already shown in [Bott], see also [K2, Theorem 11, eq. (84)] and recall (2)) we get

∑_{w∈W_{G}/W_{K}} Td_{t}(T G_{c}/K)χ_{ρ_{K}+λ−kα}^{K}(wt) = χ_{ρ+λ−kα}(t) ;

furthermore, ⟨α^{∨}, ρ_{K} + λ⟩ = ⟨α^{∨}, ρ + λ⟩ for α ∈ Σ_{K} by [K2, p. 106]. By the definition of T̃d we get

∫_{(G_{c}/H)^{t}} T̃d_{t}(T G_{c}/H, T G_{c}/K, g_{X_{0}}, g_{X_{1}})ch_{t}(E_{ρ_{H}+λ}^{H})

= ∑_{w∈W_{G}/W_{H}} \frac{χ_{ρ_{H}+λ}^{H}(wt)}{∏_{β∈Σ^{+}\Σ_{H}^{+}} (1 − e^{−2πiβ(wt)})} ∑_{α∈Σ^{+}\Σ_{K}^{+}} \frac{log α(X_{0})/α(X_{1})}{e^{2πiα(wt)} − 1}

= ∑_{α∈Σ^{+}\Σ_{K}^{+}} ζχ_{ρ+λ−kα}(t) log \frac{α(X_{0})}{α(X_{1})} .

Thus, we find

T_{t}(G_{c}/H, \overline{E}_{ρ_{H}+λ}^{H}) = (C′ + C′′)χ_{ρ+λ}(t) + ∫_{(G_{c}/H)^{t}} Td_{t}(T G_{c}/H)R_{t}(T G_{c}/H)ch_{t}(E_{ρ_{H}+λ}^{H})

− ∑_{α∈Σ^{+}\Σ_{K}^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα}(t) log k − ∑_{α∈Σ_{K}^{+}\Σ_{H}^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα}(t) log k

− ∑_{α∈Σ^{+}\Σ_{K}^{+}} ζχ_{ρ+λ−kα}(t) log α^{∨}(X_{1}) − ∑_{α∈Σ_{K}^{+}\Σ_{H}^{+}} ζχ_{ρ+λ−kα}(t) log α^{∨}(X_{0})

− ∑_{α∈Σ^{+}\Σ_{K}^{+}} ζχ_{ρ+λ−kα}(t) log \frac{α(X_{0})}{α(X_{1})} − χ_{ρ+λ}(t) ∑_{α∈Σ_{K}^{+}\Σ_{H}^{+}} log α^{∨}(X_{0})

= Cχ_{ρ+λ}(t) + ∫_{(G_{c}/H)^{t}} Td_{t}(T G_{c}/H)R_{t}(T G_{c}/H)ch_{t}(E_{ρ_{H}+λ}^{H})

− ∑_{α∈Σ^{+}\Σ_{H}^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα}(t) log k − ∑_{α∈Σ^{+}\Σ_{H}^{+}} ζχ_{ρ+λ−kα}(t) log α^{∨}(X_{0}) .

Q.E.D.

Proof of Theorem 5.4: Consider the fibration G_{c}/T_{c} → G_{c}/K with fiber K/T_{c}. We equip G_{c}/T_{c} by a Kähler metric associated to X_{0} ∈ 𝔱_{c} with stabilizer T_{c}. Using again theorem 4.1 we get the formula

T_{t}(G_{c}/K, \overline{E}_{ρ_{K}+λ}^{K}) = − log |σ|_{L^{2},t}^{2} + T_{t}(G_{c}/T_{c}, \overline{E}_{λ}^{T})
− ∫_{(G_{c}/K)_{t}} Td_{t}(T G_{c}/K)T_{t}(K/T_{c}, \overline{E}_{λ}^{T})

+ ∫_{(G_{c}/T)_{t}} T̃d_{t}(T G_{c}/T_{c}, T G_{c}/K, g_{X_{0}}, g_{X_{1}})ch_{t}(E_{λ}^{T}) .

By applying the formulae for T_{t}(G_{c}/T_{c}, \overline{E}_{λ}^{T}) and T_{t}(K/T_{c}, \overline{E}_{λ}^{T}) shown in Proposition 5.6, we get the value of T_{t}(G_{c}/K, \overline{E}_{ρ_{K}+λ}^{K}) by the analogue of the calculation above. Q.E.D.

6 Description of the L^{2}-metric

Consider a vector bundle E_{ρ_{K}+λ}^{K} on M associated to a Σ_{K}^{+}-dominant weight λ. In this section we describe the L^{2}-metric on H^{∗}(M, E_{ρ_{K}+λ}^{K}) induced by embedding this space into the C^{∞} differential forms with coefficients in E_{ρ_{K}+λ}^{K} via Hodge theory. As the cohomology is an irreducible G-representation, the metric is uniquely determined by the norm of one element.

Classically [Bott], Γ^{∞}(G_{c}/K, E_{ρ_{K}+λ}^{K}⊗Λ^{q}T^{∗0,1}G_{c}/K) has the L^{2}-dense subspace

⨁_{μ∈I_{+}} Hom_{K}(V_{ρ+μ}, V_{ρ_{K}+λ}^{K} ⊗ Λ^{q}Ad_{G/P}^{1,0} ) ⊗ V_{ρ+μ}  (12)

where the section associated to f ⊗ v ∈ Hom_{K}(V_{ρ+μ}, V_{ρ_{K}+λ}^{K} ⊗ Λ^{q}Ad_{G/P}^{1,0} ) ⊗ V_{ρ+μ} is given by the K-equivariant C^{∞}-function

s : G_{c} → V_{ρ_{K}+λ}^{K} ⊗ Λ^{q}Ad_{G/P}^{1,0}
g ↦ f(g^{−1}v) .

The direct sum (12) is a direct sum of complexes. The summand for μ is the set of K-invariants in the tensor product of the complex Hom(V_{ρ+μ}, Λ^{q}Ad_{G/P}^{1,0} ) with the space V_{ρ_{K}+λ}^{K} ⊗ V_{ρ+μ}. This last complex is canonically isomorphic to the standard complex calculating the Lie algebra cohomology for the nilpotent radical [SB, p. 89, eq. (***)], [Bott, equation (15.3)]. Set w such that w^{−1}(ρ + λ) =: ρ + λ_{0} ∈ I_{+}. Then the cohomology has either highest weight λ_{0} or it vanishes in case λ_{0} ∉ I_{+}. Assume that λ_{0} ∈ I_{+}. According to the proof of Kostant’s theorem [V, Corollary 3.2.11] the first tensor factor of the summand associated to μ = λ_{0} in (12) consists only of a one-dimensional space in degree l(w). Thus the cohomology is embedded uniquely in the complex (12) by the one-dimensional space

Hom_{K}(V_{ρ+λ_{0}}, V_{ρ_{K}+λ}^{K} ⊗ Λ^{l(w)}Ad_{G/P}^{1,0} ) .

