A Proof of Connes-Kasparov Conjecture on Flag Varieties ofReal Semisimple Lie Groups

Zhaoting Wei∗

May 7, 2012

Abstract

We consider the equivariant K-theory of the real semisimple Lie group which acts on the (complex)flag variety of its complexification group. We construct a Dirac index map in the framework of KK-theory. Then we prove that it is an isomorphism. The prove relies on a careful study of the orbitsof the real group action on the flag variety. This result can be considered as a special case of theConnes-Kasparove conjecture with coefficient.

1 IntroductionLet G be a locally compact topolocial group and A is a C∗-algebra equipped with a continous action

of G by C∗-algebra automorphisms. Following [2] section 4, we define the equivariant K-theory of A tobe the K-theory of the reduced crossed product algebra:

K∗G(A) := K∗(C∗r (G,A)).

In particular, K∗G(pt) = K∗(C∗r (G)). It is well-known that C∗r (G) reflects the tempered unitary dual whenG is a reductive Lie group, see [2], 4.1.

When G is compact, this definition coincides with the usual equivariant K-theoy, see [7].Back to general G. Let U be its maximal compact subgroup. We have the Dirac index map [6]

K∗+dimG/UU (pt)

D−→ K∗G(pt).

The Connes-Kasparov conjecture claims that the composition map is an isomorphism of abeliangroups. A good reference for the statement is [11]. In 2003, J. Chabert, S. Echterhoff, R. Nest [6] provedthis conjecture for almost connected and for linear p-adic G.

Moreover, we have the Connes-Kasparov conjecture with coefficients, that is, the Dirac index map

K∗+dimG/UU (A)

D−→ K∗G(A)

is an isomorphism. This is still an open problem for general G and A.When G is a real semisimple Lie group, we can also consider the Connes-Kasparov conjecture on flag

varieties: Let GC be the complexification of G, BC be the Borel subgroup of GC. We have the flag varietyF of GC

F := GC/BC.

GC (hence G and U) acts on F , so we also have maps

K∗+dimG/UU (F )

D−→ K∗G(F ) (1)

∗Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania, 19104. zhaotwei@sas.upenn.edu.

1

The main result of this paper is Theorem 2.1, which claims that the map in (1) is an isomorphism forreal semisimple Lie group G.

This paper is organized as follows: In Section 2 and 3 we give the constructions of the Dirac indexmap. In Section 4 we study the Dirac index map on a single orbit of the flag variety and in Section 5 weprove the Theorem 2.1. In Section 6 we give an example to illustrate the idea of this paper.

This work is inspired by [10] and [2] and hopefully it will be useful in representation theory, e.g. inthe constructions of discrete series, see [3].

Acknowledgement The author would like to thank his advisor Jonathan Block for introducing meto this topic, for his encouragement and helpful discussions. I am also very grateful to Nigel Higson forhis introduction to Connes-Kasparov conjecture. I would like to thank Yuhao Huang and Eric Kormanfor helpful comments about this work.

2 Notations and First ConstructionsWe will use the following notations in this paper

1. Let G be a connected real semisimple Lie group, U be the identity component of its maximalcompact subgroup.

2. Let GC be the complexification of G, BC be the Borel subgroup of GC and F be the flag varietyF := GC/BC.

3. Let X be any space with continuous G-action. Let C0(X) be the space of continuous functions onX which vanishes at infinity. If X is compact, then C0(X) = C(X) is the space of all continuousfunctions on X. We define

K∗G(X) := K∗(C∗r (G,C0(X))). (2)

K∗U (X) can be defined in the same way. In particular we can define K∗U (F ) and K∗G(F ).

Obviously G acts on the flag variety F . Unlike GC, the G-action is not transitive.For any G-space X we will construct a Dirac index map

K∗+dimG/UU (X)

D−→ K∗G(X)

in the next section. Our goal is to prove the following

Theorem 2.1. Let G be a connected complex semisimple Lie group, U be its maximal compact subgroup,F be the flag variety. The following map

K∗+dimG/UU (F )

D−→ K∗G(F ) (3)

is an isomorphism.

