Cyclic cohomology of crossed products byalgebraic groups

Victor Nistor

April 27, 2004

Contents

1 Introduction 2

2 Precyclic objects 62.1 Cyclic and Quasicyclic Objects . . . . . . . . . . . . . . . . . 62.2 Examples of quasicyclic objects . . . . . . . . . . . . . . . . . 82.3 Cyclic and Hochschild homology for precyclic objects . . . . . 92.4 The Main Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 12

3 General Results 143.1 The action of G on L(U,G1) . . . . . . . . . . . . . . . . . . . 143.2 Cyclic cohomology and AdG-invariant functions in the com-

pact case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.3 G-homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Reduction to the Maximal Compact Subgroup 194.1 The ring C∞inv(G) . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Localization at x and at Gx . . . . . . . . . . . . . . . . . . . 204.3 Definition of X (j) and δ . . . . . . . . . . . . . . . . . . . . . . 224.4 The definition of σ . . . . . . . . . . . . . . . . . . . . . . . . 234.5 The definition of η and the equation ∇ = σδ + δσ . . . . . . . 234.6 The condition of the Main Lemma are satisfied . . . . . . . . . 254.7 The Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . 274.8 Some nonalgebraic groups . . . . . . . . . . . . . . . . . . . . 28

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5 More on the Case G Compact 29

6 Appendix 32

1 Introduction

“Quantum space” is one of the buzz words of the last couple of years inmathematics and physics.

A “quantum space” X is not a space, in the same way as a “quantumgroup” is not a group. It is not defined as a set but rather by means ofFunct(X), “the algebra of functions” defined on it [8, 21, 10].

It has been known for quite a while what meaning to give to notions suchas a “measure on a quantum space” or even “the K-theory of a quantumspace”. In the former case this is because K0(C(X)) = K0(X), where K0

is the algebraic K-theory of the algebra C(X) of continuous functions on acompact space X, and K0 is the usual K-theory defined using vector bundles[2]. One can see that the following pattern appears in the generalization of aconcept from the classical case to “quantum spaces”: first define that conceptin terms of Funct(X) and, if that definition nowhere requires commutativityof the algebra, we are entitled to promote that definition to the case of“quantum spaces”.

In [8] Connes defined the cyclic cohomology groups HC∗(A) of an algebraA over the complex numbers and a pairing K∗(A) ⊗ HC∗(A) → C. Healso proved that if A = C∞(X) for a compact smooth manifold X thenHCn(C∞(X)) ' Ωn(X)/dΩn−1(X)⊕Hn−2

DR (X)⊕ . . .. This shows that the DeRham cohomology of X can be expressed in terms of the cyclic cohomologyof Funct(X), for Funct(X) = C∞(X). Furthermore the pairing Connes hasdefined recovers the usual Chern-Weyl definition of the Chern classes (i.e. interms of the curvature form). This tells us that cyclic cohomology fits nicelyinto the above pattern and defines the (De Rham-) cohomology of “quantumspaces”. Anticipating a little, from this point of view our results imply,among other things, a determination of the cohomology of the representationspace of some Lie groups. Cyclic homology was introduced independently byTsygan [30]. This pairing (or quantum Chern character) was successfullyused by Connes and Moscovici to obtain important geometric applications.Their approach was to prove a “cohomology index formula” for a generalized

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index theorem.Generalized index theorems are related to the K-theory groups of crossed

product C∗-algebras. The form of some of these K-theory groups, one of thecentral problem in operator algebras and index theory, is conjectured by theBaum and Connes in [3]. In spite of the fact that important results havebeen obtained [9, 18, 27, 28, 29], a solution to this problem is still unknown.

For this reason the determination of the cyclic cohomology groups ofthese “quantum spaces” has become quite important. In fact, in all knowncases, these groups, for suitable “smooth” crossed product algebras, wereshown to behave very much like the K-theory groups of the correspondingC∗-algebras. The known cases are: Z [23, 28], R [9, 11] and groups havinga special manifold as classifying space [18, 24]. In case of (crossed productsby) discrete groups our knowledge of cyclic cohomology goes far beyond thatof their K-theory, and in fact it was shown [24] that they behave as expectedfrom the Baum-Connes conjecture, which in its original setting for K-theoryis still open.

It is the purpose of this paper to study the cyclic cohomology of crossedproducts by Lie groups. The main result is the Connes-Kasparov conjecturein cyclic cohomology (the K-theory case is still unsolved).

Let G be a Lie group acting smoothly on a locally convex algebra Aover the complex numbers. Then C∞c (G,A) with the convolution productbecomes a locally convex algebra, denoted here A o G. This is the crossedproduct we shall be concerned with. Note that as in [24] we have chosen themost restrictive behavior at infinity, i.e., the vanishing in a neighborhood ofinfinity. This choice is justified philosophically in the following way. In caseA is the convolution algebra of compactly supported smooth functions ona Lie group G, the underlying “quantum” (or “noncommutative”) space isthe dual of that group, that is, the space of all irreducible representationsof G. Chosing a different convolution algebra means ruling out some ofrepresentations of G and hence does not seem to be reasonable. This alsoexplains why our result implies a determination of the cohomology of therepresentation space of certain Lie groups.

We now indicate our main result. By G we shall mean a real algebraicgroup, that is the set of real points of a complex algebraic group definedover R. C∞inv(G) will denote the ring of smooth AdG-invariant functions onG (class functions).

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Theorem. 1.1 Let G be a connected real algebraic group, K a maximal com-pact subgroup and q = dim(G/K). The periodic cyclic homology groupsPHC∗(AoG) and PHC∗(AoK) are modules over C∞inv(G) such that

PHC∗(AoG)m ' PHC∗+q(AoK)m

for any maximal ideal m of C∞inv(G). So in particular PHC∗(AoG)m vanishesif m is not elliptic. ut

If the group G is not connected the theorem is still true, but q dependson the ideal m1.

This fits with Mackey’s method of orbits, except that now, for reasonswe do not yet understand, we obtain orbits in (Lie G) rather than in (LieG)∗. An interesting feature of the result is worthwhile stressing: there is noγ-obstruction [18] in cyclic cohomology.

The proof is based on the study of L(G,G), a global form of the con-struction we have used in [24]. There we have shown that for G discretethe Connes-Tsygan complex decomposes as a direct sum of subcomplexes in-dexed by the conjugacy classes of G. For Lie groups such a result is no longertrue and is replaced by a C∞inv(G)-module structure on the Connes-Tsygancomplex. So instead of looking at a given conjugacy class we use localiza-tion at the corresponding maximal ideal. That is why localization enters thestatement of the theorem. Actually it also follows that PHC∗(A o G) andPHC∗+q(AoK) are isomorphic as vector spaces; however we do not obtainthis canonically but simply by counting dimensions. Typically the situationis comparable to that of modules of sections in two vector bundles over acompact space having the same dimension at each point.

One of the problems that appears is that L(G,G) is not a cyclic object(i.e. a simplicial object plus a compatible action of the cyclic groups), butrather a quasicyclic object as defined in the first section (i.e. a presimplicialobject with a compatible action of the infinite cyclic group Z which cyclicallypermutes the face operators). What makes L(G,G) interesting and, as weshall see, also useful, is that there exists a smooth G-action on L(G,G)such that L(G,G) ⊗G C ' (A o G)\. One would like then to deduce theusual spectral sequence relating the cohomology of a complex and that of

1in the published paper this condition was omitted, the proof being correct non theless

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its tensor products [20]. Technically this uses a dual form of the van Est’scomplex to obtain a resolution of AoG by quasicyclic objects. We can nowsee what the problem is. One can define the cyclic homology groups onlyfor quasicyclic objects for which the action of Z descends to an action ofthe appropriate cyclic group and this is not the case for the natural objectL(G,G). The Main Lemma contains the technique necessary to overcomethis problem. It uses a “dual Dirac element”, which is however more similarto a construction in Lie algebra homology (“The Poincare duality”) than tothe usual dual Dirac element in bivariant K-theory [18]. The difference is thefact that it is G-invariant rather than only K-invariant. The main drawbackof our construction is that can be performed only locally and that’s why wehad to include localization in the statement of the main result. It is worthmentioning Connes’ observation that for the most interesting applicationssuch as the Selberg Principle localization is enough. It might be possiblethat using computations of [26, 25] one could remove the localization in thestatement of the main theorem.

Proofs of the Selberg principle using cyclic cohomology were obtained byJulg and Valette [16, 17] and Blanc and Brylinski [4] for the case of connectedreductive groups over nonarchimedean local fields. The first authors usethe homotopy invariance of the Chern character in cyclic homology and adeformation of a canonical Fredholm module associated to a group acting ona tree. The second authors use methods closer to ours in that they also definea localization map and use Shapiro’s lemma to reduce the problem from tothe centralizer groups of various elements g ∈ G. In case G is reductive, thecentralizer is commutative (g is a regular element in a Cartan subgroup ofG), and g is not compact, they prove that S = 0 using the computation of thecyclic homology of commutative algebras. Our method does not require suchassumptions. In order to generalize this method from totally disconnectedgroups to our case (G in a class of real Lie groups) we would need to usean algebra larger than C∞c (G), at least containing the coefficients of squareintegrable representations.

