Twisted limit formula for torsion and cyclic base change

Twisted limit formula for torsion and cyclic base change Tome 4 (2017), p. 435-471.

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TWISTED LIMIT FORMULA FOR TORSION AND CYCLIC BASE CHANGE

by Nicolas Bergeron & Michael Lipnowski

Abstract. — LetG be the group of complex points of a real semi-simple Lie group whose fundamental rank is equal to1, e.g.G= SL2(C)×SL2(C)orSL3(C). Then the fundamental rank ofGis2, and according to the conjecture made in [3], lattices inGshould have ‘little’

— in the very weak sense of ‘subexponential in the co-volume’ — torsion homology. Using base change, we exhibit sequences of lattices where the torsion homology grows exponentially with thesquare root of the volume. This is deduced from a general theorem that compares twisted and untwistedL²-torsions in the general base-change situation. This also makes uses of a precise equivariant ‘Cheeger-Müller Theorem’ proved by the second author [23].

Résumé(Formule de multiplicité limite tordue pour la torsion et changement de base cyclique) SoitGle groupe des points complexes d’un groupe de Lie semi-simple réel dont le rang fon- damental est égal à 1, par exempleG= SL2(C)×SL2(C)ouSL3(C). Alors le rang fondamental deGest égal à2et, selon la conjecture faite dans [3], les réseaux dansGdevraient avoir « peu »

— dans le sens très faible de « sous-exponentiel en le co-volume » — de torsion homologique.

En utilisant le changement de base, nous exhibons des suites de réseaux le long desquelles la torsion homologique croît exponentiellement avec la racine carrée du volume. Ce comportement est déduit d’un théorème général qui compare les torsionsL² tordues et non tordues dans la situation générale d’un changement de base. Nous utilisons également une version équivariante précise du « Théorème de Cheeger-Müller » démontrée par le second auteur [23].

Contents

1. Introduction. . . 436

2. The simple twisted trace formula. . . 439

3. Lefschetz number and twisted analytic torsion. . . 442

4. L²-Lefschetz number,L²-torsion and limit formulas . . . 446

5. Computations on a product. . . 451

6. Computations in the caseE=C. . . 453

7. General base change. . . 461

8. Application to torsion in cohomology. . . 462

References. . . 470

Mathematical subject classification (2010). — 11F75, 11F70, 11F72, 58J52.

Keywords. — Homological torsion, limit multiplicities, base change.

1. Introduction

1.1. Asymptotic growth of cohomology. — Let Γ be a torsion-free uniform lattice in a semisimple Lie group G with maximal compact subgroup K. Let Γ_n ⊂Γ be a decreasing sequence of normal subgroups with trivial intersection. It is known that

n→∞lim

dimH^j(Γ_n,C) [Γ : Γn]

converges tob⁽²⁾_j (Γ), thejth L²-Betti number of Γ. Ifb⁽²⁾_j 6= 0 for somej, it follows that cohomology is abundant. However, it is often true that b⁽²⁾_j (Γ) = 0 for all j;

this is the case whenever δ(G) := rank_CG−rank_CK 6= 0. What is the true rate of growth ofbj(Γn) = dimH^j(Γn,C) whenδ(G)6= 0? In particular, is bj(Γn)non-zero for sufficiently largen?

We address this question for ‘cyclic base-change.’ Before stating a general result, let’s give two typical examples of this situation.

Examples

(1) The real semisimple Lie groupG= SL2(C)satisfies δ= 1. Letσ:G→Gbe the real involution given by complex conjugation.

(2) The real semisimple Lie group G= SL₂(C)× · · · ×SL₂(C)(n times) satisfies δ=n. Letσ:G→Gbe the order2nautomorphism of Ggiven byσ(g1, . . . , gn) = (g_n, g1, . . . , g_n−1).

Now letΓ_n ⊂Γ be a sequence of finite index, σ-stable subgroups ofG. It follows from the general Proposition 1.2 below that

X

j

bj(Γn)vol(Γ^σ_n\SL2(R)).

Note that when the Γn’s are congruence subgroups of an arithmetic latticeΓ, then vol(Γ^σ_n\SL2(R))grows like vol(Γn\G)^1/order(σ).

In this paper, we shall more generally consider the case whereGis obtained from a real algebraic group by ‘base change.’ Let Gbe a connected semisimple quasi-split algebraic group defined overR. LetEbe an étaleR-algebra such thatE/Ris a cyclic Galois extension with Galois group generated by σ ∈ Aut(E/R). Concretely, E is eitherR^sorC^s. In the first caseσis of ordersand acts onR^sby cyclic permutation. In the second caseσis of order2sand acts onC^sby(z₁, . . . , z_s)7→(z_s, z₁, . . . , z_s−1). The automorphismσinduces a corresponding automorphism of the groupGof real points ofRes_E/RG.⁽¹⁾ We will furthermore assume thatH¹(σ, G) ={1}; see Section 2.4 for comments on this condition. The following proposition is ‘folklore’ (see e.g. Borel- Labesse-Schwermer [6], Rohlfs-Speh [30] and Delorme [14]).

(1)The assumption that G/Ris quasi-split is used in the case where E =C, where we quote results of Delorme [14] concerning base change from G(R)toG(C). We emphasize that assuming G/Ris quasi-split is unnecessary in the case whereE=R^sand σacts by cyclic permutation. See Section 5.

1.2. Proposition. — Let Γn ⊂ Γ be a sequence of finite index, σ-stable subgroups of G. Suppose that δ(G^σ) = 0. Then we have:

X

j

dimb_j(Γ_n)vol(Γ^σ_n\G^σ).

We prove Proposition 1.2 for certain families {Γn} in Section 4.6 but our real interest here is rather how thetorsion cohomologygrows.

1.3. Asymptotic growth of torsion cohomology. — Let (ρ, F) be a finite dimen- sional representation of G defined overR and suppose that the Γn’s stabilize some fixed latticeO⊂F. The first named author and Venkatesh [3] prove that for ‘strongly acyclic’ [3, §4] representationsρ, there is a lower bound

X

j

log|H^j(Γn,O)tors| c(G, ρ)·[Γ : Γn] for some constantc(G, ρ). In fact, they prove a limiting identity (1.3.1)

P

j(−1)^jlog|H^j(Γn,O)tors|

[Γ : Γn] −→c(G, ρ)

and prove thatc(G, ρ)is non-zero exactly when δ(G) = 1. The numerator of the left side of (1.3.1) should be thought of as a ‘torsion Euler characteristic.’ The purpose of this article is to prove an analogous theorem about ‘torsion Lefschetz numbers.’