3 The Dirac-Dual Dirac Method and the Dirac Index MapBefore we begin the rather involved construction, let us take a brief look at the geometric picture and

get some intuition.The following isomorphism is very natural, see [12]

Proposition 3.1 (The induction map). For any group G, H ⊂ G a closed subgroup, and an H-spaceX,there is an induction map

K∗H(X)→ K∗G(G×H X)

which is an natural isomorphism.Here the G-action on G×H X is the left multiplication on the first component.

Proof: Just notice that C∗r (H,C0(X)) and C∗r (G,C0(G×H X)) are Strongly Morita equivalent. �

2

Proposition 3.2. If X is also a G-space in the above setting then, G×HX is G-isomorphic to G/H×X,where G acts on G/H ×X by the diagonal action. Hence

K∗G(G×H X) ∼= K∗G(G/H ×X).

Proof: See [2], Section 2.5. In fact, both sides are quotient spaces of G × X by right actions of U .The point is that the actions are different.

For G×H X(g, x) · h := (gh, h−1x).

For G/H ×X(g, x)̃·h := (gh, x).

Then we see that the following map

G×H X →G/H ×X(g, x) 7→(g, gx)

intertwines the action · and ·̃. Moreover it is equivariant under the left G-action. �

Remark 1. The sprit of Proposition 3.1 and 3.2 will appear later in Theorem 3.6 and Proposition 3.7.

Let H be U . In the special case which G/U is spin ( e.g., G = SL(2,R) ) we can take the Dirac index(take X as the parameter space) and get a map

K∗G(G/U ×X)D−→ K∗G(X)

henceK∗U (X)

D−→ K∗G(X) (in this case dimG/U is even.)

Of course, G/U may be not spin. In the general case the construction of the map

K∗+dimG/UU (F )

D−→ K∗G(F )

and the proof of Proposition 4.4 requires more careful exam of Kasparov’s Dirac-Dual Dirac methodin the framework of equivariant KK-theory, see [8]. His work plays an important role not only in theconstruction of D but also in the proof that D is an isomorphism.

Here we give a brief summary of his construction.For a group G and G-C∗ algebras A, B we have the equivariant KK-group ([8] 2.4)

KKG(A,B)

and for a σ-unital G-C∗ algebra E (We avoid the letter D to keep it for the Dirac index map) we have anatural map

σE : KKG(A,B) −→ KKG(A⊗ E,B ⊗ E). (4)

which maps 1A to 1A⊗E .The Kasparov product is given as ([8] 2.12)

KKG(A1, B1 ⊗ E)⊗KKG(E ⊗A2, B2) −→ KKG(A1 ⊗A2, B1 ⊗B2). (5)

Now we define the Dirac element and the dual Dirac element. For a group G and a G-manifold X,let Cτ (X) denote the algebra of continous sections of the Clifford bundle over X vanishing at infinity.Kasprov defines the Dirac element [8] 4.2:

dG,X ∈ KKG(Cτ (X),C)

3

Remark 2. We do not require that X is spin in the definition of dG,X .

Kasparov also introduce the concept of G-special manifold in [8] 5.1. For G-special manifold X, thereexists an element

ηG,X ∈ KKG(C, Cτ (X))

such that the Kasparov product

KKG(Cτ (X),C)×KKG(C, Cτ (X)) −→ KKG(Cτ (X), Cτ (X))

givesdG,X ⊗C ηG,X = 1Cτ (X).

When the group G is obvious we will denote them by dX and ηX .Kasparov also showed that for the maximal compact subgroup U of G, the homogenous space G/U

is a G-special manifold.

Remark 3. Although dG,X ⊗C ηG,X = 1Cτ (X) for special manifold, the Kasparov product in the otherway

ηX ⊗Cτ (X) dX

need not to be 1.

We denote ηX ⊗Cτ (X) dX by γX .

Remark 4. If γX = 1, then dX and ηX are invertible elements in the Kasparov product.

Kasparov proved that γX = 1 in some special cases, which is sufficient for our purpose. To state theresult in full generality we need to introduce the concept of restriction homomorphism

Let f : G1 → G2 be a homomorphism between groups, the restriction homomorphism

rG2,G1 : KKG1(A,B) −→ KKG2(A,B)

Proposition 3.3. If X is a G2 manifold and f : G1 → G2 as above. Then under the map rG2,G1 wehave

rG2,G1(dG1,X) =dG2,X

rG2,G1(ηG1,X) =ηG2,X

rG2,G1(γG1,X) =γG2,X .