The output of the Main Lemma is an exact sequence of reduced quasi-cyclic objects the extreme terms of which are the ones we are interested in(localizations of (AoG)\ and (AoK)\). The middle terms of this sequencehave vanishing PHC∗-groups.

Here, briefly, is the contents of the sections. In the first section we in-troduce quasicyclic objects, we discuss their properties and prove the Main

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Lemma. The second and the third sections contain the constructions we needto use the Main Lemma. The second section also contains a theorem for thecyclic cohomology of coverings: If G1 → G is a finite covering then

HC∗(AoG) ' HC∗(AoG1)H

where H is the covering group. The fourth section contains a Weyl theoremin cyclic cohomology: PHC∗(A o G) ' PHC∗(A o T )W if G is a compactconnected Lie group, T ⊂ G a maximal torus and W the Weyl group ofthe pair (G, T ). These last two theorems are generalizations of well-knownresults in the theory of compact group representation.

The third section also contains results for nonalgebraic groups: for exam-ple we prove that if G is the universal covering group of SL2(R) then

PHC∗(AoG) ' PHC∗+3(A).

The Appendix collects some technical constructions related to C∞inv(G).I would like to express my gratitude to Professor A. Buium for useful

discussions about algebraic groups and to the referee for useful comments.The present paper has been circulated as a preprint of the MathematicalInstitute of the Romanian Academy (INCREST) No.18/1990.

2 Precyclic objects

In this section we introduce the class of quasicyclic objects, a class includingthe class of cyclic objects and whose definition is inspired by [24]. We care-fully study its properties and show how to exploit certain exact sequencesof quasicyclic objects. We will be only interested in quasicyclic objects in acategory of vector spaces over C.

2.1 Cyclic and Quasicyclic Objects

Let us recall the definition of a cyclic object in an abelian categoryM [7]. It isa simplicial object (Xn)n≥0 inM with an extra structure given by an actionof Zn+1 on Xn. The face and degeneracy operators di : Xn → Xn−1, si :Xn → Xn+1, i = 0, . . . , n satisfy the usual simplicial identities [20]:

(S1) didj = dj+1di for i < j (1)

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(S2) sisj = sj−1si for i ≤ j (2)

(S3) disj =

sj−1di for i < j1 for i = j, j + 1sjdi−1 for i > j + 1

(3)

Moreover, if tn+1 denotes the action of the canonical generator of Zn+1 on Xn

then

(C1) ditn+1 =

tndi−1 for 1 ≤ i ≤ ndn for i = 0

(4)

(C2) sitn+1 =

tn+2si−1 for 1 ≤ i ≤ nt2n+2sn for i = 0

(5)

(C3) tn+1n+1 = 1 (6)

The main example [8] is A\ = (A⊗n+1)n≥0 where A is a unital algebra over C

and

di(a0⊗. . .⊗an) =

a0 ⊗ . . .⊗ aiai+1 ⊗ . . .⊗ an for i = 0, . . . , n− 1ana0 ⊗ a1 ⊗ . . .⊗ an−1 for i = n

(7)

si(a0 ⊗ . . .⊗ an) = a0 ⊗ . . .⊗ ai ⊗ 1⊗ ai+1 ⊗ . . .⊗ an, i = 0, . . . , n

tn+1(a0 ⊗ . . .⊗ an) = an ⊗ a0 ⊗ . . .⊗ an−1

If A is a locally convex algebra we replace ⊗ by ⊗ = the projective tensorproduct [12] as in [8].

We now define the class of quasicyclic objects, a class defined such thatit contains objects like A\ even if A is nonunital, and the objects L(A,G, x)defined in [24] (see also 2.2 iii).

Definition. 2.1 A quasicyclic object in an abelian category M is a gradedobject (Xn)n≥0, Xn ∈ Ob(M) together with morphisms di : Xn → Xn−1 fori = 0, . . . , n and Tn+1 : Xn → Xn satisfying (S1), (C1). ut

Observe that we do not require (C3) but that (C1) implies

(Q) djTn+1n+1 = T nn dj

We will be only interested in the case when M is a category of complexvector spaces.

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2.2 Examples of quasicyclic objects

i) Cyclic objects.ii) A\ for a not necessarily unital algebra. We will be interested in the fol-lowing situation. Let A be a complete locally convex algebra, 1 ∈ A, G a Liegroup acting smoothly on A, in the sense that a morphism α : G→ Aut(A)is given such that αg is continuous and unital for each g ∈ G and themap G 3 g → αg(a) ∈ A is smooth for each a ∈ A. Then A o G,the smooth crossed product of A with G is defined as C∞c (G,A) = ϕ ∈C∞(G,A), supp (ϕ) is compact with the convolution product

ϕ ∗ ψ(g) =

∫G

ϕ(h)αh(ψ(h−1g))dh

for ϕ, ψ ∈ AoG and dh a fixed left Haar measure on G.The explicit formulae in this case are A\n = C∞c (Gn+1, A⊗n+1), A = AoG,

(djϕ)(g0, . . . , gn−1) =∫Gdj(1⊗ . . .⊗ 1⊗ αh ⊗ 1⊗ . . .⊗ 1)(ϕ(g0, . . . , gj−1, h, h

−1gj, . . . , gn−1))dh

for j = 0, . . . , n− 1, and

(dnϕ)(g0, . . . , gn−1) =

∫G

dn(αh ⊗ 1⊗ . . .⊗ 1)(ϕ(h−1g0, g1, . . . , gn−1, h))dh

and(tn+1ϕ)(g0, . . . , gn) = tn+1(ϕ(g1, . . . , gn, g0)), ϕ ∈ (AoG)\n.

In these formulae the left hand side dj acts on (AoG)\, while the righthand side dj acts on A\. Also, h appears on the jth position in the firstformula.

iii) Let A and G be as in ii) and suppose that there are given f : G→ G1,a morphism of Lie groups, and U ⊂ G an open set.

Denote Ln(U,G1) = C∞c (U ×Gn+11 , A⊗n+1) and define

(d0ϕ)(γ, g0, . . . , gn) =

∫G

d0 (1⊗ αγ ⊗ 1⊗ . . .⊗ 1)(ϕ(γ, g0, g1, . . . , gn))dg0.

(Tn+1ϕ)(γ, g0, . . . , gn) = (1⊗α−1γ ⊗ 1⊗ . . .⊗ 1) tn+1(ϕ(γ, g1, . . . , gn, ρ(γ)g0)

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for γ ∈ U, g0, . . . , gn ∈ G1, ϕ ∈ Ln(U,G1). (The “hat” means, as usuallythat the corresponding symbol is to be omitted.) Also let dj = T jnd0T

−jn+1 for

j = 1, . . . , n. Explicitly

(djϕ)(γ, g0, . . . , gj, . . . , gn) =

∫G

dj(ϕ(γ, g0, . . . , gn))dgj.

Then (Ln(U,G1))n≥0 is a quasicyclic object.Examples ii) and iii) will be effectively used in computation.

2.3 Cyclic and Hochschild homology for precyclic ob-jects

Definition. 2.2 A quasicyclic object ((Xn)n≥0, dj, Tn+1) will be called pre-cyclic if T n+1

n+1 = 1 for any n ≥ 0. ut

If (Xn)n≥0 is an arbitrary quasicyclic object then (Xn/(1− T n+1n+1 )Xn)n≥0

and (ker(T n+1n+1 : Xn → Xn))n≥0 are examples of precyclic objects.

Given a precyclic object X = ((Xn)n≥0, dj, Tn+1) we can form the Connes-Tsygan complex C(X) as in the cyclic case [8, 19, 30]. Let us recall itsdefinition.

Let b, b′ : Xn → Xn−1 be defined by b′ =∑n−1

j=0 (−1)jdj, b = b′+ (−1)ndn.

Also let ε = 1 − (−1)nTn+1, N =∑n

j=0(−1)njT jn+1. The Connes-Tsygancomplex C(X) associated with X is the total complex associated with theperiod 2 bicomplex

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. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

? ? ?

b −b′ b

Xn+1 Xn+1

Xn+1 · · ·

ε N ε

? ? ?

b −b′ b

Xn Xn

Xn · · ·

ε N ε

? ? ?

b −b′ b

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

? ? ?

b −b′ b

X0 X0

X0 . . .

ε N ε

Then HC∗(X), the cyclic homology of X, is by definition, the homologyof C(X). The homology of the infinite two-sided periodic version of thisbicomplex (with direct products replacing direct sums) defines PHC∗(X),the periodic cyclic homology of X.

If X = A\ we shall write HC∗(A) and PHC∗(A) instead of HC∗(A\) or

PHC∗(A\).

The above definitions are the same as in the cyclic case [7, 19]

We now briefly investigate the extension of Connes’ exact sequence [8] inthe form proved by Loday and Quillen [19].

