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On the Connes-Kasparov isomorphism, I: The reduced C*-algebra of a real reductive group and the K-theory of the tempered dual

This is the first of two papers dedicated to the computation of the reduced C*-algebra of a connected, linear, real reductive group up to Morita equivalence, and the verification of the Connes-Kasparov conjecture for these groups. These results were originally announced by Antony Wassermann in 1987. In Part I we shall give details of the C*-algebraic Morita equivalence, and then compute the Connes-Kasparov morphism subject to some results in tempered representation theory that we shall prove in Part II using tools from David Vogan's classification of the tempered dual.


INTRODUCTION
If π is a unitary representation of a locally compact group G on a Hilbert space H, then the formula defines a representation of the group C * -algebra C * (G) as bounded operators on H. In this way the category of unitary representations of G becomes equivalent to the category of (nondegenerate) representations of C * (G). See [Dix77,Chap. 13].
The unitary dual of G is the set of irreducible unitary representations of G, up to equivalence. The C * -algebra point of view equips the unitary dual with a topology whose closed sets are in bijection with closed, two-sided ideals J ⊳ C * (G): the closed set determined by J is the set of all irreducible unitary representations that vanish on J. If G is a real reductive group, then Harish-Chandra's tempered dual may be identified with the closed subset of the unitary dual associated with the kernel of the left regular representation (1.2) λ : C * (G) −→ B(L 2 (G)).
See [CHH88]. To a first approximation, the goals of this paper are to determine the tempered dual as a topological space, and then compute its K-theory.
Date: 02/15/2022. 1 The tempered dual may be identified with the (topological) space of irreducible representations of the reduced C * -algebra C * r (G), which is the quotient of the full group C * -algebra by the kernel of the regular representation (1.2). Our precise goals are to determine C * r (G) up to Morita equivalence, and then compute its K-theory.
It is more common in tempered representation theory to study the dual as a set, or as a measure space, ignoring its topological structure. But the extra effort required to study the tempered dual as a topological space (and at the same time study the reduced C * -algebra) is rewarded in spectacular fashion by a beautiful isomorphism statement in K-theory that was conjectured by Connes [Ros84,Conjecture 4.1] and Kasparov [Kas84, Section 5, Conjecture 1]. A proof of the conjecture was announced by Wassermann in [Was87], following pioneering work by Penington and Plymen [PP83] and Valette [Val84,Val85]. Later, Lafforgue gave an entirely different proof using his work on the Baum-Connes conjecture [Laf02b].
We shall present full details of a proof of the Connes-Kasparov isomorphism for connected, linear, real reductive groups. We shall more or less follow Wassermann's announced approach, which went through a determination of the reduced C * -algebra up to Morita equivalence. An earlier paper [CCH16] gave a detailed account of most of the steps in Wassermann's description of C * r (G). Our first task here will be to complete that description and use it to compute the K-theory of the reduced C * -algebra. We shall follow Wassermann's outline closely. It depends very much on the Knapp-Stein theory of R-groups [KS71,KS80], and we shall make explicit the results from this theory that we require. Then we shall analyze the Connes-Kasparov index homomorphism in C * -algebra K-theory, and prove it to be an isomorphism. Here we shall depart from Wassermann's approach and use instead results about Dirac operators in representation theory, along with the theory of minimal K-types, not all of which were available to Wassermann. The necessary facts about Dirac operators in representation theory will be stated in this paper, and the way in which they lead to a proof of the Connes-Kasparov isomorphism conjecture, will be explained. But some of the purely representation-theoretic arguments pertaining them will be presented in a second paper since they require a quite different set of mathematical tools to those that we shall use here (we shall also defer the proof one fact about the Knapp-Stein R-groups to that paper). As for the present paper we have tried to make it accessible to both C * -algebra specialists and representation theorists.
Notes on Terminology. Throughout the paper, by a real reductive group, we shall always mean the group G of real points in a connected complex reductive linear algebraic group that is defined over R. We shall also assume that G is itself connected. We shall often refer to the text [Kna86], so let us note here that our groups are the same (up to isomorphism) as the linear connected reductive groups in [Kna86, §I.1].
We shall use fraktur letters such as g, etc, to refer to the Lie algebras of Lie groups such as G, etc, and not to the complexifications of these Lie algebras. This is because we shall have no use for the complexified Lie algebras in this first paper. But in the second paper we shall use the complexifications extensively and we shall follow a different convention.