�If the manifold X is G/U , where U is the maximal compact subgroup of G, we denote dG/U by d(G),

ηG/U by η(G) and γG/U by γ(G)

Theorem 3.4 ([8], 5.9). Let f : G1 → G2 be a homomorphism between almost connected groups with thekernel amenable and the image closed. Then the restriction homomorphism

rG2,G1(γ(G2)) = γ(G1).

In particular, γ(G) = 1 for every amenable almost connected group G.

�

Remark 5. Remember Remark 4, we know that if G is an amenable almost connected group, thendG/U = d(G) and ηG/U = η(G) are invertible elements in the KK-groups.

Now we want to find the relation between equivariant KK-theory and the K-theory of crossed-productalgebras.

First remember we have the map

σX : KKG(A,B) −→ KKG(A⊗ C0(X), B ⊗ C0(X)).

Apply this to dG/U ∈ KKG(Cτ (G/U),C) we get

σX(dG/U ) ∈ KKG(Cτ (G/U)⊗ C0(X), C0(X)).

Next, we know that Kasparov gives the following definition-theorem (Theorem 3.11 in [8] )

4

Theorem 3.5. There is a natural homomorphism

jGr : KKG(A,B) −→ KK(C∗r (G,A), C∗r (G,B))

which is compatible with the Kasparov product. Moreover, for 1A ∈ KKG(A,A)

jGr (1A) = 1C∗r (G,A) ∈ KK(C∗r (G,A), C∗r (G,A)).

�Apply the map jGr to σX(dG/U ) which is in KKG(Cτ (G/U)⊗ C0(X), C0(X)) we get

jGr (σX(dG/U )) ∈ KK(C∗r (G,Cτ (G/U)⊗ C0(X)), C∗r (G,C0(X))).

We denote jGr (σX(dG/U )) by DG,X or simply by D.

Now we move on to define the Dirac index map. First let V be the tangent space at e ·U ∈ G/U andCl(V ) be the (complex) Clifford algebra of V .

ThenK∗+dimG/UU (X) = K∗(Cr(U,C0(X)⊗ Cl(V ))) (6)

The following theorem is standard in the theory of C∗-algebras. ([8], 3.15)

Theorem 3.6. If Γ is a subgroup of G and A is a Γ-algebra, let A(G) = A⊗ C0(G) denote continuousfunctions from G to A which vanish at infinity, then C∗r (G, (A(G))Γ) is strongly Morita equivalent toC∗r (Γ, A).

�

In the above theorem let A = C(X)⊗Cl(V )) and Γ = U , we then know that C∗r (U,C0(X)⊗Cl(V )))is strongly Morita equivalent to C∗r (G, (C0(G)⊗ Cl(V ))⊗ C0(X))U ).

C∗r (U,C0(X)⊗ Cl(V )))Strongly Morita−−−−−−−−−−→

∼C∗r (G, (C0(G)⊗ Cl(V ))⊗ C0(X))U ) (7)

Next we need the following isomorphism

Proposition 3.7. (C0(G)⊗ Cl(V ))⊗ C0(X))U is ( G-equivariantly) isomorphic to Cτ (G/U)⊗ C0(X),hence

C∗r (G, (C0(G)⊗ Cl(V ))⊗ C0(X))U ) ∼= C∗r (G,Cτ (G/U)⊗ C0(X)). (8)

Proof: First we see thatf ⊗ v ⊗ a 7→ f ⊗ f ∗ v ⊗ f ∗ a

gives an isomorphism

C0(G)⊗ Cl(V ))⊗ C0(X))U∼−→ C0(G)U ⊗ Cl(V ))⊗ C0(X). (9)

On the other hand by definition

Cτ (G/U) ∼= (C0(G)⊗ Cl(V )))U

andf ⊗ v 7→ f ⊗ f ∗ v

gives an isomorphismC0(G)⊗ Cl(V )))U

∼−→ C0(G)U ⊗ Cl(V )

and hence(C0(G)⊗ Cl(V ))U ⊗ C0(X)

∼−→ C0(G)U ⊗ Cl(V )⊗ C0(X).