Let X = ((Xn)n≥0, dj, Tn+1) be a quasicyclic object a category of complexvector spaces. Denote by E(X) the bicomplex

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. . . . . . . . . . . . . .

? ?

b −b′

Xn+1ε

Xn+1

? ?

b −b′

Xn

ε Xn

? ?

b −b′

. . . . . . . . . . . . . .

? ?

b −b′

X0

ε X0

and by HH∗(X) its homology. Here b and b′ are as before and ε = 1 −(−1)nTn+1.

If X is a precyclic object then, as for cyclic objects [8], there exists anexact sequence

· · · I−→HCn(X)S−→HCn−2(X)

B−→HHn−1(X)I−→HCn−1(X)

S−→· · ·

Moreover HC∗(X), PHC∗(X) and S are related by the following exactsequence

0→ lim←

1(HCn+2k+1(X), S)→ PHCn(X)→ lim←

(HCn+2k(X), S)→ 0.

This shows that PHC∗(X) = 0 whenever there exists a positive m such thatSm = 0.

The following vanishing criterion for S was proved in [24].

Proposition. 2.3 Let X = (Xn, n ≥ 0, dj, Tn+1) be a quasicyclic object ina category of complex vector spaces, such that 1 − T n+1

n+1 is injective. ThenS = 0 on HC∗(Xn/(1− T n+1

n+1 )Xn). ut

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Proof. Consider the composition of

E(X)→ E(Xn/(1− T n+1n+1 )Xn)n≥0 −→ C(Xn/(1− T n+1

n+1 )Xn)n≥0.

The line filtration of the first and third bicomplex shows that the abovemap induces an isomorphism on homology. This proves that I is onto andhence that S = 0 for the Connes’ exact sequence of the precyclic object(Xn/(1− T n+1

n+1 )Xn)n≥0. ut

2.4 The Main Lemma

We are now going to introduce a sort of “dual Dirac element”, an importantdevice in establishing isomorphisms of PHC∗-groups.

Suppose the following data is given:a) Two exact sequences of quasicyclic complete locally convex spaces

(E1) 0→ X (q) δ−→X (q−1) δ−→· · · δ−→X (0) → Y → 0

(E2) 0→ X (0) σ−→X (1) σ−→· · · σ−→X (q) → Z → 0

b) A C∞-action of R, η : R → GL(X (j)) for any j = 0, . . . , q such thatη1 = T n+1

n+1 and if ∇ denotes the derivative of ηt at t = 0 then ∇ = δσ + σδ(we define δ(X (0)) = σ(X (q)) = 0).

c) 1− T n+1n+1 is injective with on X (j)

n for any j = 0, . . . , q and any n ≥ 0.

Define X (0) = X (0)/∇X (0), X (j) = X (j)/(∇X (j) + σX (j−1)) for j =1, . . . , q − 1.

Main Lemma. 1 (i) X (j) is a precyclic object for any j = 0, . . . , q − 1.(ii) The complex

0→ Z δ−→ X (q−1) δ−→ · · · δ−→ X (1) δ−→ X (0) → Y → 0

is acyclic, (after identifying Z = X (q)/σX (q−1)).

(iii) PHC∗(X (j)) = 0 for any j = 0, . . . , q − 1.(iv) PHC∗(Y) ' PHC∗+q(Z). ut

Proof. Since a short exact sequence 0 → X ′ → X → X ′′ → 0 of precyclicobjects gives rise to the six term exact sequence

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PHC0(X ′) −→ PHC0(X) −→ PHC0(X ′′)

?

6

PHC1(X ′′)←− PHC1(X)←− PHC1(X ′)

we immediately see that iv) is a consequence of i), ii) and iii). (PHC∗ isa Z2-graded theory.)

The relation 1−T n+1n+1 =

∫ 1

0∇ηtdt = ∇(

∫ 1

0ηtdt) shows that Im(1−T n+1

n+1 ) ⊂Im(∇) (Im means “Image”) and hence X (j) for j = 0, . . . , q − 1 is precyclic.The fact that Z and Y are precyclic will follow from ii). We also obtain that∇ is injective.

Let us first observe that δ∇ = ∇δ. This and δσ = ∇ − σδ show thatδσX (q) ⊂ ∇X (q−1) + σX (q−2), δ(∇X (j) + σX (j−1)) ⊂ ∇X (j−1) + σX (j−2) and∇X (0) ⊂ δX (1). We obtain that the complex in ii) is well defined. We

now prove its exactness. At Y and X (0): this follows immediately from theexactness of (E1). At X (j), j = 1, . . . , q − 1, and Z: observe first thatδX (j+1)∩σX (j−1) = 0. Indeed if x ∈ δX (j+1)∩σX (j−1) then δx = σx = 0 andhence ∇x = 0. Since ∇ is injective we obtain x = 0. Let vj ∈ X (j) be suchthat δvj = ∇vj−1+σvj−2 for some vj−1 ∈ X (j−1) and vj−2 ∈ X (j−2). It followsas in the above discussion that δ(vj − σvj−1) = 0. Then vj = σvj−1 + δvj+1

for some vj+1 ∈ X (j+1). (The undefined terms are to be replaced by 0). Sowe have proved i) and ii).

For iii) let us note that X (j)/σX (j−1) is a quasicyclic object with R actingon it by quasicyclic endomorphisms. This follows since ∇ commutes withσ and with the structural morphisms of X (j) (use ∇ = δσ + σδ). We showthat 1− T n+1

n+1 is injective on X (j)/σX (j−1) for j = 0, . . . , q− 1. Suppose that(1−T n+1

n+1 )vj = σvj−1 for some vj ∈ X (j), vj−1 ∈ X (j−1). Then σ(1−T n+1n+1 )vj =

0 and hence σvj = 0 since 1 − T n+1n+1 was assumed to be injective. The

exactness of (E2) shows that there exists v′j−1 ∈ X (j−1) such that vj = σv′j−1.

The action of R on X (j)/σX (j−1) factors to an action of the compact groupT = R/Z on

(X (j)n /σX (j−1)

n )/(1− T n+1n+1 )(X (j)

n /σX (j−1)n )

whose set of fixed vectors is just χ(j). The proof is completed using Propo-sition 1.6 and the fact that F −→ F T is an exact functor. ut

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3 General Results

In this section we shall be concerned with the properties of the quasicyclicobjects L(U,G1) defined in 2.2 iii). We shall also establish the technical factsneeded for the proof of the exactness of (E1) appearing in the Main Lemma.

3.1 The action of G on L(U,G1)

Let A,G,G1, α : G → Aut(A) and ρ : G → G1 be as in 2.2 iii). Also letU ⊂ G be an AdG-invariant open set. Define β : G→ GL(L(U,G1)) by

(βγ(ϕ))(γ1, g0, . . . , gn) = α⊗n+1γ (ϕ(γ−1γ1γ, ρ(γ)−1g0, . . . , ρ(γ)−1gn)

for any γ, γ1 ∈ G, g0, . . . , gn ∈ G1 and ϕ ∈ Ln(U,G1). Then β defines asmooth action of G.

In the case G = G1, ρ = id and U = G we define p : L(G,G)→ (AoG)\

by

(pϕ)(h0, . . . , hn) =

∫G

Ψ((βγϕ)(gn, g0, g1, . . . , gn))dγ

where g0 = h0, g1 = h0h1, . . . , gn = h0h1 . . . hn, Ψ = (αgn⊗αg0⊗. . .⊗αgn−1)−1

and ϕ ∈ Ln(U,G1).Let C∆ be the G-module having C as underlying space and the G-module

structure given by g ·λ = ∆(g)λ for any g ∈ G and any λ ∈ C. Here ∆ is themodular function of G, it is defined by d(gh) = ∆(h)dg. Let L = L(G,G).

Proposition. 3.1 i) βγ and p are morphisms of quasicyclic objects.ii) p∗ : (A × G)\∗ → L∗ establishes an isomorphism of (A o G)\∗ onto

HomG(C∆, L∗).

(The action of G on L is 〈gf, ϕ〉 = 〈f, g−1ϕ〉.) ut

Proof. The proof of i) is a tedious but straightforward computation whichwe shall omit.

Let Φ : Ln → C∞c (G,C∞c (Gn+1, A⊗n+1))) = C∞c (G, (AoG)\n) be given by

Φ(ϕ)(g−1, h0, . . . , hn) = Ψ((βgϕ)(gn, g0, g1, . . . , gn))

(here Ψ, g0, . . . , gn have the same meaning as above). Φ is obviously anisomorphism which is also G-equivariant since

Φ(βγ(ϕ)(g−1, h0, . . . , hn)) = Ψ(βgγ(ϕ(gn, g0, . . . , gn)))

= Φ(ϕ)(γ−1g−1, h0, . . . , hn).

14

The proof of the proposition will be accomplished after the following lemmais proved.ut

Let G be as above, X a smooth principal G-bundle and B = G \X. Also letE be a complete locally convex space. Then C∞c (X,E) is a complete smoothG-module. Let F = C∆ ⊗ C∞c (X,E).