Now assume that V_{ρ_{K}+λ}^{K} is one-dimensional and equipped with an Hermitian metric. Fix one v′ ∈ V_{ρ_{K}+λ}^{K} ⊗ Λ^{l(w)}Ad_{G/P}^{1,0} of weight wλ_{0} and norm 1. Choose v ∈ V_{ρ+λ_{0}} of weight wλ_{0} and let v^{∨} ∈ V_{ρ+λ_{0}}^{∨} be of weight −wλ_{0} with v^{∨}(v) = 1. Then v^{∨} ⊗ v′ ∈ Hom_{K}(V_{ρ+λ_{0}}, V_{ρ_{K}+λ}^{K} ⊗ Λ^{l(w)}Ad_{G/P}^{1,0} ), hence there is an element of H^{l(w)}(G_{c}/K, E_{ρ_{K}+λ}^{K}) of weight wλ_{0} given by the section

s_{0} : g ↦ v^{∨}(g^{−1}v) ⊗ v′.

which is independent of the choice of v.

Lemma 6.1 Let M be equipped with the metric associated to X_{0} ∈ 𝔱_{reg}. Then the L^{2}-norm of s_{0} is given by

|s_{0}|_{L^{2}}^{2} = ∏_{α∈Ψ} \frac{α^{∨}(X_{0})}{⟨α^{∨}, ρ + λ⟩} .

Proof: According to [BeGeV, Lemma 7.23], the computation reduces to that for the measure d\overline{g} on M induced by the Haar measure dg via the formula

|s_{0}|_{L^{2}}^{2} = ∏_{α∈Ψ} \frac{α^{∨}(X_{0})}{⟨α^{∨}, ρ⟩} ∫_{G_{c}/K} |s_{0}(\overline{g})|^{2}d\overline{g}

(by [K2, (71)], our normalized Haar measure differs from the measure considered in [BeGeV, Lemma 7.23] by the factor in the denominator). By the K-invariance of |s_{0}|^{2}, this equals

|s_{0}|_{L^{2}}^{2} = ∏_{α∈Ψ} \frac{α^{∨}(X_{0})}{⟨α^{∨}, ρ⟩} · ∫_{G} |v^{∨}(g^{−1}v)|^{2}dg

= ∏_{α∈Ψ} \frac{α^{∨}(X_{0})}{⟨α^{∨}, ρ⟩} · (dimV_{ρ+λ})^{−1} .

The last equality follows by the orthogonality relations ([BtD, Th. 4.5]). As dim V_{ρ_{k}+λ}^{K} = 1, the Weyl dimension formula shows

dim V_{ρ+λ} = ∏_{α∈Σ^{+}} \frac{⟨α^{∨}, ρ + λ⟩}{⟨α^{∨}, ρ⟩} = ∏_{α∈Ψ} \frac{⟨α^{∨}, ρ + λ⟩}{⟨α^{∨}, ρ⟩} .

Thus the Lemma follows. Q.E.D.

7 The main result

Theorem 7.1 Let G be a Chevalley group having tiny weights and let P be a standard parabolic subgroup. Consider a P -module A, free of rank one over Z, equipped with an Hermitian metric on A_{C}. Let λ be the weight of A and choose w ∈ W_{G} such that w^{−1}(ρ+λ) = ρ+λ_{0} ∈ I_{+}. We denote the induced line bundle on the n + 1-dimensional variety X = G/P by L_{λ}. There is a constant C ∈ R such that the following identity of formal linear combinations of weights holds

−(−1)^{l(w)} ∑_{μ∈X^{∗}(T)} μ log covol \overline{H^{l(w)}(X, L_{λ})}_{μ}

+ ∑_{q=0}^{n}(−1)^{q} ∑_{μ∈X^{∗}(T)} μ log #H^{q}(X, L_{λ})_{μ,tor}

= −\frac{1}{2} ∑_{α∈Ψ^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα} · log k + \frac{1}{2} ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩} χ_{ρ+λ+kα} · log k

+(C − log covol \overline{A})χ_{ρ+λ} .

Furthermore,

C = −\frac{1}{2} ∑_{α∈Ψ^{+}} log α^{∨}(X_{0})

if either G_{c}/K is Hermitian symmetric or if λ ∈ I_{+}.

Proof: Since T (C)_{Reg} := {t ∈ T (C)_{tor}|α(t) ≠ β(t) ∀α, β ∈ Σ, α ≠ β} is Zariski-dense in T (C), it is enough to prove the identities of complex numbers obtained by substituting μ with μ(t) ∈ S^{1} for all t ∈ T (C)_{Reg}. We apply [KR2, Th. 6.14] for imbeddings μ_{N} ↪ T satisfying property (Reg) (see definition 2.4) and for t ∈ μ_{N}(C) being a generator. We adopt the notations from [KR2, section 4].

Classically, the free part of the cohomology H^{q}(X, L_{λ}) vanishes in all degrees except at most for q = l(w) [Bott]. According to [KR2, Th. 6.14],

−(−1)^{l(w)} ∑_{μ∈X^{∗}(μ_{N})} μ(t) log covol \overline{H^{l(w)}(X, L_{λ})}_{μ}

+ ∑_{q=0}^{n}(−1)^{q} ∑_{μ∈X^{∗}(μ_{N})} μ(t) log #H^{q}(X, L_{λ})_{μ,tor}

= \frac{1}{2}T_{t}(X_{C}, \overline{L}_{λ,C}) − \frac{1}{2} ∫_{X(C)_{t}} Td_{t}(\overline{T X}_{C})R_{t}(\overline{T X}_{C})ch_{t}(\overline{L}_{λ})

+d̂egf_{∗}(T̂d_{μ_{N}} (\overline{T f})ĉh_{μ_{N}} (\overline{L}_{λ})) .

Let f : X → Spec Z be the structure morphism. As X^{μ_{N}} = X^{T}, T acts on T f_{|X^{μ_{N}}}. Let T f_{α} denote the direct summand of T f_{|X^{μ_{N}}} associated to a weight α. By Corollary 2.3, the components of the fixed point scheme are indexed by W_{G}/W_{P}. When restricted to the component X_{w}^{μ_{N}} for [w] ∈ W_{G}/W_{P}, the equivariant characteristic classes are given by

T̂d_{μ_{N}} (\overline{T f})_{|X_{w}^{μ_{N}}} = ∏_{α∈Ψ}(1 − e^{−2πiwα(t)})^{−1} · (1 + ∑_{α∈Ψ} \frac{ĉ_{1}(\overline{T f}_{wα})}{1 − e^{2πiwα(t)}} )

(compare the expansion of the equivariant Todd-class in [KR2, section 3.3]) and

ĉh_{μ_{N}} (\overline{L}_{λ})_{|X_{w}^{μ_{N}}} = e^{2πiwλ(t)}(1 + ĉ_{1}(\overline{L}_{λ})) .