i.e.Cτ (G/U)⊗ C0(X)

∼−→ C0(G)U ⊗ Cl(V )⊗ C0(X). (10)

5

Combine the two isomorphisms in 9 and 10 we get

C0(G)⊗ Cl(V ))⊗ C0(X))U ∼= Cτ (G/U)⊗ C0(X)

henceC∗r (G, (C0(G)⊗ Cl(V ))⊗ C0(X))U ) ∼= C∗r (G,Cτ (G/U)⊗ C0(X)). �

Combine the isomorphisms in 7 and 8 we get

C∗r (U,C0(X)⊗ Cl(V ))) ∼= C∗r (G,Cτ (G/U)⊗ C0(X))

so

K∗+dimG/UU (X) =K∗(Cr(U,C0(X)⊗ Cl(V )))

∼=K∗(C∗r (G,Cτ (G/U)⊗ C0(X)))

=KK∗(C, Cr(G,Cτ (G/U)⊗ C0(X)))

(11)

Apply the element

DG,X = jGr (σX(dG/U )) ∈ KK(C∗r (G,Cτ (G/U)⊗ C(X)), C∗r (G,C(X))).

via the Kasparov product

KK(C, Cr(G,Cτ (G/U)⊗ C(X)))⊗KK(C∗r (G,Cτ (G/U)⊗ C(X)), C∗r (G,C(X))) −→KK(C, C∗r (G,C(X)))

we get the desired map

KKdimG/U (C, Cr(G,Cτ (G/U)⊗ C(X))) −→ KK(C, C∗r (G,C(X)))

i.e.K∗+dimG/UU (X)

D−→ K∗G(X).

Definition 3.1 (The Dirac index map). The above map

K∗+dimG/UU (X)

D−→ K∗G(X).

is called the Dirac index map.

Now we can immediately get a isomorphic result in the almost connected amenable case. The followingresult is implicitly given in [8],5. 10.

Theorem 3.8. If A is an almost connected amenable group, T is the maximal compact subgroup of A,X is an A space, then

K∗+dimA/TT (X)

D−→ K∗A(X)

is an isomorphism.

Proof: By Theorem 3.4 we know that γ(A) = 1 hence d(A) is invertible. Since D is got from d(A)

and remember Formula 4 and Theorem 3.5, invertible elements go to invertible elements. So D is anisomorphism. �

Remark 6. Although the above theorem does not directly lead to the main result of this paper, it is veryinteresting and nevertheless it will be useful in the example in the final section of this work.

6

4 The Dirac Index Map on a Single G-Orbit of FAccording to [10], there are finitely many G-orbits on F . Let us denote α+ to be one of them. Let S

be the isotropy group of G at a critical point (see [10] for the definition of critical points) x ∈ α+.We are going to prove that K∗U (α+) ∼= K∗+dimG/U

G (α+). More precisely, we want to prove

Proposition 4.1. The Dirac index map

K∗+dimG/UU (α+)

D−→ K∗G(α+)

is an isomorphism.

Remark 7. Proposition 4.1 is the building block of the main theorem of this paper-Theorem 2.1. We willpiece together the blocks in the next section.

The proof of Proposition 4.1 consists of several steps. First notice that α+ can by identified withG/S. We prove the lemma

Lemma 4.2 (Interchange subgroups). There is an isomorphism:

K∗+dimG/UU (G/S)

∼−→ K∗+dimG/US (G/U).

Proof: First by definition

K∗+dimG/UU (G/S) ∼= K(C∗r (U,C(G/S)⊗ Cl(V ))) ∼= K(C∗r (G,C(G/S)⊗ Cτ (G/U))) (12)

here V is a vector space have the same dimension as G/U and Cl(V ) is the complex Clifford algebra.Then we know that C∗r (G,C(G/S) ⊗ Cτ (G/U)) is Strongly Morita equivalent to C∗r (S,Cτ (G/U)),

henceK(C∗r (G,C(G/S)⊗ Cτ (G/U))) ∼= K(C∗r (S,Cτ (G/U))) ∼= K∗+dimG/U

S (G/U) (13)

�Now we are ready to have the following

Proposition 4.3. We have the following commuting diagram:

K∗+dimG/UU (G/S)

∼−−−−→ K∗+dimG/US (G/U)yD

yD

K∗G(G/S)∼−−−−→ K∗S(pt)

Proof: To prove the proposition we need to investigate the maps. First we look at the right verticalmap. At the beginning we have the Dirac element

dG,G/U ∈ KKG(Cτ (G/U),C)

apply the restriction homomorphism rG,S we get

rG,S(dG,G/U ) ∈ KKS(Cτ (G/U),C).