When discussing Dirac operators we shall follow conventions appropriate to index theory on manifolds. These are a bit different from the conventions in representation theory, where Dirac cohomology is studied. But in the second paper we shall switch and follow the Dirac operator conventions that are used in representation theory.

PARABOLIC INDUCTION AND THE REDUCED GROUP C*-ALGEBRA
In this section we shall review the description of the reduced C * -algebra of a real reductive group that was obtained in [CCH16] using results in tempered representation theory due to Harish-Chandra, Langlands and others. Then, following Wassermann [Was87], we shall refine that description so as to determine the reduced group C * -algebra up to Morita equivalence.
We shall fix, once and for all in this paper, a maximal compact subgroup K⊆G and Cartan decomposition g = k ⊕ s. We shall also fix a maximal abelian subspace a⊆s and a compatible Iwasawa decomposition G = KAN. The associated minimal parabolic subgroup is Since we shall be working with convolution algebras we shall also fix a Haar measure on G, as well as normalized Haar measure on K.
Parabolic Induction. We begin by reviewing some essential points about parabolic induction. A standard parabolic subgroup of G is any closed subgroup P of G that includes P min . It decomposes as a semi-direct product P = L P N P of a Levi component L P that is mapped to itself by the Cartan involution and the unipotent radical N P . Furthermore, the Levi component L P is the product of its compactly generated part M P and split component A P . This leads to a Langlands decomposition P = M P A P N P . See [Kna86,§V.5] If π is a unitary representation of L P , then we may form the (unitarily) parabolically induced representation Ind G P π, which is the unitary action of G by left translation on the Hilbert space completion of the vector space of smooth functions The completion is taken with respect to the inner product and ρ ∈ a * P is defined by ρ(X) = 1 2 Trace ad X : n P → n P .
2.3. Definition. Let P be a standard parabolic subgroup and let σ be an irreducible, square-integrable representation of M P and let ϕ ∈ a * P . The formula σ⊗e iϕ : ma −→ e iϕ(log a) σ(m) defines a unitary representation of L P =M P A P on the Hilbert space of the representation σ. The associated (P, σ)-principal series representation of G is the unitary representation π σ,ϕ that is obtained from σ⊗e iϕ by parabolic induction. We shall denote by Ind G P H σ ⊗C ϕ the Hilbert space on which it acts.
If we restrict the functions in (2.1) to K, then all the representations in the (P, σ)-principal series can be regarded as acting on the following common Hilbert space (and we shall mostly do so from now on): 2.4. Definition. We shall denote by Ind G P H σ the Hilbert space completion of the space of smooth functions in the inner product (2.2).
Intertwining Operators. The following concept gives us a first, large-scale view of the tempered dual: 2.6. Definition. Let P 1 = M 1 A 1 N 1 and P 2 = M 2 A 2 N 2 be standard parabolic subgroups and let σ 1 , σ 2 be irreducible square-integrable representations of M 1 and M 2 respectively. The pairs (P 1 , σ 1 ) and (P 2 , σ 2 ) are associate if there exists an element k ∈ K such that Ad k [L P 1 ] = L P 2 and Ad * k σ 1 ≃ σ 2 . We shall call an equivalence class of pairs under this relation an associate class and use the notation [P, σ].
2.7. Theorem. The tempered dual admits a disjoint union decomposition as a topological space, where G P,σ consists of the irreducible components of the (P, σ)-principal series representations and the union is indexed by associate classes. Each part G P,σ is a connected and open subset G temp (and it follows that each part is also closed, too).
The theorem summarizes several important results of Harish-Chandra and others, and it is the foundation for the study of the tempered dual and the reduced group C * -algebra. See [CCH16, §5] for a discussion.
To probe the equivalences within a single principal series family, as well as the possible reducibility of the representations within that family, one studies intertwining operators.
Here w(ϕ)(X) = ϕ(Ad w −1 (X)). The operator U w,ϕ varies strongly-continuously with ϕ ∈ a * P . The Knapp-Stein intertwining operators completely account for the decomposition of principal series representations into irreducible representations, and for equivalences among these irreducible summands. We refer to [CCH16,§6] for a summary that is adapted to our purposes; the same information will be encoded in the description of the reduced C * -algebra in Theorem 2.14 below.