Hence we get by Lemma 2.7

d̂egf_{∗}(T̂d_{μ_{N}} (\overline{T f})ĉh_{μ_{N}} (\overline{L}_{λ}))

= ∑_{[w]∈W_{G}/W_{P}} \frac{e^{2πiwλ(t)}}{∏_{α∈Ψ}(1 − e^{−2πiwα(t)})} [∑_{α∈Ψ} \frac{d̂egf_{∗}ĉ_{1}(\overline{T f}_{wα})}{1 − e^{2πiwα(t)}} + d̂egf_{∗}ĉ_{1}(\overline{L}_{λ})]

= ∑_{[w]∈W_{G}/W_{P}} \frac{e^{2πiwλ(t)}}{∏_{α∈Ψ}(1 − e^{−2πiwα(t)})} [\frac{1}{2} ∑_{α∈Ψ} \frac{− log α^{∨}(X_{0})}{1 − e^{2πiwα(t)}} − log covol \overline{A}]

Furthermore, [K2, Th. 11, eq. (84)] shows

∑_{[w]∈W_{G}/W_{P}} \frac{−e^{2πiwλ(t)} log covol \overline{A}}{∏_{α∈Ψ}(1 − e^{−2πiwα(t)})} = −χ_{ρ+λ}(t) · log covol \overline{A}

Similarly we shall proceed as in the proof of [K2, theorem 10] to obtain a formula for the other term. For a class [w] ∈ W_{G}/W_{P} we shall denote by l(w) the minimal length of its representents.

∑_{[w]∈W_{G}/W_{P}} \frac{e^{2πiwλ(t)}}{∏_{α∈Ψ}(1 − e^{−2πiwα(t)})} ∑_{α∈Ψ} \frac{− log α^{∨}(X_{0})}{1 − e^{2πiwα(t)}}

= ∑_{[w]∈W_{G}/W_{P}} \frac{e^{2πiw(ρ−ρ_{K}+λ)(t)}(−1)^{l(w)} ∑_{w′∈W_{P}} e^{2πiww′ρ_{K}(t)}}{∏_{β∈Σ^{+}} 2i sin πβ(t)} ∑_{α∈Ψ} \frac{log α^{∨}(X_{0})}{e^{2πiwα(t)} − 1}

= ∑_{[w]∈W_{G}/W_{P}} \frac{1}{∏_{β∈Σ^{+}} 2i sin πβ(t)}

· ∑_{α∈Ψ} ∑_{w′∈W_{P}} (−1)^{l(w)+l(w′)}e^{2πiww′(ρ+λ)(t)} \frac{log(w′α)^{∨}(X_{0})}{e^{2πiww′α(t)} − 1}

(in the last equation we used the facts that λ and Ψ are W_{P} -invariant)

= ∑_{α∈Ψ} ∑_{w∈W_{G}} \frac{(−1)^{l(w)}e^{2πiw(ρ+λ)(t)}(e^{2πiwα(t)} − 1)^{−1}}{∏_{β∈Σ^{+}} 2i sin πβ(t)} log α^{∨}(X_{0})

(as X_{0} is K-stable)

= ∑_{α∈Ψ} ζχ_{ρ+λ−kα}(t) · log α^{∨}(X_{0})

(by equation (11)). Thus

d̂egf_{∗}(T̂d_{μ_{N}} (\overline{T f})ĉh_{μ_{N}} (\overline{L}_{λ}))
= \frac{1}{2} ∑_{α∈Ψ} ζχ_{ρ+λ−kα}(t) · log α^{∨}(X_{0}) − χ_{ρ+λ}(t) · log covol \overline{A} .

The proof is finished by combining this result with Th. 5.4. For Hermitian symmetric spaces, combine instead with the more precise corollary 5.3 which does not involve the unknown constant C. Now consider the case λ ∈ I_{+}. In this case, by the Kempf vanishing theorem [J2, Prop. 4.5] the cohomology has no torsion. Furthermore Ψ = Ψ^{+}, thus the formula in theorem 7.1 simplifies to

− ∑_{μ∈X^{∗}(T)} μ log covol \overline{H^{0}(X, L_{λ})}_{μ}

= −\frac{1}{2} ∑_{α∈Ψ} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩} χ_{ρ+λ−kα} · log k + (C − log covol \overline{A})χ_{ρ+λ} .

In particular, the component associated to μ = λ verifies the equality

− log covol \overline{H^{0}(X, L_{λ})}_{λ} = \frac{1}{2} ∑_{α∈Ψ} log⟨α^{∨}, ρ + λ⟩ + (C − log covol \overline{A}) ,

as the weight λ occurs only in the characters χ_{ρ+λ} and χ_{ρ+λ−⟨α^{∨},ρ+λ⟩α}, with multiplicity 1 and −1, respectively. By combining Proposition 2.12 and Lemma 6.1, we find on the other hand

− log covol \overline{H^{0}(X, L_{λ})}_{λ} = −\frac{1}{2} ∑_{α∈Ψ} log α^{∨}(X_{0})+ \frac{1}{2} ∑_{α∈Ψ} log⟨α^{∨}, ρ+λ⟩−log covol \overline{A}

(in Lemma 6.1, \overline{A} was normalized to have covolume 1). Thus we get the value of the constant C for dominant λ. Q.E.D.

Corollary 7.2 Let A_{ρ+λ} := H^{l(w)}(X, L_{λ})_{free} be a G-module induced by a line bundle L_{λ}. If A_{ρ+λ} ≠ 0 (i.e. λ_{0} ∈ I_{+}) equip A_{ρ+λ} with the unique G_{c}-invariant metric such that the generator of the weight space to the highest weight λ_{0} has norm 1. Then

−(−1)^{l(w)} ∑_{μ∈X^{∗}(T)} μ log covol \overline{A}_{ρ+λ,μ}

+ ∑_{q=0}^{n}(−1)^{q}( ∑_{μ∈X^{∗}(T)} μ log #H^{q}(X, L_{λ})_{μ,tor} − (−1)^{l(w)}χ_{ρ+λ} log #H^{q}(X, L_{λ})_{λ_{0},tor})

= −\frac{1}{2} ∑_{α∈Ψ^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩−1} χ_{ρ+λ−kα} · log k + \frac{1}{2} ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩−1} χ_{ρ+λ+kα} · log k

Proof: This follows from theorem 7.1 by comparing the components of weight λ_{0}. The weight λ_{0} occurs only in the characters χ_{ρ+λ} and χ_{ρ+λ−⟨α^{∨},ρ+λ⟩α}, with multiplicity (−1)^{l(w)} and −(−1)^{l(w)}, respectively. Q.E.D.