Nevertheless we have the Diract element

dS,G/U ∈ KKS(Cτ (G/U),C).

In fact from the definition it is easy to see that

rG,S(dG,G/U ) = dS,G/U .

Then we apply the map

jSr : KKS(Cτ (G/U),C) −→ KK(C∗r (S,Cτ (G/U)), C∗r (S)).

7

we get jSr (dS,G/U ) ∈ KK(C∗r (S,Cτ (G/U)), C∗r (S)) and we denote it by DS . Right multiplication of DS

gives the vertical map in the diagram

K∗+dimG/US (G/U)

D−→ K∗S(pt).

On the other hand we have the map

σG/S : KKG(Cτ (G/U),C) −→ KKG(Cτ (G/U)⊗ C(G/S), C(G/S))

so we getσG/S(dG,G/U ) ∈ KKG(Cτ (G/U)⊗ C(G/S), C(G/S))

then via jGr we get

jGr (σG/S(dG,G/U )) ∈ KK(C∗r (G,Cτ (G/U)⊗ C(G/S)), C∗r (G,C(G/S)))

which we denote by DG,G/S . Right multiplication of DG,G/S gives the other vertical map

K∗+dimG/UU (G/S)

D−→ K∗G(G/S).

The horizontal maps in the diagram are given by Strongly Morita equivalence. Now, under theStrongly Morita equivalence, DG,G/S

∼= DS , so the diagram commutes. �

According to Propostion 4.3, in order to prove Proposition 4.1, i.e.

K∗+dimG/UU (α+)

D−→ K∗G(α+)

is an isomorphism, it is sufficient to prove the following proposition

Proposition 4.4.K∗+dimG/US (G/U)

D−→ K∗S(pt)

is an isomorphism.

Proof: It is sufficient to prove

DS = jSr (dS,G/U ) ∈ KK(C∗r (S,Cτ (G/U)), C∗r (S))

is invertible. In fact, we can prove that dS,G/U ∈ KKS(Cτ (G/U),C) is invertible.As in the construction in Section 3, we have the dual Dirac element

ηS,G/U ∈ KKS(C, Cτ (G/U))

and

dS,G/U⊗CηS,G/U = 1 ∈ KKS(Cτ (G/U), Cτ (G/U)),

ηS,G/U⊗Cτ (G/U)dS,G/U = γS,G/U ∈ KKS(C,C).

We want to prove γS,G/U = 1. Remember that S is an almost connected amenable group and we haveTheorem 3.4, which claims that

γ(S) = 1.

where by definition γ(S) = γS,S/U∩S . We need to prove γ(S) is equal to γS,G/U .Notice that γS,G/U is nothing but the image of the element γG,G/U under the restriction homomor-

phismrG,S : KKG(C,C) −→ KKS(C,C).

i.e.rG,S(γG,G/U ) = γS,G/U .

8

Other other hand, in the notation of Theorem 3.4, γG,G/U is just γ(G), and again by Theorem 3.4 wehave

rG,S(γ(G)) = γ(S)

Compare the last two identity we getγS,G/U = γ(S)

so γS,G/U = 1.Now we proved that dS,G/U hence DS , is invertible, so

K∗+dimG/US (G/U)

D−→ K∗S(pt)

is an isomorphism.�

Combine Proposition 4.3 and 4.4 we know get

K∗+dimG/UU (α+)

D−→ K∗G(α+)

is an isomorphism, which finishes the prove of Proposition 4.1.�

5 The Isomorphism on the Flag VarietyWe have proved the isomorphism on an arbitary orbit of G. Now we need to "piece together orbits".The study of the orbits in [10] is important to our purpose, so we summarize their result here

Theorem 5.1 ([10] 1.2, 3.8). On the flag variety F there exists a real value function f such that

1. f is a Morse-Bott function on F .