The Reduced Group C*-Algebra. Let P = M P A P N P be a parabolic subgroup of G, and let σ be an irreducible square-integrable representation of M P . The (P, σ)-principal series representations are tempered, and they therefore determine representations π σ,ϕ of the reduced group C * -algebra using formula (1.1). We now introduce the C * -algebra (2.11) C 0 (a * P , K(Ind G P H σ )) of norm-continuous functions, vanishing in norm at infinity, from the locally compact space a * P to the C * -algebra of compact operators on the Hilbert space Ind G P H σ . 2.12. Proposition ([CCH16, Cor 4.12]). There is a (unique) C * -algebra homomorphism and every ϕ ∈ a * P . 2.13. Remark. In [CCH16], the right-hand side is described in terms of functions on A P , which identifies with a * P through the exponential map as in Definition 2.3.
There is an action of the intertwining group W σ on the C * -algebra (2.11) that is characterized by the formula w(f)(w(ϕ)) = U w,ϕ f(ϕ)U * w,ϕ for all w ∈ W σ and all ϕ ∈ a * P . It follows from the intertwining property (2.10) that the image of the morphism π σ in Proposition 2.12 is fixed pointwise by this action of W σ . The description of the reduced C * -algebra given in [CCH16] is as follows: 2.14. Theorem ([CCH16, Thm 6.8]). The morphisms in Proposition 2.12 combine to give an isomorphism of C * -algebras The direct sum is the C 0 -direct sum of C * -algebras over a choice of representatives of the associate classes [P, σ].
The Principal Series as an Equivariant Bundle. Form the trivial bundle of Hilbert spaces with fiber Ind G P H σ over the locally compact space a * P . The Knapp-Stein intertwiners determine an action on this bundle, Following Wassermann [Was87], we shall give a simpler description, up to isomorphism, of this W σ -equivariant Hilbert space bundle.
2.17. Remark. The bundle a * P × Ind G P H σ is infinite-dimensional, but it decomposes canonically as the orthogonal Hilbert direct sum of its finitedimensional K-isotypic components, which are finite-dimensional W σ -equivariant bundles in their own right. One could, if one preferred, work with these finite-dimensional bundles.
Define a second W σ -action on the bundle a * P × Ind G P H σ by the formula (2.18) w · (ϕ, v) = (w(ϕ), U w,0 v). Proof. The single-point subset {0} ⊆ a * P is a W σ -equivariant deformation retract. It therefore follows from elementary vector bundle theory that any two W σ -equivariant bundles over a * P whose fibers over 0 are unitarily equivalent as representations of W σ are in fact unitarily equivariantly isomorphic as bundles.
2.20. Corollary. The W σ -actions on the C * -algebra C 0 (a * P , K(Ind G P H σ )) defined by the formulas (2.16) and (2.18) are conjugate by a C * -algebra automorphism. In particular, the corresponding fixed-point C * -subalgebras are isomorphic.
The Reduced C*-Algebra up to Morita Equivalence. In this section we shall construct a Morita equivalence between each summand in the decomposition of Theorem 2.14 and a still more elementary C * -algebra. We shall continue to follow Wassermann [Was87] closely.
We shall use Corollary 2.20, and throughout this subsection we shall work with the action of W σ on the C * -algebra C 0 (a * P , K(Ind G P H σ )) that is derived from (2.18). We shall determine the fixed-point C * -subalgebra up to Morita equivalence.
The intertwiner U w,0 acts as a scalar on Ind G P H σ . Denote by R σ the quotient group W σ /W ′ σ .
The group R σ acts on the C * -algebra is an isomorphism of C * -algebras. We shall therefore concentrate on the R σ -fixed point C * -algebra.
To proceed, we shall use further information about the intertwining group W σ .
2.23. Theorem. [KS80,Thm 13.4] The quotient group homomorphism W σ → R σ splits, and the intertwining group W σ therefore admits a semi-direct product decomposition W σ = W ′ σ ⋊ R σ . We shall fix a splitting and its associated semi-direct product decomposition from now on. See the next section for more details on the splitting (those details are not needed by us yet).