Remark. Let P′ ⊆ P be another standard parabolic subgroup. Then A is a P′-representation, and both sides of the above formula remain the same. Namely, H^{•}(G/P, L_{λ}) = H^{•}(G/P′, L_{λ}) and χ_{ρ+λ−kα} = 0 for any α ∈ Σ_{K}^{+}, 1 ≤ k ≤ ⟨α^{∨}, ρ + λ⟩ − 1.

For two Z-free G-modules A, A′ with A_{Q} and A′_{Q} isomorphic and irreducible we denote by [A, A′] ∈ Q^{+} the index obtained by embedding A′ in such a way in A ⊗ Q that the weight spaces of highest weight are identified. Then [A, A′] = covol A′/covol A. Let w_{0} be the Weyl group element of maximal length. When applied to w_{0}. λ instead of λ, the above corollary reads

−\frac{1}{2} ∑_{α∈Ψ^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩−1} χ_{ρ+λ−kα} · log k + \frac{1}{2} ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩−1} χ_{ρ+λ+kα} · log k

= −\frac{(−1)^{l(w_{0})}}{2} ∑_{α∈Ψ^{+}} ∑_{k=1}^{−⟨−w_{0}α^{∨},w_{0}(ρ+λ)⟩−1} χ_{w_{0}(ρ+λ−kα)} · log k

+\frac{(−1)^{l(w_{0})}}{2} ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−w_{0}α^{∨},w_{0}(ρ+λ)⟩−1} χ_{w_{0}(ρ+λ+kα)} · log k

= −\frac{(−1)^{l(w_{0})}}{2} ∑_{α∈−w_{0}Ψ^{+}} ∑_{k=1}^{−⟨α^{∨},w_{0}(ρ+λ)⟩−1} χ_{w_{0}(ρ+λ)+kα} · log k

+\frac{(−1)^{l(w_{0})}}{2} ∑_{α∈−w_{0}Ψ^{−}} ∑_{k=1}^{⟨α^{∨},w_{0}(ρ+λ)⟩−1} χ_{w_{0}(ρ+λ)−kα} · log k

= (−1)^{l(w)} ∑_{μ∈X^{∗}(T)} μ log covol \overline{A}_{w_{0}(ρ+λ),μ}

− ∑_{q=0}^{n}(−1)^{q+l(w_{0})}( ∑_{μ∈X^{∗}(T)} μ log #H^{q}(X, L_{w_{0}.λ})_{μ,tor}

−(−1)^{l(w_{0}w)}χ_{w_{0}(ρ+λ)} log #H^{q}(X, L_{w_{0}.λ})_{λ_{0},tor})

= (−1)^{l(w)} ∑_{μ∈X^{∗}(T)} μ log covol \overline{A}_{w_{0}(ρ+λ),μ}

− ∑_{q=0}^{n}(−1)^{q}( ∑_{μ∈X^{∗}(T)} μ log #H^{n−q}(X, L_{w_{0}.λ})_{μ,tor}

−(−1)^{l(w)}χ_{ρ+λ} log #H^{n−q}(X, L_{w_{0}.λ})_{λ_{0},tor}) .

By subtracting the equations for A_{ρ+λ} and A_{w_{0}(ρ+λ)} we get an identity which is an immediate consequence of Corollary 2.10 and Proposition 2.11. Thus corollary 7.2 is equivalent to the statement obtained by adding the equations for A_{ρ+λ} and A_{w_{0}(ρ+λ)}. We get

Corollary 7.3 (Jantzen sum formula [J2, p. 311])

(−1)^{l(w)} ∑_{μ∈X^{∗}(T)} μ log[A_{ρ+λ,μ} : A_{w_{0}(ρ+λ),μ}]

+ ∑_{μ∈X^{∗}(T)} ∑_{q=0}^{n} μ · (−1)^{q}( log #H^{q}(X, L_{λ})_{μ,tor} − log #H^{n−q}(X, L_{w_{0}.λ})_{μ,tor})

−(−1)^{l(w)}χ_{ρ+λ} ∑_{q=0}^{n}(−1)^{q}( log #H^{q}(X, L_{λ})_{λ_{0},tor} − log #H^{n−q}(X, L_{w_{0}.λ})_{λ_{0},tor})

= − ∑_{α∈Ψ^{+}} ∑_{k=1}^{⟨α^{∨},ρ+λ⟩−1} χ_{ρ+λ−kα} · log k + ∑_{α∈Ψ^{−}} ∑_{k=1}^{⟨−α^{∨},ρ+λ⟩−1} χ_{ρ+λ+kα} · log k .

As above, both sides are independent of the parabolic subgroup. In particular, for P = B we get the usual formulation of the Jantzen sum formula.

Combining these formulae with [J2, (8.7)]

#H^{q}(X, L_{λ})_{λ_{0},tor} = 0 ∀λ, q

one notices the vanishing of the multiples of χ_{ρ+λ} in the above two corollaries. We did not use this result before to point out what exactly our result says without applying further representation theory. For λ ∈ I_{+}, the Kempf vanishing theorem [J2, Prop. 4.5] states that the torsion of the cohomology vanishes. Note that the other way round the Jantzen sum formula provides the values of the equivariant Ray-Singer torsion for isolated fixed points up to a multiple of χ_{ρ+λ}, in particular for the types G_{2}, F_{4} and E_{8}.

8 The height of generalized flag varieties

For the definitions of the objects in Arakelov geometry which are used in this chapter we refer the reader to [SABK]. According to [BH, 14.7], the very ample line bundles on G_{c}/K are induced by the K-representations with highest weight λ such that

⟨α^{∨}, λ⟩ = \{ \begin{matrix}= 0 & if α ∈ Σ_{K} \\ > 0 & if α ∈ Ψ . \end{matrix}

Equip a line bundle L_{λ} → G/P to such a λ with the equivariant metric induced by normalizing the length of the generator of the P -module to 1. Thus, as the equivariant metric on L_{λ} is unique up to a factor, it coincides with the metric induced by the O(1) bundle via the projective embedding associated to L_{λ}. In particular, this metric is positive (this can be shown directly, too). In this section we compute the global height of G/P with respect to \overline{L}_{λ} defined as

h(G/P, \overline{L}_{λ}) := f_{∗} [ĉ_{1}(\overline{L}_{λ})^{n+1}] .