2. f is U invariant, hence the gradient flow φ : R× F → F is also U invariant.

3. The critical point set C consists of finitely many U -orbits α. The flow preserves the orbits of G.

4. The limits limt→±∞ φt(x) := π±(x) exist for any x ∈ F . For α a critical U -orbit, the stable set

α+ = (π+)−1(α)

is an G-orbit, and the unstable setα− = (π−)−1(α)

is an UC-orbit, where UC is the complexification of U in GC.

5. α+ ∩ α− = α.

�We will use the following corollary in [10]:

Corollary 5.2 ([10] 1.4). Let α and β be two critical U -orbits. Then the closure α+ ⊃ β+ if and only if

α+ ∩ β− 6= ∅.

�From this we can get

Corollary 5.3. Let α and β be two different critical U -orbits, i.e. α 6= β. Then α+ ⊃ β+ implies thatthe Morse-Bott function f has values

f(α) > f(β)

9

Proof: By the previous corollary,α+ ∩ β− 6= ∅.

so there exists an x ∈ α+ ∩ β−.Since limt→+∞ φt(x) ∈ α, we have

f(α) > f(x),

similarlyf(x) > f(β).

On the other hand since α 6= β we get α 6⊂ β− and β 6⊂ α+. So

x 6∈ α, x 6∈ β

sof(x) 6= f(α), f(x) 6= f(β).

So we havef(α) > f(β).

�We can now give a partial order on the set of G-orbits of F .

Definition 5.1. If f(α) > f(β), we say that α+ > β+.If f(α) = f(β), we choose an arbitary partial order on them.

Now let us list all G-orbits in F in ascendent order, keep in mind that there are finitely many of them:

α+1 < α+

2 < . . . α+k . (14)

From the definition we can easily get

Corollary 5.4. For any G-orbits α+i , the union

Zi :=⋃

α+j 6α+

i

α+j

is a closed subset of F . Notice that α+i ⊂ Zi

Proof: It is sufficient to prove that Zi contains all its limit points, which is a direct corollary ofDefinition 5.1 and Corollary 5.3.�Now we can piece together the orbits

Proposition 5.5. For 1 6 i 6 k − 1 we have a short exact sequence of crossed product algebras:

0→ C∗r (G,C0(α+i+1))→ C∗r (G,C(Zi+1))→ C∗r (G,C(Zi))→ 0.

Proof: From the construction we also get

Zi ⊂ Zi+1, α+i+1 ⊂ Zi+1,

Zi ∪ α+i+1 = Zi+1, Zi ∩ α+

i+1 = ∅,

and Zi is closed in Zi+1, α+i+1 is open in Zi+1.

Since F is a compact manifold, we get that Zi and Zi+1 are both compact.The inclusion gives a short exact sequence:

0→ C0(α+i+1)→ C(Zi+1)→ C(Zi)→ 0. (15)

Now we need to go to the crossed product C∗-algebras. The following technique result will help us:

10

Theorem 5.6 ([9] Theorem 6.8). Let G be a locally compact group and

0→ A→ B → C → 0

be a short exact sequence of G-C∗ algebra. Then we have a short exact sequence:

0→ C∗r (G,A)→ C∗r (G,B)→ C∗r (G,C)→ 0.

�Since we have

0→ C0(α+i+1)→ C(Zi+1)→ C(Zi)→ 0.

exact, Theorem 5.6 gives the short exact sequence

0→ C∗r (G,C0(α+i+1))→ C∗r (G,C(Zi+1))→ C∗r (G,C(Zi))→ 0. (16)

�From Proposition 5.5 we have the well-known six-term long exact sequence

K∗(C∗r (G,C0(α+i+1))) −−−−→ K∗(C∗r (G,C(Zi+1))) −−−−→ K∗(C∗r (G,C(Zi)))x y

K∗+1(C∗r (G,C(Zi))) ←−−−− K∗+1(C∗r (G,C(Zi+1))) ←−−−− K∗+1(C∗r (G,C0(α+i+1))).

i.e.K∗G(α+

i+1) −−−−→ K∗G(Zi+1) −−−−→ K∗G(Zi)x yK∗+1G (Zi) ←−−−− K∗+1

G (Zi+1) ←−−−− K∗+1G (α+

i+1).