To state one instance of our main result, we keep assuming that H¹(σ, G) ={1}

and furthermore assume that:

(1) σhas prime orderpandOFp is trivial.

(2) the Γn’s are σ-stable finite index subgroups of Γ such that T

nΓn ={1} (or, more generally, that satisfy the hypothesis of Section 4.3), and

(3) the representationρis strongly acyclic and can be extended to a finite dimen- sional (twisted) representation ρe of the twisted space Ge = Goσ that is strongly acyclic(see Section 2.6).

Under these hypotheses we shall prove the following:

1.4. Theorem. — We have:

(1.4.1) lim sup

P

jlog|H^j(Γ_n,O)|

vol(Γ^σ_n\G^σ) >0 wheneverδ(G^σ) = 1.

Example. — Let E/Q be an imaginary quadratic extension. Let B be a division algebra of dimension 9 over Qsuch thatB is split at infinity andBE :=B⊗_QE is a division algebra. Let o be a maximal order in B and let oE be its tensor product over Z with the ring of integers of E. Then o^×_E embeds into PGL₃(C). Let O be the set of elements in oE of trace 0, considered as an o^×_E-module by conjugation.

Let Γ be a torsion free congruence subgroup of o^×_E such that Γ is contained in the kernel ofρmod 2. Then the local system OF2 is trivial. Given a primeq, we denote

byΓq the kernel of the reduction map Γ7→(oE/qoE)^×. Theorem 1.4 applies to the sequence{Γq} and we conclude that

(1.4.2) lim sup 1

q⁸ X

j

log|H^j(Γq,O)|>0.

Here, q⁸ is the growth rate of the log of torsion in H³ of the corresponding q- congruence subgroups ofo^× embedded intoPGL₃(R). The cohomology classes that contribute to (1.4.2) should conjecturally arise by base change transfer over Z. One may regard (1.4.2) as a partial evidence for the existence of such a transfer.

Similarly, one can construct examples of lattices Γ in SL2(C)^p (p > 1 prime), in SL₃(C)or inSL₄(C)such that the torsion homology of levelqcongruence subgroups ofΓgrows exponentially with respectivelyq⁶,q⁸ orq¹⁵.

1.5. — Analogously to (1.3.1), Equation (1.4.1) follows from a limiting identity for torsion Lefschetz numbers. For example, whenσ²= 1, conditional on an assumption about the growth of the Betti numbersdim_F2H^j(Γ^σ_n,O_F2)we prove that

(1.5.1)

P(−1)^j(log|H^j(Γn,O)⁺_tors| −log|H^j(Γn,O)⁻_tors|)

|H¹(σ,Γ_n)|vol(Γ^σ_n\G^σ) −→c(G, ρ, σ), where the superscript±denotes the±1 eigenspace.

Assume that the maximal compact subgroupK⊂Gisσ-stable and letX =G/K andX^σ=G^σ/K^σ. The proof of (1.5.1) crucially uses the equivariant Cheeger-Müller theorem, proven by Bismut-Zhang [5]. This enables us to compute the left side of (1.5.1) (up to a controlled integer multiple oflog 2) by studying the eigenspaces of the Laplace operators of the metrized local system associated to ρtogether with their σ action. More precisely, the left side of (1.5.1) nearly equals the equivariant analytic torsionlogT_Γ^σ

n\X(ρ); see (3.5.1) for a definition of the latter. Using the simple twisted trace formula and results of Bouaziz [7], we prove a ‘limit multiplicity formula.’

1.6. Theorem. — Assume(1)–(3)above. Then we have:

(1.6.1) logT_Γ^σ

n\X(ρ)

|H¹(σ,Γ_n)|vol(Γ^σ_n\G^σ)−→s2^rt⁽²⁾_Xσ(ρ),

where E=R^s or C^s, and r= 0in the first case and r= rank_RG(C)−rank_RG(R) in the second case.

Here,t⁽²⁾_Xσ(ρ)is the (usual)L²-analytic torsion of the symmetric spaceX^σ twisted by the finite dimensional representationρ. It is explicitly computed in [3]. Note that it is non-zero if and only ifδ(G^σ) = 1.

The authors hope that the limit multiplicity formula (1.6.1) together with the twisted endoscopic comparison implicit in Section 7 will be of interest independent of torsion in cohomology. These computations complement work by Borel-Labesse- Schwermer [6] and Rohlfs-Speh [30].

Acknowledgements. — The authors would like to thank Abederrazak Bouaziz, Lau- rent Clozel, Colette Moeglin and David Renard for helpful conversations. They also would like to thank an anonymous referee for pointing out an error in the first version of this paper.

2. The simple twisted trace formula

LetGbe a connected semisimple quasi-split algebraic group defined overR. LetE be an étale R-algebra such that E/R is a cyclic Galois extension with Galois group generated byσ∈Aut(E/R). The automorphismσinduces a corresponding automor- phism of the group G of real points of Res_E/RG. We furthermore choose a Cartan involution θ ofGthat commutes withσ and denote byK the group of fixed points ofθ inG. Here we follow Labesse-Waldspurger [20].

2.1. Twisted spaces. — We associate to these data thetwisted space Ge=Goσ⊂GoAut(G).

The left action ofGonG,

(g, xoσ)7−→gxoσ,

turns Ge into a left principal homogeneous G-space equipped with a G-equivariant mapAd :Ge→Aut(G)given by

Ad(xoσ)(g) = Ad(x)(σ(g)).

We also have a right action ofGonGe by

δg= Ad(δ)(g)δ (δ∈G, ge ∈G).

This enables us to define an action by conjugation ofG onGe and yields a notion of G-conjugacy class in G. The left and right actions ofe Gon the twisted space Goσ are induced by the left and right multiplications in the semi-direct productGohσi.

In particular, takingδ= 1oσwe have:

(1oσ)g=δg=σ(g)δ=σ(g)oσ.