Remark. The line bundle L_{λ} is also very ample [J2, II.8.5] (we shall not need this fact). See [Z], [Abbes] for relations between the global height of a variety and the height of points on that variety in this case.

Set Ψ_{j} := {α ∈ Ψ|⟨α^{∨}, λ⟩ = j} for j ∈ N. These sets are W_{K}-invariant as Ψ and λ are W_{K}-invariant. Hence for every j there is a virtual P -representation with character equal to ∑_{α∈Ψ_{j}} e^{2πiα}, defining a virtual vector bundle E_{j} on G/P. As before, we equip G_{c}/K with the metric associated to some X_{0} in a certain subset of 𝔱_{c}.

Let Ht denote the additive topological characteristic class associated to the power series

Ht(x) := ∑_{k=0}^{∞} \frac{(−x)^{k}}{2(k + 1)(k + 1)!} .

This is the Taylor expansion of the function x ↦ \frac{1}{2x} (log |x| − Γ′(1) − Ei(−x)) at x = 0 with Ei being the exponential-integral function [N, 39 (13)]. For x < 0 Ei(x) is given by

Ei(x) = ∫_{−∞}^{x} \frac{e^{t}}{t} dt .

Theorem 8.1 The height of G/P with respect to \overline{L}_{λ} is given by

h(G/P, \overline{L}_{λ}) = (n + 1)! ∫_{M} Ht(∑_{j∈N} jψ^{j}E_{j})e^{c_{1}(L_{λ})}

where ψ^{j} denotes the j-th Adams operator.

In particular the height can be written as

∑_{l=0}^{n} \frac{(−1)^{l}}{2(l + 1)}\binom{n + 1}{l + 1} ∫_{M} l! ∑_{j∈N} j^{l+1}ch(E_{j})^{[l]}c_{1}(L_{λ})^{n−l} .  (13)

We conclude that h(G/P, \overline{L}_{λ}) has the form

∑_{ℓ=1}^{n+1} \frac{k_{ℓ}}{2ℓ}

with k_{ℓ} ∈ Z ∀ℓ. See Corollary 8.2 for a refinement and a conjecture.

Proof: As in [SABK, VIII.2.3] we have for m → ∞

r(m) := d̂eg(π_{∗}\overline{L}_{mλ}, ∥ · ∥_{L^{2}}^{2}) = \frac{m^{n+1}}{(n + 1)!}h(G/P, \overline{L}_{λ}) + O(m^{n} log m) . (14)

Here we use the fact that \overline{L}_{λ} is positive, which implies a result by Bismut-Vasserot [BV] on the asymptotics of the non-equivariant holomorphic torsion. Consider the polynomials d_{ρ+mλ−kα} := dim V_{ρ+mλ−kα} (α ∈ Ψ) in m and k, given by the Weyl dimension formula

d_{ρ+mλ−kα} = ∏_{β∈Σ^{+}} (1 + \frac{⟨β^{∨}, mλ − kα⟩}{⟨β^{∨}, ρ⟩}) .

The term r(m) equals the left hand side of theorem 7.1 evaluated at zero. By the Jantzen sum formula (or theorem 7.1 for G having tiny weights), for m ≥ 1

r(m) = −\frac{1}{2} ∑_{α∈Ψ} ∑_{k=1}^{⟨α^{∨},ρ+mλ⟩} d_{ρ+mλ−kα} log k

−d_{ρ+mλ} · [ log covol \overline{H^{0}(X, L_{mλ})}_{mλ} + \frac{1}{2} ∑_{α∈Ψ} log⟨α^{∨}, ρ + mλ⟩] .

Here we replaced the constant C in theorem 7.1 by the value one obtains by comparing the components of weight mλ, similar to the last section in the proof of theorem 7.1. As in the case λ ∈ I_{+} in theorem 7.1 (where only groups G with tiny weights were allowed), we find again by combining Proposition 2.12 and Lemma 6.1

log covol \overline{H^{0}(X, L_{mλ})}_{mλ} + \frac{1}{2} ∑_{α∈Ψ} log⟨α^{∨}, ρ + mλ⟩ = \frac{1}{2} ∑_{α∈Ψ} log α^{∨}(X_{0}) ,

thus this factor is independent of m. The expression for r(m) is a sum over terms which (at a first sight) look like having order O(m^{1+#Σ^{+}} log m) for m → ∞. Thus, it is not obvious that it is in fact of order O(m^{n+1}), and one has to be careful when calculating the term of highest degree. We shall need the following three facts about the polynomials d_{ρ+mλ−kα}:

1. The degree in m of d_{ρ+mλ−kα} is equal to n, since

d_{ρ+mλ−kα} = ∏_{β∈Ψ} \frac{⟨β^{∨}, ρ + mλ − kα⟩}{⟨β^{∨}, ρ⟩} · ∏_{β∈Σ_{K}^{+}} \frac{⟨β^{∨}, ρ − kα⟩}{⟨β^{∨}, ρ⟩} .

2. For all j ∈ N, the common polynomial degree in m and k of ∑_{α∈Ψ_{j}} d_{ρ+mλ−kα} is less or equal to n. This is a consequence of the Riemann-Roch theorem, which states in this case

∑_{α∈Ψ_{j}} d_{ρ+mλ−kα} = ∫_{M} Td(T M)ch(ψ^{−k}E_{j})e^{mc_{1}(L_{λ})}

(compare [K2, Th. 13]).

3. The polynomial d_{ρ+mλ−kα} is skew-symmetric in k around k = ⟨α^{∨}, ρ + mλ⟩/2. More precisely, by applying the reflection S_{α} at the hyperplane orthogonal to α we get

d_{ρ+mλ−(k+⟨α^{∨},ρ+mλ⟩)α} = −d_{ρ+mλ+kα} .