Lemma. 3.2 i) HomG(C∆, C∞c (X,E)∗) ' H0

d(G,F ∗) ' C∞c (B,E)∗,ii) Hj

d(G,F ∗) = 0 for j > 0.ut

Here Hjd(G,F0), j ≥ 0 denote the differentiable cohomology groups of G

with values in the continuous G-module F0. As in the case G discrete theyrepresent the derived functors of F0 → FG

0 (= H0d(G,F0)) [6, 13].

Proof. It is proved in [13], Observation III.1.3, that H0d(G,C∞(G,E∗)) = E∗

and Hjd(G,C∞(G,E∗)) = 0 for j > 0 see also [6]. The formulae for the homo-

topies in that proof extend by continuity to prove that H0d(G,C∞c (G,E)∗) =

E∗ and Hjd(G,C∞c (G,E)∗) = 0 for j > 0. The isomorphism C∞c (X,E) '

C∆⊗C∞c (G,E) as G-modules proves the theorem for X = G. Since C∞c (G×B,E) ' C∞c (G,C∞c (B,E)), the case X trivial reduces to the case X = G.The general case follows by a partition of unity argument and the C∞(B)-linearity of the homotopies for X = G×B. ut

The isomorphism HomG(C∆, C∞c (G,E)∗) ' C∞c (B,E)∗ is the dual of p0 :

C∞c (X,E)→ C∞c (B,E)

(p0f)(x) =

∫G

f(g−1y)dg,

where y is an arbitrary lifting of x.

3.2 Cyclic cohomology and AdG-invariant functions inthe compact case

Let U ⊂ G be an AdG-invariant open set. Define J : L(U,G1) → L(U, e)by

Jϕ(γ) =

∫Gn+1

1

ϕ(γ, g0, . . . , gn)dg0 . . . dgn.

15

Lemma. 3.3 J is a morphism of quasicyclic objects and HHn(J) is anisomorphism.ut

Proof. The first part is obvious.Let F = ker J . It is enough to show that HHn(F ) = 0. Define F (−1) = 0,

F (j) = ϕ :∫Gj+1

1ϕ(γ, g0, . . . , gn)dg0 . . . dgj = 0 ∀gj+1, . . . , gn ∈ G1 (for

∈ F ∩ Ln(U,G1)). Then F (j) ⊂ F (j+1) and ∪j≥0F(j) = F .

Let us observe that diF(j)n ⊂ F

(j)n−1 for i = 0, . . . , n, diF

(j)n ⊂ F

(j−1)n−1 for

i = 0, . . . , j. So in particular F (j) is invariant for both b and b′. Chooseχ ⊂ C∞c (G1) satisfying

∫G1χ(g)dg = 1. and then define s : F

(j)n → F

(j)n+1 by

the formula

(sϕ)(γ, g0, . . . , gn+1) = sj(ϕ(γ, g0, · · · , gj, gj+2, . . . , gn+1))χ(gj+1)

sj being the degeneracy of A\. Using b = (−1)j+1(dj+1 − dj+2 + . . . +(−1)n−j+1dn) on F (j)/F (j−1) we obtain bs + sb = b′s + sb′ = (−1)j+1 onF (j)/F (j−1). This proves the lemma.ut

We now prove a consequence of the above lemma.

Proposition. 3.4 Let K be a compact group and ρ : K → G be a continuousgroup morphism, then

HC∗(L(K,G)K) ' HC∗(AoK).

ut

Proof. The above lemma shows that J induces an isomorphism

HH∗(L(K,G)) ' HH∗(L(K, e))

Since K is compact we also get that HH∗(L(K,G)K) ' HH∗(L(K, e)K).A standard argument using Connes’ exact sequence shows that J also in-duces an isomorphism HC∗(L(K,G)K) ' HC∗(L(K, e)K). The proof iscompleted using the isomorphism (AoK)\ ' L(K,K)K proved in Proposi-tion 3.1 ii).ut

Let K be a compact Lie group. Consider C∞(K) with the algebra struc-ture given by the convolution product.

16

Corollary. 3.5 [22] HC∗(C∞(K)) ' C∞(K)inv ⊗HC∗(C). ut

Let E and E ′ be two locally convex spaces which are also continuous G-modules in the sense that there are given morphisms G→ GL(E) and G→GL(E ′) such that G 3 g → gξ ∈ E is continuous for each ξ ∈ E (respectivelyξ ∈ E ′). Then E ⊗G E ′ will denote the quotient of E⊗E ′ by the closedsubspace generated by all the tensors g−1ξ ⊗ ξ′ − ξ ⊗ gξ′.

If E is a complete smooth G-module and G is compact, then E ⊗G C isisomorphic to EG. For any G we have (E ⊗G C)∗ ' E∗G.

3.3 G-homology

Let F be a smooth G-module. We now define a complex whose homologymay be called the G-homology of F . It is closely related to van Est’s theorem[6, 13, 31].

Let K ⊂ G be a maximal compact subgroup of G, t = Lie K ⊂ g = Lie G.Define

δ : (Λj(g/t)⊗ F )⊗K C −→ (Λj−1(g/t)⊗ F )⊗K C

by

δ(X1 ∧ . . . ∧ Xj ⊗ ξ) =

j∑i=1

(−1)i+1X1 ∧ . . . ∧ Xi ∧ . . . Xj ⊗Xi(ξ)− (8)

−∑i<k

(−1)i+k[Xi, Xk]⊗ X1 ∧ . . . ∧ Xi ∧ . . . ∧ Xk ∧ . . . ∧ Xj ⊗ ξ (9)

where X1, . . . , Xj ∈ g, ξ ∈ F and X denotes the class of X ∈ g in g/t.We shall be interested only in the case F = C∆ ⊗ C∞c (G,E) for E an

arbitrary locally convex space.

Proposition. 3.6 Let G be a finite component Lie group, K,E and F be asabove. Denote q = dim (G/K). Then the complex

0→ (Λq(g/t)⊗ F )⊗K Cδ−→ . . .

δ−→(Λ0(g/t)⊗ F )⊗K C→ F ⊗G C→ 0

is well defined and acyclic. Moreover we have an isomorphism of C∞inv(G)-modules:

(C∆ ⊗ L(G,G))⊗G C ' (AoG)\.

ut

17

Proof. It is easy to verify that the complex is well defined (see [6, 13]). If wediscard the augmentation (Λ0(g/t)⊗F )⊗K C = F ⊗K C→ F ⊗G C and passto the dual complex, we obtain the complex computing Hj

d(G,F ∗) [6, 13, 31].Since Hj

d(G,F ∗) = 0 for j > 0 and H0d(G,F ∗) = F ∗G = (F ⊗GC)∗ by Lemma

3.2 we obtain the result. The last statement follows from Proposition 3.1. ut

Let H ⊂ G be a central compact subgroup acting trivially on A, f : G→G1 = G/H the quotient morphism. There exists an action of H on L(G,G)and L(G,G1) given by

(hϕ)(γ, g0, . . . , gn) = ϕ(hγ, g0, . . . , gn)

commuting with the action of G defined in section 3.1Let J0 : L(G,G)→ L(G,G1) be given by

J0ϕ(γ, g0, . . . , gn) =

∫Hn+1

ϕ(γ, h0g0, . . . , hngn)dh0 . . . dhn

(gi is the class of gi in G1).Then J0 is a morphism of quasicyclic objects commuting with the actions

of G and H. 3.3 shows HH∗(J0) is an isomorphism. Since H is compact andcentral it is contained in all maximally compact subgroups of G. This showsthat the complex of Lemma 3.6 for L(G,G1) is exact. A spectral sequenceargument shows that

HH∗(J0 ⊗G C) : HH∗((C∆ ⊗L(G,G))⊗G C)→ HH∗((C∆ ⊗L(G,G1))⊗G C)

is an isomorphism and hence also

HC∗(J0 ⊗G C) : HC∗((C∆ ⊗ L(G,G))⊗G C)→ HC∗((C⊗∆L(G,G1))⊗G C)

is an isomorphism. Since (C∆ ⊗ L(G,G)) ⊗G C ' (A o G)\ by Proposition3.1 and 3.6 and L(G,G1)H ' L(G1, G1) we obtain

Theorem. 3.7 Let H ⊂ G be a compact central subgroup, G1 = G/H, thenHC∗(AoG1) ' HC∗(AoG)H and HC∗(AoG1) ' HC∗(AoG)H .ut

This theorem may be viewed as a generalization of a well known theoremin the theory of compact group representation: “if G is compact, then C ⊗R(G1) ' (C⊗R(G))H” [1].

18

4 Reduction to the Maximal Compact Sub-

group

In this section we shall show that PHC∗(A o G) and PHC∗+q(A o K) areisomorphic when localized at the maximal ideals of C∞inv(G) = f ∈ C∞(G),f(γgγ−1) = f(g) for any γ, g ∈ G. Here K is a maximal compact subgroupof G and q = dim (G/K). We shall accomplish this by using the MainLemma, so most of this section will be concerned with obtaining the dataneeded to put us in position to use the Main Lemma.