We similarly define the twisted spaceKe =Koσ.

2.2. Twisted representations. — A representation ofG, in a vector spacee V, is the data for everyδ∈Ge of a invertible linear map

eπ(δ)∈GL(V) and of a representation ofGin V:

π:G−→GL(V) such that forx, y∈Gandδ∈G,e

π(xδy) =e π(x)eπ(δ)π(y).

In particular

eπ(δx) =π(Ad(δ)(x))π(δ).e

Thereforeπ(δ)e intertwinesπ andπ◦Ad(δ). Note thatπe is the restriction to Goσ of a genuine representation ofGohσi. As such it determinesπ; we will say thatπis the restriction ofπeto G.

Conversely eπ is determined by the data of π and of an operator A which inter- twinesπandπ◦σ:

Aπ(x) = (π◦σ)(x)A

and whosep-th power is the identity, wherepis the order ofσ. We reconstructeπby setting

eπ(xoσ) =π(x)A forx∈G.

Say that πe is essential if π is irreducible. If eπ is unitary and essential, Schur’s lemma implies thatπdeterminesAup to a p-th root of unity.

There is a natural notion of equivalence between representations ofGe— see e.g. [20,

§2.3]. This is the obvious one; beware however that, even if πe is essential, the class ofπdoes not determine the class ofπesince the intertwiner Ais only determined up to a root of unity. We have a corresponding notion of a(g,K)-module.e

Ifeπis unitary andf ∈C_c^∞(G)e we set π(fe ) =

Z

Ge

f(y)eπ(y)dy:=

Z

G

f(xoσ)eπ(xoσ)dx.

It follows from [20, Lem. 2.3.2] that eπ(f)is of trace class. Moreover:traceπ(fe ) = 0 unlessπeis essential. In the following, we denote byΠ(G)e the set of irreducible unitary representationsπof G(considered up to equivalence) that can be extended to some (twisted) representationπeofG. Note that the extension is not unique.e

2.3. Twisted trace formula (in the cocompact case). — LetΓbe a cocompact lattice of Gthat is σ-stable. Associated to Γ is the (right) regular representation ReΓ of Ge on L²(Γ\G), where the restriction R_Γ of Re_Γ is the usual regular representation in L²(Γ\G)and

(Re_Γ(σ))(f)(Γx) =f(Γσ(x)).

Note that

(Re_Γ(σ)R_Γ(g))(f)(Γx) =f(Γσ(x)g) = (R_Γ(σ(g))Re_Γ(σ))(f)(Γx).

Given δ ∈ Ge we denote by G^δ its centralizer in G (for the (twisted) action by conjugation ofGonG). Corresponding toe Γis a (non-empty) discrete twisted subspace Γe⊂G. Givene δ∈Γewe denote by{δ}itsΓ-conjugacy class (where here againΓacts by (twisted) conjugation onΓ).e

Letf ∈C_c^∞(G). The twisted trace formula is obtained by computing the trace ofe ReΓ(f)in two different ways. It takes the following form (the LHS is the spectral side and the RHS is the geometric side):

(2.3.1) X

π∈Π(G)e

m(π,eπ,Γ) traceeπ(f) =X

{δ}

vol(Γ^δ\G^δ) Z

G^δ\G

f(x⁻¹δx)dx.˙

Here, eπis some extension ofπto a twisted representation of Geand m(π,eπ,Γ) = X

πe⁰|G=π

λ(πe⁰,eπ)m(eπ⁰)

= trace σ|Hom_G(π, Le ²(Γ\G)) ,

wherem(eπ⁰)is the multiplicity ofπe⁰ in ReΓ andλ(πe⁰,π)e ∈C^× is the scalar such that, for allδ∈G, we havee πe⁰(δ) =λ(eπ⁰,eπ)eπ(δ).⁽²⁾Note thatλ(πe⁰,π)e is in fact ap-th root of unity.

The definition of traceeπ(f)depends on a choice of a Haar measuredx onG. On the geometric side the volumes vol(Γ^δ\G^δ) depend on choices of Haar measures on the groupsG^δ. We will make precise choices later on. For the moment we just note that the measuredx˙ on the quotientG^δ\Gis normalized by:

Z

G

φ(x)dx= Z

G^δ\G

Z

G^δ

φ(gx)dg dx.˙

2.4. Galois cohomology groups H¹(σ,Γ). — Let Z¹(σ,Γ) ={δ ∈ eΓ : δ^p = 1}; it is invariant by conjugation byΓ. We denote by H¹(σ,Γ)the quotient ofZ¹(σ,Γ)by the equivalence relation defined by conjugation by elements of Γ. We have similar definitions whenΓ is replaced byG.

We will assume that

(2.4.1) H¹(σ, G) ={1}.

Note that the setH¹(σ, G)is the same as the nonabelian cohomology groupH¹(hσi,Γ) or H¹(E/R,Γ) if E/R is a cyclic Galois extension with Galois group generated by σ∈Aut(E/R).

If E=R^p, Condition (2.4.1) is always satisfied. Indeed: in that case G=G(R)^p and an element

(g₁, . . . , g_p)oσ∈Ge

belongs toZ¹(σ, G)if and only ifg1g2· · ·gp=e. But then there is an equality σ(g1, g1g2, . . . , g1· · ·gp)⁻¹(g1, g1g2, . . . , g1· · ·gp) = (g1, . . . , gp).

Equivalently, (g1, . . . , gp)oσ is conjugated toσinGe by some element inG.

We furthermore note that H¹(C/R,SLn(C)) = H¹(C/R,Sp_n(C)) = {1}, see e.g. [32, Chap. X]. Therefore, Condition (2.4.1) holds ifGis a product of factorsSL_n orSp_n or of factors whose group of real points is isomorphic to a complex Lie group viewed as a real Lie group. Note however that H¹(C/R,SOp,q(C)) is not trivial; it is in bijective correspondence with the set of non degenerate real quadratic forms that are equivalent to the standard quadratic form of signature (p, q) over C. We emphasize that, in the introduction, we have assumed Condition (2.4.1) to hold.

(2)Note thatm(π,π,e Γ) traceπ(f)e does not depend on the chosen particular extensioneπbut only onπ.