Now we see that

−2r(m) + O(m^{n}) = ∑_{j} ∑_{α∈Ψ_{j}} ∑_{k=1}^{jm} d_{ρ+mλ−kα} log k

+ ∑_{α∈Ψ} ∑_{k=⟨α^{∨},mλ⟩+1}^{⟨α^{∨},ρ+mλ⟩} d_{ρ+mλ−kα} log k

= ∑_{j} ∑_{k=1}^{jm} ∑_{α∈Ψ_{j}} d_{ρ+mλ−kα} log k

− ∑_{α∈Ψ} ∑_{k=1−⟨α^{∨},ρ⟩}^{0} d_{ρ+mλ+kα} log(k + ⟨α^{∨}, ρ + mλ⟩)

using the skew-symmetry in the last equation. Notice that the last double sum is of order O(m^{n} log m). By approximating the first sum with integrals and doing partial integration we get

−2r(m) = ∑_{j} ∫_{1}^{jm} ∑_{α∈Ψ_{j}} d_{ρ+mλ−kα} log k dk + O(m^{n} log m)

= ∑_{α∈Ψ} ∫_{1}^{⟨α^{∨},mλ⟩} d_{ρ+mλ−kα} log k dk + O(m^{n} log m)

= ∑_{α∈Ψ} ∫_{0}^{x} d_{ρ+mλ−kα}dk log x|_{x=1}^{⟨α^{∨},mλ⟩}

− ∑_{α∈Ψ} ∫_{1}^{⟨α^{∨},mλ⟩} ∫_{0}^{x} d_{ρ+mλ−kα}dk \frac{dx}{x} + O(m^{n} log m)

= ∑_{α∈Ψ} ∫_{0}^{⟨α^{∨},mλ⟩} d_{ρ+mλ−kα}dk log ⟨α^{∨}, mλ⟩

− ∑_{α∈Ψ} ∫_{0}^{⟨α^{∨},mλ⟩} ∫_{0}^{x} d_{ρ+mλ−kα}dk \frac{dx}{x} + O(m^{n} log m)

By the skew-symmetry of d_{ρ+mλ−kα}, this equals

∑_{α∈Ψ} ∫_{0}^{⟨α^{∨},ρ⟩} d_{ρ+mλ−kα}dk log ⟨α^{∨}, mλ⟩

− ∑_{α∈Ψ} ∫_{0}^{⟨α^{∨},mλ⟩} ∫_{0}^{x} d_{ρ+mλ−kα}dk \frac{dx}{x} + O(m^{n} log m)

= − ∑_{α∈Ψ} ∫_{0}^{⟨α^{∨},mλ⟩} ∫_{0}^{x} d_{ρ+mλ−kα}dk \frac{dx}{x} + O(m^{n} log m)  (15)

In particular, we see that r(m) is of order O(m^{n+1}), which can also be derived from equation (14). By decomposing into the Ψ_{j}-parts again and applying the Riemann-Roch theorem we get

2r(m) = ∑_{j} ∫_{0}^{jm} ∫_{0}^{x} ∫_{M} Td(T M)ch(ψ^{−k}E_{j})e^{mc_{1}(L_{λ})}dk \frac{dx}{x} + O(m^{n} log m)

= ∫_{0}^{m} ∫_{0}^{x} ∫_{M} Td(T M)ch(∑_{j} jψ^{−jk}E_{j})e^{mc_{1}(L_{λ})}dk \frac{dx}{x} + O(m^{n} log m)

= ∫_{0}^{m} ∫_{0}^{x} ∫_{M} ch(∑_{j} jψ^{−jk}E_{j})e^{mc_{1}(L_{λ})}dk \frac{dx}{x} + O(m^{n} log m)

Hence we can express r(m) as

r(m) = ∫_{M} Ht(∑_{j} jψ^{j}E_{j})e^{c_{1}(L_{λ})} · m^{n+1} + O(m^{n} log m) .

This proves the above theorem. Q.E.D.

Remark. If G_{c}/K is Hermitian symmetric a formula for T (G_{c}/K, \overline{L}) is known [K2]. In this case we could have equally well worked with the difference T_{g}(G_{c}/K, \overline{L})− T (G_{c}/K, \overline{L}) by arguing similarly as above, but avoiding the use of [BV]. In the case of Hermitian symmetric spaces there is a unique primitive λ. Using the classification of irreducible Hermitian symmetric spaces [He], one verifies that ⟨α^{∨}, λ⟩ equals either 1 or 2 for all α ∈ Ψ.

Equation (15) provides a very effective way to compute the height. Namely, for j ∈ N consider the sum

f_{j}(m, k) := ∑_{α∈Ψ_{j}} d_{ρ+mλ−kα}  (16)

= ∑_{α∈Ψ_{j}} ∏_{β∈Σ_{K}^{+}}(1 − k \frac{⟨β^{∨}, α⟩}{⟨β^{∨}, ρ⟩}) ∏_{j′} ∏_{β∈Ψ_{j′}}(1 + \frac{j′m − k⟨β^{∨}, α⟩}{⟨β^{∨}, ρ⟩})

and replace every power k^{l} by \frac{(mj)^{l+1}}{2(l+1)^{2}} . The height is obtained by adding the coefficients of m^{n+1} and multiplying with (n + 1)!. For α ∈ Σ

#{β ∈ Σ^{+}|⟨α, β⟩ ≠ 0} ≤ 2c(G) − 3  (17)

where c(G) is the Coxeter number of G [Bour, Ch. V §6.1]. Namely, by [Bour, Ch. VI §1, Prop. 32], equality holds in (17) for root systems where all roots have the same length. Furthermore, for B_{l} and C_{l} the cardinality in (17) equals 2c(G) − 5 and c(G) − 1 (depending on the root), for F_{4} it equals 2c(G) − 9 = 15 and for G_{2} it equals c(G) − 1 = 5. Thus the degree in k of f_{j}(m, k) is less or equal to 2c(G) − 3, and we find by using the same reformulation as in formula (13)

Corollary 8.2 There are k_{ℓ} ∈ Z (1 ≤ ℓ ≤ 2c(G) − 2) such that

h(G/P, \overline{L}_{λ}) = ∑_{ℓ=1}^{2c(G)−2} \frac{k_{ℓ}}{2ℓ} .

In other words, the largest power of a prime occurring in the denominator of 2h(G/P, \overline{L}_{λ}) is less or equal to 2c(G) − 2.

Our computations of examples as well as Tamvakis’ results in [T1], [T2], [T3] strongly suggest the

Conjecture There are k_{ℓ} ∈ Z (1 ≤ ℓ ≤ c(G) − 1) such that

h(G/P, \overline{L}_{λ}) = ∑_{ℓ=1}^{c(G)−1} \frac{k_{ℓ}}{2ℓ} .

Tamvakis [T4] meanwhile verified this conjecture explicitly for several families of homogeneous spaces. Theorem 8.1 implies the following fixed point expression for h(G/P, \overline{L}_{λ}):

Lemma 8.3 Let Y ∈ 𝔤 act with isolated fixed points and let ϕ, θ_{ν} denote the angles of the action on L_{λ|p}, T M_{|p} respectively for p ∈ M^{Y}. Then

h(G/P, \overline{L}_{λ}) = ∑_{p∈M^{Y}} \frac{1}{∏_{ν} θ_{ν}} ∑_{l=1}^{n+1} ∑_{j∈N} ∑_{θ_{ν}∈Ψ_{j}} \frac{ϕ^{n+1} − ϕ^{n+1−l}(ϕ − jθ_{ν})^{l}}{2lθ_{ν}} .