We obtain from the splitting in the theorem an actual, as opposed to projective, unitary action of R σ on the Hilbert space Ind G P H σ via the Knapp-Stein operators U w,0 . Now form the space K Ind G P H σ , ℓ 2 R σ of compact Hilbert space operators from Ind G P H σ into the finite-dimensional Hilbert space ℓ 2 R σ . Use the action of R σ on Ind G P H, along with the left-translation action of R σ on ℓ 2 R σ , to define an R σ -action on K Ind G P H σ , ℓ 2 R σ . Finally, form the Banach space (2.24) C 0 a * P /W ′ σ , K(Ind G P H σ , ℓ 2 R σ ) Rσ .
It carries commuting actions of the C * -algebra C 0 (a * P /W ′ σ , K(Ind G P H σ )) Rσ on the right, by pointwise composition, and of C 0 (a * P /W ′ σ , K(ℓ 2 R σ )) Rσ on the left. We shall prove the following result, the significance of which is that the right-hand side of the equivalence depends only on the group W σ and its normal subgroup W ′ σ , and not the specific action of W σ on Ind G P H σ . 2.25. Theorem (c.f. [Was87,Cor 7]). For each associate class [P, σ], the bimodule Rσ implements a strong Morita equivalence Let us quickly recall that the C * -algebraic concept of Morita equivalence includes analytic requirements that are obviously absent from the purely algebraic theory (among other things, they help extending the reach of the theory to non-unital C * -algebras). A succinct formulation is as follows: an equivalence A-B-bimodule must have the form pCp ⊥ , where (i) C is a C * -algebra and p is a projection in the multiplier algebra [Ped79] of C; (ii) C * -algebra isomorphisms are provided between pCp and A, and between p ⊥ Cp ⊥ and B; and (iii) pCp ⊥ Cp and p ⊥ CpCp ⊥ are dense in pCp and p ⊥ Cp ⊥ , respectively. See [RW98]. In the present case, C will be the C * -algebra of R σ -fixed functions of class C 0 from a * P /W ′ σ to the C * -algebra of compact operators on the direct sum Hilbert space Ind G P H σ ⊕ ℓ 2 R σ . 2.26. Lemma. Let Γ be a finite group acting properly on a locally compact Hausdorff space X, and let H 1 and H 2 Hilbert spaces equipped with unitary representations of Γ . If for every x ∈ X, H 1 and H 2 are weakly equivalent representations of the stabilizer subgroup Γ x (that is, each is contained in a multiple of the other), then the bimodule C 0 X, K(H 2 , H 1 ) Γ implements a Morita equivalence of C * -algebras Proof. Denote the two C * -algebras in the statement of the proposition by A and B, and the A-B-bimodule by E. We need to show that the sets span dense ideals in A and B, respectively. If an ideal in a C * -algebra is not dense, then there is an irreducible representation of the C * -algebra that vanishes on the ideal. So to prove density in A we need only show that for every irreducible representation of A there is some element f∈E such that the representation is nonzero on ff * ∈ A.
The irreducible representations of A are given by evaluations at points x∈X, which give representations as compact operators on the Hilbert space H 1 , followed by compression to one of the irreducible subrepresentations of these representations, which are precisely the nonzero Γ x -isotypical subspaces H ρ 1 of H 1 (ρ ∈ Γ x ). By hypothesis, the isotypical subspace H ρ 2 is nonzero, and hence there is a nonzero Γ x -equivariant compact operator T : H 2 → H 1 whose range lies in H ρ 1 , and there is a function f ∈ E whose value at x is T . But now the value of ff * ∈ A at x is equal to TT * , which is nonzero.
We shall need to combine the simple computation above with the following more substantial result from the Knapp are linearly independent of one another.
2.28. Corollary. The representation of R σ on Ind G P H σ includes a copy of every irreducible representation of R σ .
Proof of Theorem 2.25. It follows from the corollary above that Ind G P H σ includes a copy of every irreducible representation of every subgroup of R σ , and certainly the same is true of ℓ 2 R σ . So Lemma 2.26 applies with X = a * P /W ′ σ and Γ = R σ . We shall conclude this section by showing how the statement of Theorem 2.25 can be streamlined using some standard C * -algebra language (although we shall not use this language in what follows).
Let Γ be a finite group. Denote by λ and ρ the actions of Γ on K(ℓ 2 Γ ) associated with the left and right regular representations, respectively. In addition, if γ ∈ Γ , then denote by e γ the rank-one projection onto the functions in ℓ 2 Γ that are supported on γ. If A is any C * -algebra with a Γ -action, then we denote by A ⋊ Γ the crossed product C * -algebra.