4.1 The ring C∞inv(G)

Let A,G,G1, ρ : G→ G1 and α : G→ Aut(A) be as in Example 2.2 iii). Alsolet V ⊂ G be an AdG-invariant open set. Define a C∞inv(G)-module structureon L(V,G1) by

(ϕψ)(γ, g0, . . . , gn) = ϕ(γ)ψ(γ, g0, . . . , gn)

for any ϕ ∈ C∞inv(G), ψ ∈ Ln(V,G1). It is easy to see that dj, Tn+1 andβγ(γ ∈ G) are C∞inv(G)-module endomorphisms.

We shall need the following

Lemma. 4.1 Let mx = ϕ ∈ C∞inv(G), ϕ(x) = 0. Then mx, x ∈ G exhaustthe set of all closed maximal ideals of C∞inv(G). If I ⊂ C∞inv(G) is an ideal notcontained in any mx and K0 ⊂ G is an arbitrary compact subset then thereexists ϕ ∈ I, 0 ≤ ϕ ≤ 1 such that ϕ = 1 on K0. ut

Proof. Let I ⊂ C∞inv(G) be an ideal not contained in any mx. Then for anyx ∈ G there exists ϕx ∈ I such that ϕx(x) 6= 0. Replacing ϕx by |ϕx|2 ifnecessary, we may suppose that ϕx ≥ 0. Let K0 ⊂ G be a compact subset.An easy covering argument shows that there exists 0 ≤ ϕ ∈ I such thatϕ ≥ 1 on K0. Let 0 ≤ g ∈ C∞(R), such that g(t) = t−1 for t > 1, andg(t) = 0 for t, 0. Then ψ = (g ϕ)ϕ belongs to I and ψ = 1 on K0.

If I is also closed, let G = ∪Kn, n ≥ 1 with Kn compact for any n ≥ 1.Choose ϕn ∈ I such that 0 ≤ ϕn ≤ 1, ϕ = 1 on Kn. Then Σ∞n≥1anϕn isconvergent in C∞(G) to an invertible element for some sequence an > 0. ut

19

4.2 Localization at x and at Gx

Fix x ∈ G. In order to study the localizations at mx we make some assump-tions on x. Let Gx = γ ⊂ G, γx = xγ, gx = Lie Gx ⊂ g = Lie G. If U is anAdGx-invariant neighborhood of 0 in gx we let G×Gx U = (G×U)/Gx for thefollowing right action of Gx on G × U : (γ1, X)γ = (γ1γ,Ad

−1γ (X)), γ1 ∈ G,

X ∈ U and γ ∈ Gx. We define c : G×Gx U → G by c(γ,X) = γxexp(X)γ−1.c is obviously well defined.

Assumptions Fix x ∈ G. We now introduce some conditions x and U ⊂ gx

must satisfy for our approach to work

(A1) c is a diffeomorphism onto an open set V ⊂ G.

Let R0 = ψ ∈ C∞inv(G), supp(ψ) ⊂ V . The condition (A1) implies thatW = exp(U) is an open set in Gx. Let R1 = ϕ ∈ C∞inv(Gx), supp(ϕ) ⊂ W.Define Φ : R0 → R1,Φ(ψ)(g) = 0 if g /∈ W,Φ(ψ)(g) = ψ(g) if g ∈ W . Weshall also assume

(A2) Φ is an isomorphism, and

(A3) There exist ϕ, ϕ′ ∈ R1 satisfying ϕ(x) = 1 and ϕϕ′ = ϕ.

Conditions for x, U and G to satisfy (A1)-(A3) will be given in the Ap-pendix.

Conventions Let H be a Lie group and M be a C∞inv(H)-module. We shalldenote by My the localization of M at my for y ∈ H.

If not otherwise stated, we shall assume (A1)-(A3) for fixed x ∈ G andU ⊂ gx. V and W will always have the meaning of 3.2.

A consequence of (A2) and (A3) is the following lemma.

Lemma. 4.2 L(V,G)x → L(G,G)x and L(W,Gx)x → L(Gx, Gx)x are iso-morphisms for the C∞inv(G) (respectively C∞inv(Gx))-module structure. ut

20

This justifies the study of L(V,G) and L(W,Gx).Let us observe that V × Gn+1 and W × Gn+1 are G (respectively Gx)

principal bundles for the action γ(γ1, g0, . . . , gn) = (γγ1γ−1, γg0, . . . , γgn),

where g0, . . . , gn ∈ G, γ ∈ G, γ1 ∈ V (respectively, γ ∈ Gx, γ1 ∈ W ). More-over due to (A1) the inclusion W × Gn+1 → V × Gn+1 factors to give adiffeomorphism Gx \ (W ×Gn+1)→ G \ (V ×Gn+1). Lemma 3.2 then gives(C∆⊗L(V,G))⊗GC ' (C∆′⊗L(W,G))⊗Gx C, ∆′ being the modular functionof Gx. Using also Lemma 4.2 we obtain

(A×G)\x ' ((C∆ ⊗ L(G,G))⊗G C)x

' ((C∆ ⊗ L(V,G))⊗G C)x ' ((C∆′ ⊗ L(W,G))⊗Gx C)x (10)

Let σ : Gx \ G → G be a locally bounded borelian section for G → Gx \ G[32] 5.1.1.

Lemma. 4.3 i) There exists a measure µ on Gx \G such that∫G

h(g)dg =

∫Gx×(Gx\G)

h(γσ(t)) dγdµ(t).

ii) If we let Eϕ(γ) =∫Gx\G ϕ(γσ(t))dµ(t) then we obtain a continuous

Gx-invariant linear map E : C∞c (G)→ C∞c (Gx) satisfying∫G

ϕ(g)dt =

∫Gx

Eϕ(γ)dγ.

ut

Proof. Let ψ ∈ Cc(Gx \ G). For an ϕ ∈ Cc(Gx) let Iψ(ϕ) =∫G

(ϕ × ψ) f−1(g)dg where f(γ, t) = γσ(t) defines a borelian isomorphism Gx × (Gx \G) → G. Iψ defines a continuous linear functional Cc(Gx) → C which isGx invariant. Hence there exists µ : Cc(Gx \ G) → C such that Iψ(ϕ) =µ(ψ)

∫Gxϕ(γ)dγ for ψ ∈ Cc(Gx \ G) and ϕ ∈ Cc(Gx). µ is a continuous

linear functional and hence there exists a measure dµ on Gx \ G such thatµ(ψ) =

∫Gx\G ψ(t)dµ(t). This proves i). ii) follows from the definition of dµ

and the locally boundedness of σ.ut

Let (AoG)\V = (C∆⊗L(V,G))⊗GC and define E0 : L(W,G) −→ L(W,Gx)by

E0ϕ(γ, g0, . . . , gn) =

∫(Gx\G)n+1

ϕ(γ, g0σ(x0), . . . , gnσ(xn))dµ(x0) . . . dµ(xn).

21

Proposition. 4.4 i) E0 is a morphism of quasicyclic objects commuting withthe action of Gx.

ii) If x ∈ G satisfies (A1) then HC∗((AoG)\V ) ' HC∗((AoGx)\W ).

iii) If also (A2) and (A3) are satisfied, then HC∗(A o G)x ' HC∗(A o

Gx)x. ut

Proof. i) follows from the above lemma. The above lemma and lemma 3.3show that HH∗(E0) is an isomorphism. Then a standard reasoning (see theproof of Theorem 3.7) shows that

HC∗(f) : HC∗((C∆′ ⊗ L(W,G))⊗Gx C) → HC∗((C∆′ ⊗ L(W,Gx))⊗Gx C)

is also an isomorphism if we denote f = (C∆′ ⊗ E0) ⊗Gx C. Since (C∆ ⊗L(V,G))⊗G C ' (C

′∆⊗L(W,G))⊗Gx C we obtain ii). iii) follows from ii) and

lemma 4.2.ut

4.3 Definition of X (j) and δ

We now continue to define the data needed for the use of the Main Lemma.An element y in a topological group is called a topologically torsion el-

ement if (y) =the closed group generated by y, is compact. If (y) is notcompact, then we shall say that y is topologically torsion free. Let Kx be anymaximal compact subgroup of Gx. Then x is topologically torsion preciselywhen x ∈ Kx.

Let

F =

C∆′ ⊗ L(W,Gx) if x ∈ Kx

C∆′ ⊗ L(W,Gx)/(1− x)(C∆′ ⊗ L(W,Gx)) if x /∈ Kx(11)

Also let M = Kx if x ∈ Kx; otherwise let M be a maximal compact subgroupof Gx/(x). Define t = Lie M and X (j) = (Λj(gx/t)⊗ F )⊗L C.

The definitions of δ and σ as well as their properties are closely relatedto the complex computing the gx-homology of F .