Note that if Y is an element of the Lie algebra of the maximal torus then there is a canonical isomorphism M^{Y} = W_{G}/W_{K}. The angle ϕ at [w] ∈ W_{G}/W_{K} is given by (2πwλ)(Y) and the angle θ_{wα} (α ∈ Ψ) corresponding to T M_{wα} is given by (2πwα)(Y). Thus, the formula in Lemma 8.3 reads

h(G/P, \overline{L}_{λ}) = ∑_{w∈W_{G}/W_{K}} \frac{1}{∏_{α∈Ψ} wα(Y)}

· ∑_{l=1}^{n+1} ∑_{α∈Ψ} \frac{(wλ(Y))^{n+1} − (wλ(Y))^{n+1−l}(S_{wα}wλ(Y))^{l}}{2lwα(Y)} .

A good choice for Y is the dual of ρ.

Proof: Applying the Bott residue formula to formula (13) yields

h(G/P, \overline{L}_{λ}) = ∑_{p∈M^{Y}} ∑_{l=0}^{n} \binom{n + 1}{l + 1}\frac{ϕ^{n−l}}{2(l + 1) ∏ θ_{ν}} ∑_{j∈N} ∑_{θ_{ν}∈Ψ_{j}}(−θ_{ν})^{l}j^{l+1}

(alternatively, one could apply [K2, Th. 11] to f_{j}(m, k) from equation (16) and proceed as in [KR3, proposition 3.7]). Using the formula

∑_{l=1}^{n} \binom{n}{l}\frac{x^{l}}{l} = ∑_{l=1}^{n} \frac{(1 + x)^{l} − 1}{l} (x ∈ R) ,

we get the desired result. Q.E.D.

Example. We shall express the height of the Grassmannian G(m, k) with G(m, k)(C) = U(m)/U(k) × U(m − k) using Lemma 8.3. In this case, the Weyl group of G is the permutation group S_{m} of m elements and the fixed point set can be identified with S_{m}/S_{k} × S_{m−k}. Let (ϵ_{ν})_{ν=1}^{m} be the cartesian base of C^{m}, which we identify with 𝔱^{∨} as in [BtD, section V.6]. A short look to the classical tables of roots (e.g. [Bour, VI, planche I] or [BtD, Proposition V.6.2]) reveals that Ψ = {ϵ_{μ} − ϵ_{ν}|1 ≤ μ ≤ k < ν ≤ m}. Also, there is a unique positive primitive line bundle L_{λ} on G(m, k) with λ = ∑_{ν=1}^{k} ϵ_{k}. In particular, Ψ = Ψ_{1} . Set I_{m} := {1, …, m}; when considering the action corresponding to Y = ∑_{ν=1}^{m} νϵ_{ν}^{∗}, we get by Lemma 8.3

h(G(m, k), \overline{L}_{λ}) = ∑_{{I⊆Im \atop #I=k}} \frac{(∑_{c∈I} c)^{k(m−k)+1}}{∏_{{a∈I \atop b∈Im\I}} (a − b)} ∑_{{a∈I \atop b∈Im\I}} ∑_{l=1}^{k(m−k)+1} \frac{1 − (1 − \frac{a−b}{∑_{c∈I} c})^{l}}{2l(a − b)} .

Example. Assume that G/P is embedded as a hypersurface of degree d into P^{n+1} via L_{λ}. Let N be its normal bundle. Classically, there is an exact sequence

0 → O → ⨁_{1}^{n+2} O(1) → T P^{n+1} → 0

[Ha1, Ex. 8.20.1]. Furthermore, by the adjunction formula ([GH, p. 146])

N = O(d)_{|M} .

Thus ch(T M) = ch(T P_{|M}^{n+1}) − ch(N) = (n + 2)e^{c_{1}(O(1))} − 1 − e^{dc_{1}(O(1))}. Assume now that E_{1} = T M. Then equation (13) yields

h(G/P, \overline{O(1)}) = ∑_{l=1}^{n} \frac{(−1)^{l}}{2(l + 1)}\binom{n + 1}{l + 1} ∫_{M}(n + 2 − d^{l})c_{1}(O(1))^{n}

+\frac{n + 1}{2} ∫_{M} nc_{1}(O(1))^{n}

= ∑_{l=1}^{n} \frac{d(−1)^{l}}{2(l + 1)}\binom{n + 1}{l + 1}(n + 2 − d^{l}) + \frac{n(n + 1)d}{2}

= ∑_{l=2}^{n+1} \frac{1}{2l} [d(n + 2) − 1 + (1 − d)^{l}]  (18)

By [Ko, Th. III.2.3 and next paragraph], all complex hypersurfaces of dimension n > 1 with a non-trivial holomorphic C^{∗}-action have degree d ≤ 2. We reobtain the formula for the height of P^{n} [BoGS]

h(P^{n}, \overline{O(1)}) = \frac{n + 1}{2} ∑_{k=1}^{n} \frac{1}{k} − \frac{n}{2} .

Also [Bour, VI, planche IV] shows Ψ = Ψ_{1} for the even-dimensional smooth quadric Q_{2m} with Q_{2m}(C) = SO(2m + 2)/SO(2m)SO(2). Thus we reobtain the result from [CM, Cor. 2.2.10]

h(Q_{2m}, \overline{O(1)}) = (2m + 1) ∑_{k=1}^{2m−1} \frac{1}{k} + \frac{1}{2} ∑_{k=1}^{m−1} \frac{1}{k} − 2m + 1 + \frac{1}{m} .

Now consider the case of the odd-dimensional smooth quadric Q_{2m−1} with Q_{2m−1}(C) = SO(2m + 1)/SO(2m − 1)SO(2). Identifying C^{m} with 𝔱^{∨} as in [BtD, proposition V.6.5] and denoting the cartesian base by (ϵ_{ν})_{ν=1}^{m}, we notice Ψ = {ϵ_{1} ± ϵ_{ν}|1 < ν ≤ m} ∪ {ϵ_{1}} and λ = ϵ_{1} (compare also [Bour, VI,planche II]). Thus T M = E_{1} + E_{2} with E_{2} = O(1). Hence there is an additional term to equation (18) given by

∑_{l=0}^{n} \frac{(−1)^{l}}{2(l + 1)}\binom{n + 1}{l + 1} ∫_{M}(2^{l+1} − 1)c_{1}(O(1))^{n}

= ∑_{l=0}^{n} \frac{d(−1)^{l}}{2(l + 1)}\binom{n + 1}{l + 1}(2^{l+1} − 1) = ∑_{l=1}^{n+1} \frac{(−1)^{l+1}}{l}

and we find

h(Q_{2m−1}, \overline{O(1)}) = (2m + 1) ∑_{k=1}^{2m−1} \frac{1}{k} − \frac{1}{2} ∑_{k=1}^{m−1} \frac{1}{k} − 2m + 1

which is exactly the same value as obtained in [CM, Cor. 2.2.10] for the singular quadric X_{0}^{2} + ⋯ + X_{2m}^{2} = 0.