Lemma ([Rie80, Prop.4.3]). The linear map
Combining Lemma 2.29 with Theorem 2.25, we obtain for any component [P, σ] of the tempered dual a Morita equivalence Assembling the summands using the isomorphism of Theorem 2.14, we obtain the following picture of the reduced C * -algebra up to Morita equivalence, due to Wassermann [Was87].

Theorem ([Was87, Thm 8]).
There is a Morita equivalence of C * -algebras where the sum is over representatives of the associate classes [P, σ].

FURTHER INFORMATION ABOUT THE KNAPP-STEIN INTERTWINING GROUPS
The results in the preceding section give an account of the structure of C * r (G) up to Morita eqivalence in terms of the intertwining groups W σ and their semi-direct product decompositions W σ = W ′ σ ⋊ R σ . In this section we shall summarize the additional facts about these decompositions that we shall need to complete the computations in this paper.
The W'-Group. We defined W ′ σ using the action of the Knapp-Stein intertwining operators on the representations π σ,0 . An important result is that W ′ σ is also the Weyl group of a root system: 3.1. Theorem ([KS80, §13 and §15] and [Kna82]). The subgroup W ′ σ ⊳ W σ is the Weyl group of a (possibly non-reduced) root system ∆ ′ σ spanning a subspace 1 of a * P . The action of the group W σ on a * P permutes the roots in ∆ ′ σ . See [Kna86, Ch. XIV, Sec. 9] for the definition of the root system ∆ ′ σ . It includes a natural system of positive roots ∆ ′ σ,+ ⊆ ∆ ′ σ , and one important consequence of the theorem for us is that the quotient a * P may be identified with a fundamental chamber in a * P for this root system. 3.2. Definition. We shall denote by a * σ,+ ⊆ a * P the dominant Weyl chamber in a * associated to the system of positive roots ∆ ′ σ,+ ⊆ ∆ ′ σ . The R-Group. Using the system of positive roots, Knapp and Stein define R σ as a subgroup of W σ , as follows: 3.3. Definition. The Knapp-Stein R-group R σ ⊆ W σ is the subgroup consisting of those elements that permute the positive roots ∆ ′ σ,+ ⊆ ∆ ′ σ among themselves. This is consistent with our previous terminology: the subgroup R σ normalizes W ′ σ , and since W σ acts by permutations on the Weyl chambers in a * P for the root system ∆ ′ σ , while W ′ σ acts on the chambers simply-transitively, there is a semi-direct product decomposition W σ = W ′ σ ⋊ R σ .
1 To be precise, there is an isomorphism from W ′ σ to the Weyl group of a root system ∆ ′ σ spanning a subspace, and the isomorphism gives the action of W ′ σ on that subspace. There is a complementary subspace on which the action of W ′ σ is trivial.
Since R σ permutes the positive roots among themselves, the action of R σ on a * P restricts to an action R σ × a * σ,+ −→ a * σ,+ . We shall use this action in the next section.
By a reflection of a * P we shall mean an isometric involution of a * P with a one-dimensional −1-eigenspace. Two reflections are orthogonal if their −1eigenspaces are orthogonal.
3.4. Theorem ([KS80, §13 and §15] and [Kna82]). The R-group associated to every associate class [P, σ] is abelian, and indeed a finite product of groups of order two that act by pairwise orthogonal reflections on a * P . Finally, we shall need the size of the group R σ in a crucial special case.
3.5. Definition. Denote by a max the split part of a maximally compact Cartan subalgebra of g = k ⊕ s. Thus a max is the fixed part in s of the action of a maximal torus in K.
The space a max is unique up to conjugation by elements of K, and so its dimension dim(a max ) is independent of any choices.
Proof. The action of the maximal torus associated to a max on s ⊖ a max has no nonzero fixed vectors, and so s ⊖ a max is even-dimensional. 3.7. Theorem. If [P, σ] is an associate class, and if W ′ σ = {e}, then the group R σ is generated by dim(a P )− dim(a max ) pairwise orthogonal reflections on a.
We refer the reader to [CHS22] for a proof using an alternative approach to the R-group due to Vogan [Vog81,Sec. 4.3] (which seems to be much better suited to the problem of computing R σ in the essential case).