2.5. — Under assumption (2.4.1), the map

H¹(σ,Γ)−→H¹(σ, G)

necessarily has trivial image. In other words: ifδrepresents a class inH¹(σ,Γ)thenδ is conjugate toσby some element of G. In particular, we have

vol(Γ^δ\G^δ) = vol(Γ^σ\G^σ) and Z

G^δ\G

f(x⁻¹δx)dx˙ = Z

G^σ\G

f(x⁻¹σx)dx.˙ We may therefore write the geometric side of the twisted trace formula as:

(2.5.1) |H¹(σ,Γ)|vol(Γ^σ\G^σ) Z

G^σ\G

f(x⁻¹σx)dx˙

+ X

{δ}

δ /∈Z¹(σ,Γ)

vol(Γ^δ\G^δ) Z

G^δ\G

f(x⁻¹δx)dx.˙

2.6. Finite dimensional representations of Ge. — Note that the complexification ofGmay be identified with the complex points ofRes_E/RG, i.e.,G(C)^p. Every com- plex finite dimensionalσ-stable irreducible representation(eρ, F)ofGecan therefore be realized in a spaceF =F₀^⊗p, where(ρ0, F0)is an irreducible complex linear represen- tation ofG(C). The action ofGis defined by the tensor product actionρ^⊗p₀ ifE=R^p and by⊗^p/2_i=1(ρ0⊗ρ₀), where ρ₀ is obtained by composing the complex conjugation in G(C)byρ0, ifE=C^p/2. In both cases, we choose the action ofσonF =F₀^⊗p to be the cyclic permutationA:x₁⊗ · · · ⊗x_p7→x_p⊗x₁⊗ · · · ⊗x_p−1. Note that

trace(σ|F) = dimF₀.

Let gbe the Lie algebra ofG. Say that (ρ, Fe ) isstrongly twisted acyclic if there exists a positive constant η depending only on F such that: for every irreducible unitary(g,K)-modulee V for which

trace(σ|C^j(g(C), K, V ⊗F))6= 0 for somej 6dimG(C), the inequality

ΛF−ΛV >η

is satisfied. Here,ΛF, resp.ΛV, is the scalar by which the Casimir acts onF, resp.V. Writeν for the highest weight ofF0. The following lemma can be proven analo- gously to [3, Lem. 4.1].

2.7. Lemma. — Suppose thatν is not preserved by the Cartan involution θ. Then ρe is strongly twisted acyclic.

3. Lefschetz number and twisted analytic torsion

LetG,σandΓbe as in Section 2.3 and let(ρ, Fe )be a complex finite dimensional σ-stable irreducible representation ofG. We denote bye gthe Lie algebra ofG.

3.1. Twisted(g(C), K)-cohomology and Lefschetz number. — We can define an ac- tion ofσon each cohomology groupHⁱ(Γ\X, F)and thus define aLefschetz number

Lef(σ,Γ, F) =X

i

(−1)ⁱtrace(σ|Hⁱ(Γ\X, F)).

If V is a (g,K)-module, we have a natural action ofe σ on the space of (g, K)- cochainsC^•(g(C), K, V)which induces an action on the quotientH^•(g(C), K, V). We denote by

trace(σ|H^•(g, K, V))

the trace of the corresponding operator. We then define theLefschetz numberofV by Lef(σ, V) =X

i

(−1)ⁱtrace(σ|Hⁱ(g, K, V)).

If F is a finite dimensional representation ofGe thenF⊗V is still a(g,K)-module;e we denote byLef(σ, F, V)its Lefschetz number.

Labesse [19, §7] proves that there exists a compactly supported function Lρ ∈ C_c^∞(G)e such that for every essential admissible representation(eπ, V)ofGe one has

Lef(σ, F, V) = traceπ(Le _ρ).

The functionL_ρ is called the Lefschetz functionforσand(ρ, Fe ).

We then have:

traceRe_Γ(L_ρ) =X

i

(−1)ⁱtrace(σ|Hⁱ(Γ\X, F))

= Lef(σ,Γ, F).

(3.1.1)

3.2. Twisted heat kernels. — Let H_t^ρ,i∈

C^∞(G)⊗End(∧ⁱ(g/k)^∗⊗F)^K×K

be the heat kernel for L²-forms of degree i with values in the bundle associated to (ρ, F). Note that we have a natural action ofσon∧ⁱ(g/k)^∗⊗F; we denote byA_σthe corresponding linear operator and let

h^ρ,i,σ_t :xoσ7−→trace(H_t^ρ,i(x)◦Aσ).

Eventually we shall apply the twisted trace formula toh^ρ,i,σ_t . The heat kernelH_t^ρ,i is not compactly supported. However, it follows from [2, Prop. 2.4] that it belongs to all Harish-Chandra Schwartz spacesC^q ⊗End(∧ⁱ(g/k)^∗⊗F),q >0. This is enough to ensure absolute convergence of both sides of the twisted trace formula.

3.3. Lemma. — Let eπbe an essential admissible representation of Ge and letV be its associated(g,K)-module. We have:e

traceeπ(h^ρ,i,σ_t ) =e^t(ΛV−ΛF)trace(σ|[∧ⁱ(g/k)^∗⊗F⊗V]^K).

Proof. — It follows from theK×K equivariance of H_t^ρ,i and Kuga’s Lemma that, relative to the splitting

∧ⁱ(g/k)^∗⊗F⊗V = [∧ⁱ(g/k)^∗⊗F⊗V]^K⊕ [∧ⁱ(g/k)^∗⊗F⊗V]^K⊥

, we have:

π(H_t^ρ,i) =

e^t(ΛV−ΛF)Id 0

0 0

.

We furthermore note that this decomposition is σ-invariant sinceK is σ-stable. We conclude that we have:

eπ(H_t^ρ,i) :=

Z

G

(π(g)◦A)⊗(H_t^ρ,i(g)◦Aσ)dg=

e^t(ΛV−ΛF)Aσ⊗A 0

0 0

. Here,πis the restriction ofπetoGandAis the intertwining operator betweenπand π◦σthat determinesπ.e

Now let {ξ_n}_n∈N and {e_j}_j=1,...,m be orthonormal bases ofV and ∧ⁱ(g/k)^∗⊗F, respectively. Then we have:

traceeπ(H_t^ρ,i) =

∞

X

n=1 m

X

j=1

heπ(H_t^ρ,i)(ξn⊗ej),(ξn⊗ej)i

=

∞

X

n=1 m

X

j=1

Z

G

h(π(g)◦A)ξn, ξnih(H_t^ρ,i(g)◦Aσ)ej, ejidg

=

∞

X

n=1

Z

G

h(π(g)◦A)ξ_n, ξ_nih^ρ,i,σ_t (goσ)dg

= traceπ(he ^ρ,i,σ_t ).