References

[Abbes] Abbes, A.: Hauteurs et discrétitude [d’après L. Szpiro, E. Ullmo et S. Zhang]. Sém. Bourbaki, no. 825 (1997).

[A1] Andersen, H. H.: Filtration of cohomology modules for Chevalley groups. Ann. Scient. Éc. Norm. Sup. 16, 495-528 (1983).

[A2] Andersen, H. H.: Torsion in the cohomology of line bundles on homogeneous spaces for Chevalley groups. Proc. Amer. Math. Soc. 96, 537-544 (1986).

[AB] Atiyah, M., Bott, R.: A Lefschetz fixed point formula for elliptic complexes II: Applications. Ann. of Math. 88, 451-491 (1968).

[BeGeV] Berline, N., Getzler, E., Vergne, M.: Heat kernels and Dirac operators: Springer 1992.

[Be] Besse, A.: Einstein manifolds: Erg. der Math. 10, Springer 1987.

[BV] Bismut, J.-M., Vasserot, E.:The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle. Comm. Math. Phys. 125, 355-367 (1989).

[BH] Borel, A., Hirzebruch, F.: Characteristic classes and homogeneous spaces, I. Am. J. Math. 80, 458-538 (1958).

[BoGS] Bost, J.-B., Gillet, H., Soulé, C.: Heights of projective varieties and positive Green forms. J. Amer. Math. Soc. 7, n. 4, 903-1027 (1994).

[Bott] Bott, R.: Homogeneous vector bundles. Ann. Math. 66, 203-248 (1957).

[Bour] Bourbaki, N.: Groupes et algèbres de Lie I-VIII: Hermann 1971-1975.

[BtD] Bröcker, T., tom Dieck, T.: Representations of Compact Lie Groups: GTM 98, Springer 1985.

[CM] Cassaigne, J., Maillot, V.: Hauteurs des hypersurfaces et fonctions zêta d’Igusa. J. Number Theory 83, 226–255 (2000).

[Do] Donnelly, H.:Spectrum and the fixed point set of isometries I. Math. Ann. 224, 161-176 (1976).

[GH] Griffith, P., Harris, J.: Principles of algebraic geometry: Wiley Interscience 1978.

[GS3] Gillet, H., Soulé, C.: Characteristic classes for algebraic vector bundles with hermitian metrics I, II. Ann. of Math. 131, 163-203, 205-238 (1990).

[GS5] Gillet, H., Soulé, C.: Analytic torsion and the arithmetic Todd genus. Topology 30, 21-54 (1991). With an Appendix by Zagier, D.

[Ha1] Hartshorne, R.: Algebraic geometry: Graduate texts in math. 52, Springer 1977.

[Ha2] Hartshorne, R.: Residues and duality: Springer Lecture Notes 20, Springer 1966.

[He] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces: Academic Press 1978.

[Hu] Humphreys, J. E.: Introduction to Lie Algebras and Representation Theory: GTM 9, Springer 1972.

[J1] Jantzen, J. C.: Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariant Formen. Bonner math. Schriften 67 (1973).

[J2] Jantzen, J. C.: Representations of Algebraic Groups: Academic Press 1987.

[Ko] Kobayashi, S.: Transformation Groups in Differential Geometry: Springer 1972.

[K1] Köhler, K.: Equivariant analytic torsion on P^{n}(C). Math. Ann. 297, 553-565 (1993).

[K2] Köhler, K.: Holomorphic torsion on Hermitian symmetric spaces. J. Reine Angew. Math. 460, 93-116 (1995).

[K3] Köhler, K.: Equivariant Reidemeister torsion on symmetric spaces. Math. Ann. 307, 57-69 (1997).

[KR1] Köhler, K., Roessler, D.: Un théorème du point fixe de Lefschetz en géométrie d’Arakelov. C. R. Acad. Sci. Paris 326, Série I, 719-722 (1998).

[KR2] Köhler, K., Roessler, D.: A fixed point formula of Lefschetz type in Arakelov geometry I: statement and proof. Inv. Math. 145, 333-396 (2001).

[KR3] Köhler, K., Roessler, D.: A fixed point formula of Lefschetz type in Arakelov geometry II: a residue formula. Preprint SFB 256 no. 604 (1999), Universität Bonn. To appear in Ann. Inst. Fourier.

[Ma1] Ma, Xiaonan: Formes de torsion analytique et familles de submersions I. Bull. Soc. Math. France 127, 541–621 (1999). Formes de torsion analytique et familles de submersions II. Asian J. Math. 4, 633–667 (2000).

[Ma2] Ma, Xiaonan: Submersions and equivariant Quillen metrics. Ann. Inst. Fourier (Grenoble) 50, 1539–1588 (2000).

[M] Maillot, V.: Un calcul de Schubert arithmétique. Duke Math. J. 80, 195-221 (1995).

[N] Nielsen, N.: Theorie des Integrallogarithmus und verwandter Transcendenten: Teuber, Leipzig 1906.

[SGA3] Demazure, M., Grothendieck, A.: Schémas en Groupes, SGA3: Lecture Notes in Math. 151-153, Springer 1970.

[SB] Schmid, W., Bolton, V.: Discrete series. In: Bailey, T. N. (ed.) et al., Representation theory and automorphic forms. Providence, AMS. Proc. Symp. Pure Math. 61, 83-113 (1997).

[SABK] Soulé, C., Abramovich, D., Burnol, J.F. and Kramer, J.: Lectures on Arakelov Geometry (Cambridge studies in math. 33): Cambridge university press 1992.

[T1] Tamvakis, H.: Schubert calculus on the arithmetic Grassmannian. Duke Math J. 98, 421-443 (1999).

[T2] Tamvakis, H.: Arithmetic intersection theory on flag varieties. Math. Ann. 314, 641-665 (1999).

[T3] Tamvakis, H.: Arakelov theory of the Lagrangian Grassmannian. J. Reine Angew. Math. 516, 207–223 (1999).

[T4] Tamvakis, H.: Height formulas for homogeneous varieties. Michigan Math. J. 48, 593-610 (2000).

[V] Vogan, D.: Representation of real reductive Lie groups: Progress in Math. 15, Birkhäuser 1981.

[Z] Zhang, Shouwu: Positive line bundles on arithmetic varieties. J. Amer. Math. Soc. 8, 187-221 (1995).