K-THEORY OF THE REDUCED C*-ALGEBRA
In this section we shall compute the K-theory [RLL00] of C * r (G) as an abstract abelian group. Since it is a basic feature of K-theory that for any family of C * -algebras {A α } the natural map is an isomorphism, we can and shall focus on the individual fixed-point algebras C 0 (a * P , K(Ind G P H σ )) Wσ that make up the reduced group C * -algebra. We have seen that these are Morita equivalent to the C * -algebras Since K-theory is a Morita invariant it suffices to study the latter.
The computations below are very simple from a K-theoretic point of view, but they require the difficult results about the R-group that we surveyed in the last section.
Essential and Inessential Components. We start from the following partition of the set of associate classes [P, σ].
Proof. Identify the quotient a * P /W ′ σ with the dominant Weyl chamber a * σ,+ ⊆ a * P . The half-sum of the positive roots is a nonzero vector ρ in the chamber that is fixed under the action of R σ . The translations by nonnegative multiples of ρ map a * σ,+ into itself and give an R σ -equivariant homotopy between the identity morphism on the C * -algebra C 0 (a * σ,+ , K(ℓ 2 R σ )) and the zero morphism. So the R σ -fixed-point algebra is homotopy equivalent to zero.
The essential components have nonzero K-theory, and their treatment requires more of the R-group results from Section 3.
Rσ is a free abelian group on one generator, which lies in degree dim(G/K) (mod 2).
Actually, for the sake of a later calculation we shall make a more precise statement directly in terms of the K-theory of C 0 (a * P , K(Ind G P H σ )). The assumption that [P, σ] is essential implies that the group R σ decomposes as a direct product where q= dim(a)− dim(a max ). The generators of the factors act on a * P as pairwise orthogonal reflections and the fixed subspace a * ,Rσ P ⊆ a * P for the action of R σ has dimension d= dim(a max ).
It follows from Theorem 2.27 and Proposition 2.19 that for ν ∈ a * ,Rσ P the elements of the group R σ act linearly independently on the representation space Ind G P H σ ⊗C ν , which therefore decomposes into a direct sum of |R σ | distinct irreducible subrepresentations, on each of which R σ acts as multiples of a distinct character. So we can write (4.5) C 0 a * ,Rσ where X µ is the bundle of Hilbert spaces with fibers X µ,ν . We can therefore form the C * -algebra morphism in which the first map is restriction to a * ,Rσ P ⊆ a * P and the second is projection to a single summand in (4.5). The target C * -algebra is Morita equivalent to C 0 (a * ,Rσ P ) via the bimodule C 0 (a * ,Rσ P , X µ ) and so we can formulate a more precise version of Theorem 4.3 as follows: 4.7. Theorem. If [P, σ] is essential, then for every µ the restriction-projection morphism Proof. We shall prove the Morita equivalent version of the theorem that uses ℓ 2 R σ . The C * -algebra C 0 (a * P , K(ℓ 2 R σ )) Rσ admits a tensor product decomposition Now each of the fixed point algebras in the factorization above fits in an extension 0 −→ J −→ C 0 (R, K(ℓ 2 Z 2 )) Z 2 π −→ C → 0 where the quotient map π is evaluation at 0 ∈ R, followed by compression to the subspace of constant functions in ℓ 2 Z 2 . The kernel J is Morita equivalent to the contractible C * -algebra of C 0 -functions on [0, ∞). It follows that the tensor product of the morphisms π above gives an isomorphism in K-theory from the right-hand side in (4.8) to C 0 (R d ), and since the tensor product of the morphisms π is the same as (4.6), the theorem follows.
Let us summarize: 4.9. Theorem. The group K dim(G/K) (C * r (G)) is a free abelian group on the set of essential associate classes, while the group K dim(G/K)+1 (C * r (G)) is zero. More precisely, the K-theory of each essential summand of C * r (G) is free abelian in one generator in degree dim(G/K), while the K-theory of each inessential summand of C * r (G) is zero.

THE CONNES-KASPAROV INDEX HOMOMORPHISM
In this section we shall review the construction of Dirac operators on the symmetric space K\G of right K-cosets in G and the definition of the Connes-Kasparov index homomorphism. Spin Modules. Let K be a compact Lie group and let s be a finite-dimensional Euclidean vector space that is equipped with an orthogonal action of K. Later on, the example of interest will be where g = k ⊕ s is the Cartan decomposition of the Lie algebra of G. With this in mind we shall denote the K-action by (k, X) −→ Ad k (X) for k ∈ K and X ∈ s (even though this will be a slight abuse of notation in general). Form the Clifford algebra Cliff(s) using the convention that the square of any element from s is minus the norm-squared of that element. 2 If X ∈ s, then we shall denote by c(X) the corresponding element in the Clifford algebra, so that the convention reads: c(X) 2 = − X 2 · 1.