The lemma follows.

Denoting byH_t⁰∈

C^∞(G)⊗End(∧⁰(g/k)^∗)K×K

the heat kernel forL²-functions onX, the following proposition follows from [27, Prop. 5.3] and the definition of strong twisted acyclicity.

3.4. Proposition. — Assume that (ρ, Fe ) is strongly twisted acyclic. Then there ex- ist positive constants η and C such that for every x ∈ G, t ∈ (0,+∞) and i ∈ {0, . . . ,dimX}, one has:

|h^ρ,i,σ_t (xoσ)|6Ce^−ηtH_t⁰(x).

We define the kernelk_t^ρ,σ by

k^ρ,σ_t (g) =X

i

(−1)ⁱih^ρ,i,σ_t (g);

it defines a function in C^q(G), for alle q >0.

3.5. Twisted analytic torsion. — Thetwisted analytic torsionT_Γ\X^σ (ρ)is then defi- ned by

(3.5.1) logT_Γ\X^σ (ρ)

=1 2

d ds

_s=0

1 Γ(s)

Z ∞ 0

t^s−1

traceReΓ(k_t^ρ,σ)−X

i

(−1)ⁱi·trace(σ|Hⁱ(Γ\X, F))

dt

.

Note that if (ρ, Fe ) is strongly twisted acyclic eachtrace(σ |Hⁱ(Γ\X, F))is trivial.

From now on we will assume that(ρ, Fe )is strongly twisted acyclic. In particular, we have:

logT_Γ\X^σ (ρ) =1 2

d ds

_s=0

1 Γ(s)

Z ∞ 0

t^s−1traceReΓ(k^ρ,σ_t )dt

.

3.6. Twisted(g, K)-torsion. — IfV is a(g,K)-module ande F is a finite dimensional representation ofGe thenF⊗V is still a(g,K)-module; we define thee twisted (g, K)- torsionofF⊗V by

Lef⁰(σ, F, V) =X

i

(−1)ⁱitrace(σ|Cⁱ(g, K, F⊗V))

=X

i

(−1)ⁱitrace(σ|[∧ⁱ(g/k)^∗⊗F⊗V]^K).

Remark. — We should explain the notationLef⁰. Given a group Gand a G-vector space V, we denote by det[1 −V] the virtual G-representation (that is to say, element of K₀ of the category of G-representations) defined by the alternating sum P

i(−1)ⁱ[∧ⁱV]of exterior powers. This is multiplicative in an evident sense:

(3.6.1) det[1−V ⊕W] = det[1−V]⊗det[1−W].

Now giveng∈G, the derivative _dt^d

_t=1det(t1−g)is equal to the character ofgacting onP

i(−1)ⁱi∧ⁱV. We therefore definedet⁰[1−V] =P

i(−1)ⁱi∧ⁱV. Considering the virtualK-representatione det⁰[1−(g/k)^∗]we have:

(3.6.2) Lef⁰(σ, F, V) = trace

σ|[det⁰[1−(g/k)^∗]⊗F⊗V]^K . This explains our notationLef⁰ for the twisted(g, K)-torsion.

For future reference we note that we have:

(3.6.3) det⁰[1−V ⊕W] = det⁰[1−V]⊗det[1−W]⊕det[1−V]⊗det⁰[1−W].

We also note that Labesse’s proof of the existence ofLρ can easily be modified to get a function L⁰_ρ ∈ C_c^∞(G)e such that for every essential admissible representation (eπ, V)ofGe one has

Lef⁰(σ, F, V) = traceπ(Le ⁰_ρ).

3.7. — It follows from Lemma 3.3 that the spectral side of the twisted trace formula evaluated ink_t^ρ,σis

(3.7.1) traceRe_Γ(k_t^ρ,σ) = X

π∈Π(G)e

m(π,π,e Γ) Lef⁰(σ, F, V_π)e^{t(Λπ−ΛF)},

whereV_π is the(g,K)-module associated to the extensione π.e

4. L²-Lefschetz number,L²-torsion and limit formulas

LetG,σandΓbe as in Section 2.3 and let(ρ, Fe )be as in the preceding sections.

Suppose that(ρ, Fe )is strongly twisted acyclic. Letf ∈C_c^∞(G).e

4.1. — Giveng ∈Gwe definer(g) = dist(gK, eK)with respect to the Riemannian symmetric distance ofX =G/K. We extendrtoGohσiby settingr(goσ) =r(g).

Note thatr(gg⁰)6r(g) +r(g⁰).

Now, given x∈G we set `(x) = inf{r(gxg⁻¹) : g ∈G}; it only depends on the conjugacy class ofxin G. Recall that theinjectivity radius ofΓ,

rΓ =1

2inf{`(γ) : γ∈Γ− {e}},

is strictly positive. If δ ∈ Γe with δ /∈ Z¹(σ,Γ), then δ^p ∈ Γ− {1} and therefore

`(δ^p)>2r_Γ. In particular, for anyx∈Gwe have:

(4.1.1) 2rΓ 6`(δ^p)6r(xδ^px⁻¹) =r((xδx⁻¹)· · ·(xδx⁻¹))6p·r(xδx⁻¹).

4.2. Lemma. — There exist constants c1, c2>0, depending only on G, such that for any x∈G, we have:

N(x;R) :=

δ∈Γ :e δ /∈Z¹(σ,Γ) and r(xδx⁻¹)6R 6c1p^dr^−d_Γ e^c2R, wheredis the dimension of X.

Proof. — It follows from (4.1.1) that it suffices to prove the lemma forR >2r_Γ/p.