(17)

Similarly we have

K∗+dimG/UU (α+

i+1) −−−−→ K∗+dimG/UU (Zi+1) −−−−→ K∗+dimG/U

U (Zi)x yK∗+dimG/U+1U (Zi) ←−−−− K∗+dimG/U+1

U (Zi+1) ←−−−− K∗+dimG/U+1U (α+

i+1).

(18)

Proposition 5.7. We have the following commuting diagram:

K∗+dimG/UU (α+

i+1) K∗+dimG/UU (Zi+1) K∗+dimG/U

U (Zi)

K∗G(Zi)K∗G(Zi+1)K∗G(α+i+1)

K∗+dimG/U+1U (α+

i+1) K∗+dimG/U+1U (Zi+1) K∗+dimG/U+1

U (Zi)

K∗+1G (Zi)K∗+1

G (Zi+1)K∗+1G (α+

i+1)

DDD

DDD

(19)

where the top and bottom are the six-term exact sequences and the vertical arrows are the Dirac indexmap D.

Proof: The diagram commutes because all the vertical maps D come from the same element

dG,G/U ∈ KKG(Cτ (G/U),C)

11

as in Section 4. �

After all these work we can prove Theorem 2.1 Proof of Theorem 2.1: We use induction on the Zi.First for Z1 = α+

1 , by Proposition 4.1,

K∗+dimG/UU (Z1)

D−→ K∗G(Z1)

is an isomorphism.Assume that for Zi,

K∗+dimG/UU (Zi)

D−→ K∗G(Zi)

is an isomorphism.By Proposition 4.1, the vertical maps on the left face of the commuting diagram 19 are isomorphisms.

Moreover by induction we can get that the vertical maps on the right face are isomorphism too, henceby a 5-lemma-argument we get the middle vertical maps are also isomorphisms, i.e. for Zi+1,

K∗+dimG/UU (Zi+1)

D−→ K∗G(Zi+)

is an isomorphism.There are finitely many orbits and α+

k is the largest orbit, so

Zk =⋃

all orbits

α+i = F

henceK∗+dimG/UU (F )

D−→ K∗G(F )

is an isomorphism. we finished the proof Theorem 2.1. �

6 An ExampleWe look at the case when G = SL(2,R) and GC = SL(2,C). Hence

BC =

{(a b0 a−1

)∣∣∣∣ a ∈ C∗, B ∈ C}

andF = GC/BC = CP 1 ∼= S2.

It is well-known that the GC (hence) G acts on F = CP 1 by fractional linear transform, i.e. usingprojective coordinate (

a bc d

)·(uv

):=

(au+ bvcu+ dv

).

Or let z = u/v, then (a bc d

)· z :=

az + b

cz + d. (20)

From the formula 20 we can see that the action of G on F is not transitive. In fact, it has three orbits

α+1 =R ∪∞ ∼= S1 the equator,

α+2 ={x+ iy|y > 0} ∼= C the upper hemisphere,

α+2 ={x+ iy|y < 0} ∼= C the lower hemisphere.

α+1 is a closed orbit with dimension 1; α+

2 and α+3 are open orbits with dimension 2.

12

Let’s look at α+1 first. Take the point 1 ∈ α+

1 . The isotropy group at 1 is the upper triangular groupB in SL(2,R). So

K∗G(α+1 ) = K∗B(pt).

B is solvable hence amenable and Z/2Z is the maximal compact group of B. By Theorem 3.8

K0B(pt) = R(Z/2Z)

is the representation ring of the group with two elements and

K1B(pt) = 0.

SoK0G(α+

1 ) = R(Z/2Z)

andK1G(α+

1 ) = 0.

For α+2 and α+

3 , the isotropy groups are both

T =

{(cos θ sin θ− sin θ cos θ

)}hence by the similar reason to α+

1 we have

K0G(α+

2 ) = K0G(α+

3 ) = K0T (pt) = R(T )

is the representation ring of T and

K1G(α+

2 ) = K1G(α+

3 ) = K1T (pt) = 0.