Let Cj = Λjgx⊗F and δ0 : Cj → Cj−1 be defined by a formula similar tothat of δ in 3.6:

δ0(X1 ∧ . . . ∧Xj ⊗ ξ) =

j∑i=1

(−1)i+1X1 ∧ . . . ∧ Xi ∧ . . . ∧Xj ⊗Xi(ξ)

−∑

i<k(−1)i+k[Xi, Xk] ∧X1 ∧ . . . ∧ Xi ∧ . . . ∧ Xk ∧ . . . ∧Xj ⊗ ξ (12)

22

As it is well known, (Cj, δ0) computes H∗(gx, F ) = the homology of thegx-module F .

One can see that X (j) is isomorphic to Cj/C′j, C

′j being the closed subspace

of Cj generated by t ∧ Cj−1 and (γ − 1)Cj for γ ∈ Kx ∪ (x). Also observethat t(Cj) ⊂ C ′j. Recall that δ0(X ∧ ω) = X(ω)−X ∧ δ0(ω) for X ∈ gx andω ⊂ Cj−1, this equation and the Gx-invariance of δ0 show that δ0(C ′j) ⊂ C ′j.The morphism δ defined in the previous section, coincides with the quotientof δ0.

Let dj : Ck → Ck, dj(X1 ∧ . . . ∧Xk ⊗ λ ⊗ ξ) = X1 ∧ . . . ∧Xk ⊗ λ ⊗ djξif X1, . . . , Xk ∈ g, λ ∈ C∆, ξ ∈ F . Define Tn+1 to act only on the factorF . Then Ck becomes a quasicyclic object. Moreover C ′k is invariant for thestructural morphisms and we shall endow X (j) = Cj/C

′j with the quotient

quasicyclic structure.

4.4 The definition of σ

Let Z ∈ C∞(W, gx), Z(xexp(X)) = X. We define σ0 : Ck → Ck+1 byσ0(ω) = Z ∧ ω.

Lemma. 4.5 i) σ0(Y ∧ ω) = −Y ∩ σ0(ω),ii) Y (σ0(ω)) = σ0(Y (ω)) for an Y ∈ gx, ω ∈ Cj.iii) σ0(C ′j) ⊂ C ′j.ut

Proof. i) follows from the definition. Z is AdGx-invariant and hence Y (Z) =0. Then Y (σ0(ω)) = Y (Z ∧ ω) = Y (Z) ∧ ω + Z ∧ Y (ω) = σ0(Y (ω)). Thisproves ii). The last part follows from i), ii) and the Gx-invariance of σ0. ut

We let σ : X (j) → X (j+1) to be the morphism defined by σ0 after modingout C ′j, according to iii) of the lemma.

4.5 The definition of η and the equation ∇ = σδ + δσ

Let η′ : R→ GL(L(W,Gx)).

(η′tϕ)(xexp(X), g0, . . . , gn) = (βexp(tX)ϕ)(xexp(X), g0, . . . , gn)

for any ϕ ∈ Ln(W,Gx), t ∈ R. Then extend this action to Cj by

t · (X1 ∧ . . . ∧Xk ⊗ λ⊗ ξ) = X1 ∧ . . . ∧Xk ⊗ λ⊗ η′t(ξ)

23

for X1, . . . , Xk ∈ gx, λ ∈ C∆ and ϕ ∈ L(W,Gx). Let ∇0 be the derivative ofthis action at 0.

Lemma. 4.6 ∇0 = δ0σ0 + σ0δ0. ut

Proof. Let ω = λ⊗ϕ ∈ C0 = C∆′⊗L(W,Gx). Choose a basis Y1, . . . , Ym of gx

and let y1, . . . , ym be the dual basis. Then Z =∑m

j=1 fjYj if f j(xexp(X)) =

yj(X).Let [Yi, Yj] =

∑mk=1 C

kijYk, then Yi(f

j) = −∑m

k=1 Cjikf

k. Using the rela-tion d∆ = −tr ad we obtain for ω as above:

δ0(σ0(ω)) = δ0(m∑i=1

Yi ⊗ λ⊗ f iϕ)

= −m∑i=1

tr(adYi)λ⊗ f iϕ+

m∑i=1

λ⊗ Yi(f i)ϕ+m∑i=1

λ⊗ f iYi(ϕ).

It is easy to see that∑m

i=1 Yi(fi) = −

∑mi,j=1 C

iijf

j = tr(adZ) and that∑mi=1 tr(adYi

)λ⊗ f iϕ = λ⊗ tr(adZ)ϕ. Since ∇0ϕ =∑m

i=1 fiYi(ϕ) we obtain

the statement of the lemma on C0.Next we proceed by induction. Using Lemma 4.5, we get:

(δ0σ0 + σ0δ0)(X ∧ ω) = −δ0(X ∧ σ0(ω)) + σ0(X(ω)−X ∧ σ0(ω))

= −X(σ0(ω)) +X ∧ δ0σ0(ω) + σ0(X(ω)) +X ∧ σ0(δ0(ω))

= X ∧∇0(ω) = ∇0(X ∧ ω).

ut

Either from this lemma or directly from the definition we obtain that C ′jis invariant for the action of R and hence, moding out C ′j we get η : R →GL(X (j)), desired action of R on X (j).

Finally let Y = (C∆′ ⊗ L(W,Gx))⊗Gx C = (A,Gx)\W . Also let

Z =

L(W ∩Kx, Gx)⊗Kx C if x ∈ Kx

L(W ∩M0, Gx)⊗M C if x /∈ Kx(13)

Here M0 is the inverse image of M in Gx (recall that M is a maximal compactsubgroup of Gx/(x)).

24

4.6 The condition of the Main Lemma are satisfied

We have defined all the objects needed for the use of the Main Lemma so wecan now concentrate on proving that the conditions of the Main Lemma aresatisfied.

First observe that η1 = T n+1n+1 since x acts trivially on F and ∇ = σδ+ δσ

if ∇ is the derivative of ηt at 0.Let q = dim(G/M).

Proposition. 4.7 Suppose Gx is a finite component Lie group. For X (j),Y, Z, δ, σ and q as above all conditions of the Main Lemma are satisfied(except eventually the injectivity of 1− T n+1

n+1 ).ut

If Gx is a finite component Lie group the exactness of (E1) was treatedin 3.1 3.2 and especially 3.6. For x /∈ Kx we also use the Serre-Hochschildspectral sequence for the normal subgroup (x) ⊂ Gx [6, 13].

Let Y1, . . . , Ym be a basis for g such that Yq+1, . . . , Ym is a basis of t =Lie M . Also let y1, . . . , ym be the dual basis and f j(x exp(X)) = yj(X).Then (Λj(gx/t) ⊗ F, σ0) is the Koszul complex associated with the regularsequence (f 1, . . . , f q) and the C∞(W )-module F . The ideal I generated by(f 1, . . . , f q) in C∞(W ) is the ideal of Kx ∩W if x ∈ Kx, respectively, theideal of M0 ∩W if x /∈ Kx.

Now TorC∞(W )j (I, F ) = 0 for j > 0 and Tor

C∞(W )0 (I, F ) = I ⊗C∞(W ) F =

L(W ∩Kx, Gx) if x ∈ Kx, respectively, L(W ∩M,Gx)/(1− x)L(W ∩M,Gx)if x /∈ Kx. (Note that we have omitted “C∆′⊗” since it does not affect theunderlying locally convex space, it only changes the action of Gx. Since inthe above formulae such an action does not exist or is not important, ourprocedure is justified.) The exactness of (E2) follows from the exactness of⊗MC (M is compact).

We are going now to study conditions for the injectivity of 1− T n+1n+1 . As

one can easily see, this is not always the case, this happens for example, ifGx is compact or G = SL2(R) and x = e. However 1− T n+1

n+1 will turn out tobe injective for all groups G of the form G = H × R and any x ∈ G.

Lemma. 4.8 i) Suppose x ∈ Kx. Let ϕ ∈ Ln(W,Gx) satisfy (1−T n+1n+1 )ϕ = 0

then ϕ(γ, g0, . . . , gn) = 0 for any topologically torsion free γ.ii) If G ' H × R then 1− T n+1

n+1 is injective for any x ∈ G.ut

25

Proof. The assumption of i) shows that

ϕ(γ, g0, . . . , gn) = (α⊗n+1γ )−m(ϕ(γ, γmg0, . . . , γ

mgn))

for any m ∈ Z. Since ϕ has compact support and γ is topologically torsionfree this shows that ϕ(γ, g0, . . . , gn) = 0. In order to prove ii) we use i) andthe fact that for G of the form H × R the topologycaly torsion free elementsare dense in G.ut

We are now ready to draw some conclusions from the above discussionand the Main Lemma.

Let G be a finite component Lie group, K ⊂ G a maximal compactsubgroup. Also fix x ∈ G and U ⊂ gx satisfying (A1) - (A3). V,W,Kx,Mand M0 will have the same meaning as before. Let q = dim(G/K), q′ =dim(Gx/Kx), K being chosen such that Kx ⊂ K.