Definition.
If s is even-dimensional, then a spin module for the pair (K, s) is a finite-dimensional, Z 2 -graded, complex Hilbert space S that is equipped with (i) a representation of Cliff(s), written for X ∈ s and s ∈ S, in which each X acts as a grading-degree one, skew-adjoint operator, and (ii) a grading-degree zero, unitary representation of K that is compatible with the representation of Cliff(s) in the sense that k · (X · s) = Ad k (X) · (k · s) for every k ∈ K, every X ∈ s, and every s ∈ S. If s is odd-dimensional, then a spin module for (K, s) is a spin module for (K, s ⊕ R), where R is equipped with the trivial action of K.

Definition.
We shall denote by R spin (K, s) the abelian group generated by isomorphism classes of spin modules subject to the relations where S opp is obtained from S by reversing the Z 2 -grading.
where the sum is over an orthonormal basis {X a } of s.
The fundamental morphism is a Lie algebra morphism (for the commutator bracket on the Clifford algebra) and moreover for all X ∈ s and all Z ∈ k. If S is a spin module, then by composing the Clifford algebra action on S with α we obtain a representation of k on S.
Suppose for a moment that s is even-dimensional. Fix an irreducible representation S irr of the Clifford algebra on a finite-dimensional Z 2 -graded Hilbert space (S irr is unique up to a possibly grading-reversing unitary equivalence). The fundamental morphism endows S irr with a k-action, and so to any spin module S we can associate the Z 2 -graded k-module (the morphisms in mod(S) need not be grading-preserving). We can reconstruct S from mod(S) via the canonical isomorphism where the tensor product is given the diagonal k-action. If s is odd-dimensional, then we repeat the above with Cliff(s ⊕ R) in place of Cliff(s). In either case, the k-module mod(S) does not necessarily integrate to a representation of K. However if we define a compact group K by means of the pullback diagram in which the bottom morphism comes from the adjoint action of K on s, then mod(S) integrates to a representation ofK. The pullback groupK may or may not be connected, but in any case the kernel of the morphismK→K is a two-element group, and there is a unique morphism fromK into the group of invertible elements in Cliff(s) whose associated Lie algebra morphism is α, and which maps the nontrivial element of the kernel to minus the identity. 5.6. Definition. We shall say that a representation ofK is genuine if the nontrivial element in the kernel ofK→K acts as −I in the representation. 5.7. Theorem. The abelian group R spin (K, s) is isomorphic via the correspondence S → mod(S) to the free abelian group on the set of equivalence classes of irreducible and genuine representations ofK.
The Dirac Operator and its Square. For the rest of this section K will be the given maximal compact subgroup of our real reductive group G, and s will be the complementary subspace to k in the Cartan decomposition g = k ⊕ s.
Fix a G-invariant symmetric bilinear form that is positive-definite on s and negative-definite on k, and form the associated positive-definite inner product where θ is the Cartan involution. Given a spin module S, form the space [C ∞ c (G) ⊗ S] K , where K acts diagonally, and where the K-action on C ∞ c (G) is by left translation. 5.10. Definition. The Dirac operator associated to a spin module S is the linear operator in which the sum is over an orthonormal basis for s, and X a acts on C ∞ c (G) via the left-translation action of G on C ∞ c (G). Compare [Par72, AS77]. 5.11. Definition. The Casimir element for G is the element in the enveloping algebra 3 U (g), where the sum is over any basis {X a } for g with dual basis {X a } for the invariant form B (so that B(X a , X b ) = δ a b ). Similarly the Casimir element for K is the element where the sum is over any basis {Z b } for k and dual basis {Z b } for the invariant form B, restricted to K.

Definition. The diagonal morphism
is the morphism of associative algebras for which for all Z ∈ k, where α is the fundamental morphism from Definition 5.3.