Setε:=rΓ/p. By definition ofr(xδx⁻¹)we haveB(γ(σ(x)), ε)⊂B(x, R+ε), for all δ = γoσ ∈ eΓ with δ /∈Z¹(σ,Γ) and r(xδx⁻¹)6 R. Now, since ε < rΓ, the balls B(γ(σ(x)), ε),γ∈Γ, are all disjoint of the same volume. We conclude that

N(x;R)·volB(σ(x), ε)6volB(x, R+ε)6volB(x,2R).

We conclude using standard estimates on volumes of balls (see e.g. [1, Lem. 7.21] for

more details).

4.3. — Now, let{Γ_n}be a normal chain withT

nΓ_n ={1}or more generally a family of finite index normal subgroups such that for every16=γ∈Γ, the set{n:γ∈Γn} is finite. We have:

4.4. Lemma. — Let δ /∈ Z¹(σ,Γ). There exists a constant nδ such that for every n>n_δ,

|{γ∈Γn\Γ : γδγ⁻¹∈eΓn}|= 0.

Proof. — Recall that forh∈Γ, we define

Norm(h) :=hσ(h)· · ·σ^p−1(h).

Writingδ=hoσ, the conditionδ /∈Z¹(σ,Γ)is equivalent toNorm(h)6= 1. Suppose thatγδγ⁻¹∈Γen for someγ∈Γn\Γ. Equivalently,:=γhσ(γ⁻¹)∈Γn. Therefore,

Norm() =γNorm(h)γ⁻¹∈Γ_n.

Since Γ_n is normal in Γ, this implies that Norm(h) ∈ Γ_n. Since Norm(h) 6= 1, it follows that Norm(h)∈Γ_n for at most finitely manyn. Therefore,

γ∈Γ_n\Γ : γδγ⁻¹∈eΓ_n = 0

for all but finitely manyn.

We will also need the following:

4.5. Lemma. — There is a uniform upper bound

|{γ∈Γn\Γ :γδγ⁻¹∈Γen}|

[Γ^σ: Γ^σ_n]· |H¹(σ,Γ_n)| 61.

Proof. — Writeδ=hoσ. Letγ0∈Γn\Γsatisfyγ0σγ₀⁻¹∈Γen. Equivalently, we have γ0hσ(γ₀⁻¹)∈Γn. Then we have

hσ(γ₀⁻¹)γ0∈γ₀⁻¹Γnγ0= Γn ⇐⇒ hΓn=γ₀⁻¹σ(γ0)Γn.

Supposeγ∈Γn\Γis another element satisfyingγδγ⁻¹∈Γen. Then similarlyhΓn= γ⁻¹σ(γ), from which it follows thatσ(γγ⁻¹₀ ) =γγ₀⁻¹. Conversely, ifγ=gγ₀for some g∈(Γn\Γ)^σ, thenγhσ(γ⁻¹)∈Γn. Therefore,

|{γ∈Γ_n\Γ :γδγ⁻¹∈Γe_n}|=|(Γ_n\Γ)^σ|.

To estimate the size of(Γ_n\Γ)^σ, we use the long exact sequence of pointed sets 1−→Γ^σ_n −→Γ^σ−→(Γ_n\Γ)^σ d

−−→H¹(σ,Γ_n)−→H¹(σ,Γ).

The fibers of the connecting map d are orbits of Γ^σ acting on (Γn\Γ)^σ [32, I, §5.4, Cor. 1]. BecauseΓ_nis normal inΓ, the action ofΓ^σon(Γ_n\Γ)^σfactors throughΓ^σ_n\Γ^σ. Therefore, all fibers ofdhave size at most#(Γ^σ_n\Γ^σ). Also,

#image(d) = # ker[H¹(σ,Γn)−→H¹(σ,Γ)]

6#H¹(σ,Γn).

It follows that

|(Γn\Γ)^σ|

[Γ^σ: Γ^σ_n]· |H¹(σ,Γ_n)| 6#image(d)·#largest fiber [Γ^σ: Γ^σ_n]· |H¹(σ,Γ_n)|

6#H¹(σ,Γ_n)·#(Γ^σ_n\Γ^σ) [Γ^σ: Γ^σ_n]· |H¹(σ,Γn)|

61.

We next give a typical example of a sequence ofσ-stable normal subgroupsΓn CΓ satisfying the hypotheses of Section 4.3.

Example. — Let G/OF,S be a semisimple group, where OF,S denotes the ring of S-integers in a number field F. Let E/F be a cyclic Galois extension with Gal(E/F) = hσi. We fix an integral structure on G so we may speak of G(O_F) andG(O_E).

Fix a finite index,σ-stable subgroupΓ⊂G(OE). Letn⊂OF be an ideal. Let (4.5.1) Γ_n:={γ∈Γ :γ≡1 modnO_E}.

EveryΓn⊂G=G(E_R)is a Galois-stable lattice. Ifnvaries through any sequenceF of ideals satisfying Norm(n) → ∞, then {Γn}n∈F satisfies the hypothesis of Sec- tion 4.3.

4.6. Proposition. — Let f ∈ C_c^∞(G)e and let {Γn} be a sequence of finite index σ-stable subgroups ofΓ that satisfies the hypothesis of Section 4.3. Then

traceReΓ_n(f)

|H¹(σ,Γ_n)|vol(Γ^σ_n\G^σ) −→

Z

G^σ\G

f(x⁻¹σx)dx.˙ Proof. — It follows from (2.5.1) that we have:

traceReΓ_n(f) =|H¹(σ,Γn)|vol(Γ^σ_n\G^σ) Z

G^σ\G

f(x⁻¹σx)dx˙

+ X

{δ}Γn

δ /∈Z¹(σ,Γ_n)

vol(Γ^δ_n\G^δ) Z

G^δ\G

f(x⁻¹δx)dx˙

=|H¹(σ,Γ_n)|vol(Γ^σ_n\G^σ) Z

G^σ\G

f(x⁻¹σx)dx˙

+ X

{δ}Γ

δ /∈Z¹(σ,Γ)

cΓ_n(δ) vol(Γ^δ\G^δ) Z

G^δ\G

f(x⁻¹δx)dx,˙

where

cΓ_n(δ) =

γ∈Γn\Γ : γδγ⁻¹∈eΓn .