Proposition. 4.9 i) [11] PHC∗(Ao R) ' PHC∗+1(A).ii) PHC∗(AoR)0 ' PHC∗(AoR) and PHC∗(AoR)t = 0 for any t ∈ R,

t 6= 0.iii) If x ∈ Kx, Gx is a finite component Lie group then

PHC∗((AoG)\V ) ' PHC∗+q((AoK)\U ′) (14)

for U ′ = c(K ×Kx (t ∩ U)).iv) If x /∈ Kx,W∩Kx = ∅ and Gx ' H×Z

n where H is a finite componentLie group then PHC∗((A o G)\V ) = 0. (Kx is a maximal compact subgroupof H.)ut

Proof. Proposition 4.4 ii) implies that

PHC∗((AoG)\V ) ' PHC∗((AoGx)\W ) (15)

Suppose first that 1 − T n+1n+1 is injective on each X (j). Then the Main

Lemma shows that

PHC∗((AoGx)\W ) ' PHC∗+q′(Z) (16)

If x ∈ Kx the Proposition 3.2 shows that

PHC∗(Z) ' PHC∗((AoKx)\W∩Kx

) (17)

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Using once again Proposition 3.6 we obtain

PHC∗((AoKx)\W∩Kx

) ' PHC∗((AoK)\U ′)

For G = R and x = e we may take U = R and we obtain i) and ii). The proofof i) for (AoG)oR shows that PHC∗(Ao(GoR)\U×R

) ' PHC∗+1((AoG)\U).This shows that we may replace G by G × R. Since for G × R, 1 − T n+1

n+1 isinjective, the above discussion proves iii) (q and q′ have the same parity).

iv) It is enough to prove that PHC∗((AoGx)\W ) = 0. Suppose first that

W ∩H = ∅. Then the statement follows from the vanishing of the periodiccyclic homology of the inhomogeneous components of crossed products by Z

n

[24]. If W ⊂ H then we show that PHC∗((AoGx)\WZ

n) = 0.Let us observe that PHC∗(A o Gx)

\WZ

n) ' PHC∗(((A o Zn) o H)\W ).

Denote as before by M a maximal compact subgroup of H/(x) which existssince H has a finite number of components. Let z be the center of t = Lie M0.There exists a suitable power of x such that xn = exp(X) for some X ∈ z.Then R 3 s → exp(sX) ∈ M0 identifies R with a closed central subgroup ofM0 such that M0/R is compact. Then M0 'M0/R×R and this isomorphismis unique if we require M0/R→M0/R×R 'M0 →M0/R to be identity (useH1(M0/R,R) = H2(M0/R,R) = 0). We identify M0/R to the correspondingsubgroup of M0. We still have (15) and (16) for q′ = dimH/M0.

However (17) has to be replaced with

PHC∗(Z) ' PHC((AoM0)\M0∩W ).

Since M0 ∩W ⊂ (M0/R)× (R \ 0) iv) follows from ii).ut

Corollary. 4.10 [11] Let G be a connected nilpotent Lie group, K ⊂ G amaximal compact subgroup then PHC∗(AoG) ' PHC∗+q(AoK). ut

Proof. K is a central subgroup. Then use i) and induction. ut

4.7 The Main Theorem

We are now ready to prove the theorem stated in the introduction.

Proof. Let m ⊂ C∞inv(G) be a maximal ideal. Assume m is not closed,thenL(G,G)m = 0 by Lemma 4.1. Suppose now m = my for some y ∈ G. Let

27

y = xxu be the Jordan decomposition of y: x is semisimple, xu unipotentand xxu = xux [5]. Choose f : G → GL(V ) an injective morphism ofalgebraic groups [5]. Then ρ(x) and Adx are semisimple and hence x andU = Uε(ρ) for small ε > 0 satisfy (A1)-(A3) (see the Appendix). HereUε(ρ) = X ∈ gx, σ(dρ(X)) ⊂ B(0, ε). Moreover the function ϕ appearingin (A3) satisfies ϕ(y) = ϕ(x) = 1 6.2. Observe also that Gx, being analgebraic group, has a finite number of components.

If x ∈ Kx the statement of the theorem follows localizing the isomorphismof 4.9 iii) at my.

If x /∈ Kx then σ(ρ(x)) is not contained in T = |z| = 1 and the samewill be true of c(γ,X) for any γ ∈ G, X ∈ Uε(ρ) for small ε. Then sincePHC∗(AoG)y ' PHC∗((AoG)\V )y = 0 and PHC∗(AoK)y = 0 the proofis complete.ut

4.8 Some nonalgebraic groups

We are now going to discuss some more examples. Their purpose is to showhow to use the techniques we have developed for some nonalgebraic semisim-ple groups.

i) Let G = ˜SL2(R)=the simply connected covering group of SL2(R). Letx be a generator of the center, U = X ∈ sl2(R), det(X) < π2. TheProposition 4.9 iii) shows that PHC∗((A o G)\V ) ' PHC∗+3(A) if V =exp(U). Then proposition 4.9 iv) shows that PHC∗((A o G)\V ) = 0 if V =xnexp(U) and n 6= 0. Since every orbit is contained in one of the setsxnexp(U), we obtain

PHC∗(AoG) ' PHC∗+3(A).

ii) Let G = ˜SL2(R)×R/N where N = (xn, n), n ∈ Z, x being as above.Let X ∈ sl2(R) be such that exp(X) = x, then K = exp(s(X,−1)), s ∈R/Z is a maximal compact subgroup of G. Let U be as above and letV = exp(U × (−1/2, 1/2)), V ′ = exp(U × (0, 1)).Using the same reasoningas before we obtain that PHC∗((AoG)xnV )\) = 0 (respectively PHC∗((Ao

G)\xnV ′) = 0 for n 6= 0 and PHC∗((AoG)\V ) ' PHC∗+3((AoK)\I) where I =

exp s(X,−1), s ∈ (−12, 1

2) and PHC∗(A o G)\V ′) ' PHC∗+3((A o K)\I′),

where I ′ = exp s(X,−1), s ∈ (−1, 0). This shows the local isomorphism ofPHC∗(AoG) and PHC∗+3(AoK).

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5 More on the Case G Compact

In this section G will denote a connected compact Lie group, T a maximaltorus of G, and W the Weyl group of the pair (G, T ) [1].

Let L(G) = L(G, e)G. Restriction defines a map L(G, e)→ L(T, e)and hence also a map rG,T : L(G)→ L(T ). Since HC∗(L(G)) ' HC∗(AoG)(by Proposition 3.2) we obtain a morphism

r∗G,T : PHC∗(Ao T )→ PHC(AoG)

Our aim is to prove the following.

Theorem. 5.1 r∗G,T defines an isomorphism

PHC∗(AoG) ' PHC∗(Ao T )W .

ut

The proof is divided into several steps.

Lemma. 5.2 Suppose f : G1 → G is a finite covering of connected groupsand the theorem is true for G1, then it is true also for G. ut

Proof. Let H = ker f, T1 = f−1(T ), then H ⊂ T1 and T1 is a maximal torus ofG1. Moreover the Weyl group of (G1, T1) naturally identifies with W . Recall(Theorem 3.7) that there exists a morphism IG1,G : L(G1, e) → L(G, e)defined by “integration along the fibers of G1 → G”:

IG1,Gϕ(γ) =∑h∈H

ϕ(γh)

which defines an isomorphism PHC∗(AoG1)H ' PHC∗(AoG). Moreoverthe morphisms r and I commute: rT,GIG,G1 = IT,T1rT1,G1 . Since the actionsof W and H on L(T1) commute we obtain

PHC∗(AoG) ' PHC∗(AoG1)H ' (PHC∗(Ao T1)W )H

' (PHC∗(Ao T1)H)W ' PHC∗(Ao T )W .

ut

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This lemma shows that we may replace G by any of its covering groups,so in particular we may assume that G ' H × Z with H simply-connectedsemisimple and Z abelian. Since AoG ' (AoZ) oH we may suppose thatG itself is simply-connected and semisimple.

We define now a filtration invariant for r on L(G) and L(T ).Let Xj ⊂ T,Xj = x ∈ T, dim(Gx) ≥ j. (Here Gx is the centralizer of

x in G).Let L(G)(j) = ϕ ∈ L(G), all derivatives of ϕ vanish on Xj and de-

fine L(T )(j) similarly. It is immediate from the definition that r(L(G)(j)) ⊂L(T )(j)W .

Lemma. 5.3 L(G)(j+1)/L(G)(j) → (L(T )(j+1)/L(T )(j))W defines an isomor-phism for the PHC∗-groups.ut

Proof. Let T ′ be a torus contained in Xj but not in Xj+1. Consider the cylicmodules A = A′/A′′, B = B′/B′′, where

A′ = ϕ ∈ L(T ), all derivatives of ϕ vanish on W (T )′ ∩Xj+1,

A′′ = ϕ ∈ L(T ), all derivatives of ϕ vanish on W (T ′),B′ = r−1A′ and B′′ = r−1A′′.