Now α+2 ∪ α

+3 is open in F so as in the last section we have the short exact sequence

0 −→ C0(α+2 ∪ α

+3 ) −→ C(F ) −→ C(α+

1 ) −→ 0 (21)

and further0 −→ C∗r (G,α+

2 ∪ α+3 ) −→ C∗r (G,F ) −→ C∗r (G,α+

1 ) −→ 0.

i.e.0 −→ C∗r (G,α+

2 )⊕ C∗r (G,α+3 ) −→ C∗r (G,F ) −→ C∗r (G,α+

1 ) −→ 0. (22)

We get the six-term exact sequence

K0G(α+

2 )⊕K0G(α+

3 ) −−−−→ K0G(F ) −−−−→ K0

G(α+1 )x y

K1G(α+

1 ) ←−−−− K1G(F ) ←−−−− K1

G(α+2 )⊕K1

G(α+3 ).

(23)

Combine with the previous calculation we get

0 −→ R(T )⊕R(T ) −→ K0G(F ) −→ R(Z/2Z) −→ 0 (24)

and K1G(F ) = 0.

In conclusion we have

K0G(F ) =R(T )⊕R(T )⊕R(Z/2Z),

K1G(F ) =0.

(25)

Then we look at K0U (F ), and K1

U (F ). We know that for G = SL(2,R) the maximal compact subgroupU = T . By Bott periodicity we have

K0U (α+

2 ) = K0U (α+

3 ) ∼= K0U (C) ∼= K0

U (pt) = R(U) = R(T ) (26)

13

andK1U (α+

2 ) = K1U (α+

3 ) ∼= K1U (C) ∼= K1

U (pt) = 0. (27)

As for α+1 , we notice that U acts on α+

1∼= S1 by "square", so the isotropy group is Z/2Z. Hence

K0U (α+

1 ) = R(Z/2Z)

andK1U (α+

1 ) = 0.

By the six-term long exact sequence we have

K0U (F ) =R(T )⊕R(T )⊕R(Z/2Z),

K1U (F ) =0.

(28)

Compare 25 and 28 we haveK∗U (F ) ∼= K∗G(F ).

This is what we expect since for G = SL(2,R) and U = T we have dimG/U = 2.

References[1] P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C*-

algebras, Contemp. Math., 167, Amer. Math. Soc., Providence, RI, 1994

[2] J. Block, N. Higson, Weyl character formula in KK-theory, Available on Jonathan Block and NigelHigson’s website.

[3] V. Bolton and W. Schmid, Discrete Series, in: Representation Theory and Automorphic Forms,Proceedings of Symposia in Pure Mathematics 61, American Mathematical Society, 83-113(1997).

[4] R. Bremigan, J. Lorch, Orbit duality for flag manifolds, manuscripta math. 109, 233-261(2002).

[5] M. Brion, Lectures on the geometry of flag varieties, arXiv:math/0410240v1

[6] J. Chabert, S. Echterhoff, R. Nest, The Connes-Kasparov conjecture for almost connected groupsand for linear p-adic groups, Publ. Math. Inst. Hautes Etudes Sci. No. 97, 239-278(2003).

[7] P. Julg. K-théorie équivariante et produits croisés. C. R. Acad. Sci. Paris Sér. I Math., 292(13):629-632, 1981.

[8] G.G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. math. 91, 147-201(1988).

[9] E. Kirchberg, S. Wassermann. Permanence Properties of C*-exact Groups, Doc. Math. 4, 513-558(1999).

[10] I. Mirković, T. Uzawa and K. Vilonen, Matsuki correspondence for sheaves, Invent. Math. 109,231-245(1992).

[11] M.G. Penington, R. Plymen, The Dirac operator and the principal series for complex semisimple Liegroups, J. Functional Analysis 53 269-286(1983).

[12] M. Rieffel, Applications of strong Morita equivalence to transformation group C*-algebras, Proceed-ings of Symposia in Pure Mathematics 38 Part I, 299-310(1982).

[13] J. Rosenberg, Group C*-algebras and topological invariants, Operator algebras and group represen-tations, Vol. II (Neptun, 1980), 95-115, Monogr. Stud. Math., 18, Pitman, Boston, MA, 1984.

[14] J. Wolf, The action of a real semisimple group on a complex flag manifold. I: Orbit structure andholomorphic arc components, Bull. Amer. Math. Soc. 75, Number 6, 1121-1237(1969).

14