Sincef is compactly supported, the last sum above is finite: choosingR >0 so that the support off is contained in

BR={goσ∈Ge : r(g)6R},

we may restrict the sum on the right side of the above equation to the δ’s that are contained inB_R. It follows from Lemma 4.2 that the corresponding sum is finite. We conclude the proof of Proposition 4.6 by applying Lemma 4.4 to the finitely manyδ’s

appearing in this finite sum.

Remark. — In the untwisted case, the condition

|{γ∈Γ_n\Γ : γδγ⁻¹∈Γ_n}|

[Γ : Γn] −→0

is equivalent to the BS-convergence of thecompactquotientsΓn\X towards the sym- metric space X. See [1].

4.7. L²-Lefschetz number. — Proposition 4.6 motivates the following definition of theL²-Lefschetz numberassociated to the triple(G, σ, ρ):

Lef⁽²⁾(σ, X, F) = Z

G^σ\G

Lρ(x⁻¹σx)dx.˙

4.8. Corollary. — Let{Γn}be a normal chain of finite indexσ-stable subgroups ofΓ that satisfies the hypothesis of Section 4.3. Then we have:

Lef(σ,Γ_n, F)

|H¹(σ,Γn)|vol(Γ^σ_n\G^σ) −→Lef⁽²⁾(σ, X, F).

Proof. — Apply Proposition 4.6 to the Lefschetz functionLρ. 4.9. TwistedL²-torsion

Analogously we define thetwistedL²-torsionT_Γ\X^(2)σ(ρ)∈R⁺ by (4.9.1) logT_Γ\X^(2)σ(ρ) =|H¹(σ,Γ)|vol(Γ^σ\G^σ)t^(2)σ_X (ρ),

where t^(2)σ_X (ρ)— which depends only on the symmetric space X, the involution σ, and the finite dimensional representationρ— is defined by

(4.9.2) t^(2)σ_X (ρ) = 1 2

d ds

_s=0

1 Γ(s)

Z ∞ 0

Z

G^σ\G

k_t^ρ,σ(x⁻¹σx)dx t˙ ^s−1dt

.

Note that k_t^ρ,σ is not compactly supported and that we have to prove that the RHS of (4.9.2) is indeed well-defined. Recall however that k_t^ρ,σ belongs to C^q(G).e Lemma 4.2 therefore implies that the series

X

δ∈eΓ

k_t^ρ,σ(x⁻¹δx)

converges absolutely and locally uniformly. This implies that the integral of this series along a (compact) fundamental domain D for the action of Γ on G is absolutely convergent. Restricting the sum to theδ’s that belong to the (twisted)Γ-conjugacy class ofσwe conclude in particular that, for every positivet, the integral

Z

G^σ\G

k^ρ,σ_t (x⁻¹σx)dx˙ = 1 vol(Γ^σ\G^σ)

Z

D

X

δ∈{σ}

k_t^ρ,σ(x⁻¹δx)dx

is absolutely convergent. We postpone the proof of the fact that (4.9.2) is indeed well-defined until sections 6 and 7 where we will explicitly compute t^(2)σ_X (ρ). In the course of the computations we will also prove the following lemma.

4.10. Lemma. — There exist constants C, c >0 such that (4.10.1)

Z

G^σ\G

k^ρ,σ_t (x⁻¹σx)dx˙

6Ce^−ct, t>1.

Taking this lemma for granted, we conclude this section by the proof of the following

‘limit multiplicity theorem’.

4.11. Theorem. — Assume that(ρ, Fe ) is strongly twisted acyclic. Let {Γn} be a se- quence of finite index σ-stable subgroups of Γ that satisfies the hypothesis of Sec- tion 4.3. Then

logT_Γ^σ

n\X(ρ)

|H¹(σ,Γn)|vol(Γ^σ_n\G^σ)−→t^(2)σ_X (ρ).

Proof. — Sincek^ρ,σ_t ∈C^q(G), for alle q >0, we still have:

traceReΓ_n(k^ρ,σ_t ) =|H¹(σ,Γn)|vol(Γ^σ_n\G^σ) Z

G^σ\G

k_t^ρ,σ(x⁻¹σx)dx˙

+ X

{δ}_Γ δ /∈Z¹(σ,Γ)

c_Γn(δ) vol(Γ^δ\G^δ) Z

G^δ\G

k^ρ,σ_t (x⁻¹δx)dx.˙

Note that at this point it is not clear that the sum on the right (absolutely) converges.

This is however indeed the case: it first follows from (4.1.1) that ifδ /∈Z¹(σ,Γ) and x∈Gwe haver(xδx⁻¹)>2rΓ/p. Now, recall from [3, Lem. 3.8] or [27, Prop. 3.1 and (3.14)] that we have, fort∈(0,1],

(4.11.1) |k^ρ,σ_t (x⁻¹δx)|6Ct^−dexp

−cr(xδx⁻¹)² t

6Ce^−c0/texp −c⁰⁰r(xδx⁻¹)² . (Here, c⁰ depends on rΓ.) From this and Lemma 4.2, it follows that the geometric side of the trace formula evaluated ink^ρ,σ_t indeed absolutely converges. Moreover, it follows from (4.11.1) together with the uniform bound of Lemma 4.5 that we have:

(4.11.2) traceReΓn(k_t^ρ,σ)

|H¹(σ,Γn)|vol(Γ^σ_n\G^σ) = Z

G^σ\G

k^ρ,σ_t (x⁻¹σx)dx˙+O(e^−c0/t) (0< t61), where the implied constant in theO(e^−c0/t)is uniform inn. We conclude that

Z 1 0

t^s−1

traceReΓ_n(k^ρ,σ_t )

|H¹(σ,Γ_n)|vol(Γ^σ_n\G^σ)− Z

G^σ\G

k^ρ,σ_t (x⁻¹σx)dx˙

dt

is holomorphic issin a half-plane containing0, so that 1

2 d ds

_s=0

1 Γ(s)