Then A and B are direct summands of the modules in the statement ofthe lemma. It is enough to show that

PHC∗(A)W → PHC∗(B)

is an isomorphism. Let A(n) = ϕ ∈ A, the derivatives of ϕ of order < nvanish on W (T ′), B(n) = r−1(A(n)).

Let W1 be the Weyl group of

H = ZG(T ′) = g ∈ G : g commutes with all the elements of T ′

W1 is the stabilizer of T ′ in W . It coincides with the stabilizer in W of anyx ∈ T ′ \Xj+1 [1]. Let h = Lie H and h′ = [hh]. h′ is the fiber of the normalbundle of AdG(T ′ \Xj+1) at any point of T ′ \Xj (with respect to the metricdefined by the Killing form). Denote, for any vector space V , by Sn(V ) thespace of symmetric tensors in V ⊗n+1 then

(A(n)/A(n+1))W ' C∞0 (T ′ \Xj+1)W2⊗Hom(Sn+1(t′), A\)W1

30

B(n)/B(n+1) ' C∞0 (T ′ \Xj+1)W2⊗Hom(Sn+1(h′), A\)H′

Here C∞0 (T ′ \ Xj+1) is the space of smooth functions on T ′ all of whosederivatives vanish on Xj+1, W2 is the normalizer of T ′ in W that is W2 =w ∈ W : w(T ′) = T ′, H ′ is the commutant of H, of course h′ = Lie H ′.t′ = t ∩ h′ is the Lie algebra of a maximal torus of H ′ (it is not equal to LieT ′).

The rest of the proof is contained in the following three lemmas. ut

Lemma. 5.4 Let K be a connected compact Lie group acting on A. Supposethat this action commutes with the action of the compact group Γ, then

PHC∗(L(Γ)) ' PHC∗(L(Γ)K1)

is an isomorphism for any closed subgroup K1 of K.ut

Proof. The statement follows from the homotopy invariance of the periodiccyclic cohomology [8].ut

Lemma. 5.5 PHC∗(A(n)/A(n+1))W → PHC∗(B(n)/B(n+1)) is an isomor-phism. ut

Proof. The above lemma shows (using a splitting in isotypical components)that

C∞0 (T ′ \Xj+1)W2 ⊗ Sn+1(t′)∗W1⊗A\W1 → (A(n)/A(n+1))W1

andC0(T ′ \Xj+1)W2 ⊗ Sn+1(h′)∗H

′⊗A\H′ → (B(n)/B(n+1))H′

induce isomorphisms for the PHC∗-groups. Since C[h′]H′ ' C[t′]W1 [15] the

lemma is proved.ut

Lemma. 5.6 A∗ ' lim→

(A/A(n))∗ and B∗ ' lim→

(B/B(n))∗ .ut

Proof. Any distribution on a compact manifold has finite order.ut

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6 Appendix

In this section we introduce some more manageable conditions on x ∈ G andU ensuring (A1) - (A2) to hold true. Fix x ∈ G and denote g = Lie G.

If ρ : G→ GL(V ) is a linear representation let

Uε(ρ) = X ∈ g, σ(dρ(X)) ⊂ B(0, ε)

Here, B(0, ε) = |z| < ε and σ refers to the spectrum of the operator actingon V ⊗ C.

Lemma. 6.1 Suppose Adx is semisimple and U ⊂ Uε(Ad) is AdGx-invariant.Then for small ε > 0, c : G×Gx U → G is a local diffeomorphism.ut

Proof. We first identify TγG with g = TeG by means of left translationsand TXU with gx for X ∈ U . If (γ,X) ∈ G ×Gx U then T(γ,X)(G ×Gx

U) = g× gx/(Y, [XY ]), Y ∈ gx. We shall show that dcγ,X is injective,since dim T(γ,X)(G ×Gx U) = dimTc(γ,X)G we shall obtain that c is a localdiffeomorphism.

Now c is G-equivariant so we may suppose that γ = e. We compute

dc(e,X)(Y,X0) = Ad−1x exp(X)(Y )− Y + f(adX)(X0)

where f(t) = (1 − e−t)t−1, f(adX) is defined by analytic functional calculusand is the differential of exp at X [14].

If dc(e,X)(Y,X0) = 0 then, for small ε, it follows that Y ∈ gx and

f(adX)(X0) = (1− Ad−1x exp(X))(Y ) = f(adX)(adX(Y ))

and hence X0 = [XY ], again for ε small enough.ut

We look now for conditions implying the other assumptions on x.

Lemma. 6.2 Suppose there exists ρ : G→ GL(V ) a complex representationwith dρ injective, ρ(x) semisimple and U ⊂ Uε(ρ). If U is AdGx-invariantand ε > 0 is small enough then c is injective and (A2) is satisfied. If moreoverU = Uε(g) then (A3) is also satisfied.ut

32

Proof. Let Ω ⊂ C be a disjoint union of open balls each centered at a pointof σ(ρ(x)). We choose 0 < ε < π so small that σ(ρ(x exp(X))) ⊂ Ω for anyX ∈ Uε(ρ). We split the proof into several steps. ut

Step 1 c is injective. Let c(γ,X) = c(e,X0), then

ρ(γxγ−1)exp(ρ(γ)dρ(X)ρ(γ−1)) = ρ(x)exp(dρ(X0))

Let χ0 : Ω→ C be the unique locally constant function such that χ0|σ(ρ(x)) =idσ(ρ(x)). χ0 being an analytic function χ0(ρ(x)exp(dρ(X0))) may be definedby analytic functional calculus and is equal to ρ(x). Hence ρ(γxγ−1) = ρ(x)and exp(ρ(γ)dρ(X)ρ(γ−1)) = exp(dρ(X0)). We obtain that dρ(Adγ(X)) =dρ(X0) and hence Adγ(X) = X0 since dρ is injective. This shows thatγxγ−1 = x and hence (γ,X) = (e,X0) in G×Gx U .

Step 2 OG0(x0) = γx0γ−1, γ ∈ G0 is closed in G0. Here we have

denoted G0 = ρ(G), x0 = ρ(x).OG0(x0) is a submanifold of GL(V ) (we do not assume it to be closed).

OGL(V )(x0) is closed since it is equal to γ ∈ GL(V ), P (γ) = 0 if P is theminimal polynomial of x0. Let y ∈ OGL(V )(x0) ∩ G0. Then TyOGL(V )(x0) =(Ad−1

y −1)(gl(V )), TyOGL(V )(x0)∩TyG0 = (Ad−1y −1)(g) = TyOG0(y) since we

may choose an Ady-invariant complement of g in gl(V ). (We use here the factthatG is the real form of a complex algebraic group, in particular the previousstatement is a statement about real vector spaces and not complex vectorspaces.) This shows that OG0(y) is open in OGL(V )(x0)∩G0. The latter beinga union of such orbits it follows that OG0(x0) is closed in OGL(V )(x0) ∩ G0

and hence also in G0.Step 3 OG(x) is closed. ρ restricts to an obvious local diffeomorphism

OG(x) → OG0(x0). Since ker ρ is discrete OG(x) is closed iff OG0(x0) isclosed.

Step 4 Suppose L ⊂ U is a closed AdGx-invariant subset of gx. Thenc(G ×Gx L) is closed in G. We first assume that ρ is injective and iden-tify G with G0. Let c(γn, Xn) → y ∈ G and χ0 as in step 1. Thenχ0(c(γn, Xn)) = γnxγ

−1n → χ0(y) ∈ OG(x). Since OG(x) is closed we

may write γn = γ′nγ′′n with γ′′n ∈ Gx, γ

′n convergent to, say, γ. Then

exp(γ′′nXnγ′′−1n ) converges to x−1γ−1yγ and hence also γ”nXnγ

′′−1n is con-

vergent in gx. Let X = lim γ”nXnγ′′−1n , then X ∈ L. This shows that

y = lim c(γn, Xn) = lim c(γ′n, γ′′nXnγ

′′−1n ) = c(γ, x) ∈ c(G×Gx L).

33

In general, for ker(ρ) discrete 6= e observe that c(G×Gx L) is closed inρ−1(ρ(c(G×Gx L))).

Step 5 (A2) is satisfied. Observe that Φ is well defined and injective if(A1) holds true (see 3.2). If ψ ∈ R1 let ϕ(γxexp(X)γ−1) = ψ(x exp(X)).Since ϕ is smooth on V , step4 implies that ϕ is smooth also on G \ V . SinceΦ(ϕ) = ψ, ϕ is also onto.

Step 6 If U = Uε(ρ), then also (A3) is satisfied. Let ϕ0, ϕ′0 ∈ C∞c (C

l),where l = dim V be such that ϕ0(0) = 1, ϕ′0ϕ0 = ϕ0. Let ϕ(x exp(X)) =ϕ0(trρ(X), trΛ2ρ(X), . . . , trΛ1ρ(X)) for X ∈ U and 0 elsewhere. We defineϕ′ similarly. Then ϕ and ϕ′ have the required properties provided ϕ′0 hassmall support.ut

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