Z +∞

0

t^s−1

traceRe_Γn(k_t^ρ,σ)

|H¹(σ,Γn)|vol(Γ^σ_n\G^σ)− Z

G^σ\G

k^ρ,σ_t (x⁻¹σx)dx˙

dt

= Z +∞

0

traceReΓ_n(k_t^ρ,σ)

|H¹(σ,Γ_n)|vol(Γ^σ_n\G^σ)− Z

G^σ\G

k^ρ,σ_t (x⁻¹σx)dx˙ dt

t . Now it follows from Proposition 3.4 that there exists some positive η such that

|k^ρ,σ_t (xoσ)| e^−ηtH_t⁰(x). In particular, |k_t^ρ,σ(xoσ)| e^−ηtH₁⁰(x) if t > 1 and we have:

|traceReΓ_n(k_t^ρ,σ)|

|H¹(σ,Γn)|vol(Γ^σ_n\G^σ) e^−ηtX

{δ}Γ

Z

G^δ\G

H₁⁰(x^−σδx)dx.˙

The sum on the right-hand-side splits into an infinite sum over theδ’s not inZ¹(σ,Γ) and the finite sum of the remaining terms. By (4.11.1) the infinite sum is absolutely

convergent and independent oftandn. We use Lemma 4.10 to bound the finite sum.

And we get:

|traceReΓ_n(k^ρ,σ_t )|

|H¹(σ,Γ_n)|vol(Γ^σ_n\G^σ) e^−ηt,

where the implicit constant does not depend on n. Using Lemma 4.10, we conclude that both

Z +∞

1

traceRe_Γn(k_t^ρ,σ)

|H¹(σ,Γn)|vol(Γ^σ_n\G^σ) dt

t and

Z +∞

1

Z

G^σ\G

k_t^ρ,σ(x⁻¹σx)dx˙dt t are absolutely convergent uniformly inn.

It follows from Equation (4.11.2) and the last paragraph (the former for0< t61 and the latter fort>1) that

traceRe_Γn(k^ρ,σ_t )

|H¹(σ,Γn)|vol(Γ^σ_n\G^σ)− Z

G^σ\G

k^ρ,σ_t (x⁻¹σx)dx˙

is bounded by a function oftwhich does not depend onnand is absolutely integrable on(0,+∞). From this and Lebesgue’s dominated convergence theorem, we conclude

that Proposition 4.6 implies Theorem 4.11.

5. Computations on a product

In this section we compute the L²-Lefschetz numbers and twisted L²-torsion and in particular prove Lemma 4.10 in the case whereE=R^p(the product case).

Here, we suppose thatE=R^p. ThenGis thep-fold product ofG(R)andσcycli- cally permutes the factors ofG. We will abusively denote byG^σthe groupG(R). Let (ρ0, F0)be an irreducible complex linear representation ofG(C). We denote by(ρ, Fe ) the corresponding complex finite dimensionalσ-stable irreducible representation ofG.e Recall thatF =F₀^⊗p, thatGacts by the tensor product representationρ^⊗p₀ and thatσ acts by the cyclic permutationA:x₁⊗ · · · ⊗x_p 7→x_p⊗x₁⊗ · · · ⊗x_p−1. We finally letX andX^σ be the symmetric spaces corresponding to Gand G^σ respectively, so X = (X^σ)^p.

5.1. Heat kernels of a product. — The heat kernelsH_t^ρ,j decompose as (5.1.1) H_t^ρ,j(g1,· · · , gp) = X

a1+···+ap=j

H_t^ρ0,a1(g1)⊗ · · · ⊗H_t^ρ0,ap(gp).

Now the twisted orbital integral ofH_t^ρ,jassociated to the class of the identity element is given by

Z

G^σ\G

H_t^ρ,j(g^−σg)dg

◦Aσ.

But becauseH_t^ρ,j(g^−σg)preserves all of the summands in the decomposition of (5.1.1) andσmaps the(a1, . . . , ap)-summand to the(ap, a1, . . . , a_p−1)-summand, only those summands for whichj =pa anda1=· · ·=ap=acan contribute to the trace of the

above twisted orbital integral. Furthermore, by a computation identical to that done for scalar-valued functions in [21, §8], we see that

Z

G^σ\G

H_t^ρ0,a(g⁻¹_p g1)⊗· · ·⊗H_t^ρ0,a(g_p−1⁻¹ gp)dg

◦Aσ= (−1)^a2(p−1)H_t^ρ0,a∗· · ·∗H_t^ρ0,a(e).

This implies that Z

G^σ\G

h^ρ,pa_t (x⁻¹σx)dx˙ = trace Z

G^σ\G

H_t^ρ0,a(g_p⁻¹g1)⊗ · · · ⊗H_t^ρ0,a(g_p−1⁻¹ gp)dg˙

◦Aσ

= (−1)^a2(p−1)trace (H_t^ρ0,a∗ · · · ∗H_t^ρ0,a(e))

= (−1)^a2(p−1)trace H_pt^ρ0,a(e)

= (−1)^a2(p−1)h^ρ_pt^0,a(e).

Here, H_pt^ρ0,a is an untwisted heat kernel on X^σ. Lemma 4.10 therefore follows from standard estimates (see e.g. [3]). Moreover, computations of theL²-Lefschetz number and of the twistedL²-torsion immediately follow from the above explicit computation.

5.2. Theorem(L²-Lefschetz number of a product). — We have:

Lef⁽²⁾(σ, X, F) =







(−1)^12dimXσ(dimF0) χ(X_u^σ)

vol(X_u^σ) ifδ(G^σ) = 0,

0 if not.

Here, X_u^σ is the compact dual of X^σ whose metric is normalized such that multipli- cation byi becomes an isometryT_eKσ(X^σ)∼=p→ip∼=T_eKσ(X_u^σ).

Proof. — First note⁽³⁾that

Lef⁽²⁾(σ, X, F) = lim

t→+∞

Z

G^σ\G

k_t^ρ(x⁻¹σx)dx˙

= lim

t→+∞

X

a

(−1)^pa Z

G^σ\G

h^ρ,pa_t (x⁻¹σx)dx˙

= lim

t→+∞

X

a

(−1)^ah^ρ_pt^0,a(e).

The computation then reduces to the untwisted case for which we refer to [28].

The computation of the twistedL²-torsion similarly reduces to the untwisted case:

5.3. Theorem(TwistedL²-torsion of a product). — We have:

t^(2)σ_X (ρ) =p·t⁽²⁾_Xσ(ρ).

(3)Beware thatρis not assumed to be strongly acyclic here!