In the next result, it is convenient to view the Dirac operator algebraically, as an element of U (g) ⊗ Cliff(s); the choice of spin module S is therefore no longer immediately relevant. The expression for the square of the Dirac operator in Theorem 5.13 below is essentially due to Parthasarathy [Par72]; see [HP06, Prop. 3.1.6] for a modern account.
The inner product (5.9) extends to a complex inner product on the complexification g C , and it then restricts to an inner product on any Cartan subalgebra. The restriction induces an inner product on the vector space dual of the Cartan subalgebra. The same goes for any Cartan subalgebra of the Lie algebra of K, and the associated norms are those that appear below: 5.13. Theorem. Let ρ K and ρ G be the half-sums of the positive roots for k C and g C (formed using any choices of Cartan subalgebras in the complexified Lie algebras and any systems of positive roots). The square of the Dirac operator in U (g) ⊗ Cliff(s) is given by the formula Let us now bring the spin module S back into the picture and compute the operator ⊗ S] K arising from Theorem 5.13. 5.14. Definition. If S is an irreducible spin module, so that mod(S) is an irreducible representation ofK, and also of k, then we define where µ is the highest weight of the k-module mod(S) (both µ and ρ K depend on a choice of Cartan subalgebra of k C and system of positive roots, but the norm does not).

Lemma. If S is an irreducible spin module, then the operator
Proof. Write S ∼ = S irr ⊗ mod(S) * as in (5.5). Under this isomorphism, the Clifford algebra acts on S irr , but not on mod(S) * . So the diagonal morphism ∆ gives the action In contrast, the K-fixed part of this space is computed using the actions of K (or, strictly speakingK) on all three factors, so that We can therefore compute the action of ∆(Ω) using either the action of k on C ∞ c (G) ⊗ S irr or the action of k on mod(S). Using the latter it is well known that we obtain µ + ρ K 2 − ρ K 2 ; See [KV95,Prop 4.120] for instance.
With this, we can simplify the formula for the square of the Dirac operator: 5.16. Theorem. The square of the Dirac operator Infinitesimal Characters. Let us quickly review some basic topics in representation theory. For a further discussion of all these concepts and results, see for instance [Kna86,Chap.VIII].
If π is any unitary representation of G on a Hilbert space H π , then we shall denote by H π,fin the space of K-finite vectors in H π . According to a theorem of Harish-Chandra, if π is irreducible, then it is admissible, which means that each K-isotypical subspace in H π,fin is finite dimensional.
If π is admissible (but not necessarily unitary), then H π,fin is included within the smooth vectors in H π and so it carries a representation of the Lie algebra g. One says that π is quasi-simple if Z (g), the center of the universal enveloping algebra of g, acts as multiples of the identity operator.
Let h be any Cartan subalgebra of g. Harish-Chandra defined an isomorphism where W is the Weyl group associated to (g, h) and S (h) W is the W-invariant part of the symmetric algebra of h. We shall identify the range with the algebra of W-invariant complex polynomial functions on the complex vector space Hom R (h, C). If π is an admissible and quasi-simple representation of G, then the infinitesimal character of π is the algebra homomorphism inf. ch.(π) : Z (g) −→ C that gives the action of the center of the enveloping algebra on H π,fin . Using the Harish-Chandra isomorphism we can and will view the infinitesimal character as (any representative of) a W-orbit in Hom R (h, C).

Definition.
Let h = t h ⊕ a h be a Cartan subalgebra of g, with t h ⊆ k and a h ⊆ s. An element of Hom R (h, C) is said to be real if it belongs to and it is said to be imaginary if it belongs to The same terminology may be applied to infinitesimal characters, using the Harish-Chandra isomorphism. Whether or not an infinitesimal character is real or imaginary does not depend on the choice of representative of inf. ch.(π) within its W-orbit. Nor does it depend on the choice of Cartan subalgebra (as long as the Cartan subalgebra is stable under the Cartan involution). See [Kna86,p. 535].
For the next result, recall that the complexification process outlined in the discussion preceding Theorem 5.13 endows h * C ∼ = Hom R (h, C) with an inner product. Every element of Hom R (h, C) decomposes as a sum of real and imaginary parts, and we shall use the standard notation for these. 5.19. Lemma. Let π be a unitary, admissible and quasi-simple representation of G on a Hilbert space H π . If the real and imaginary parts of the infinitesimal character of π are orthogonal, then the operator π(Ω G ) + ρ G 2 acts on H π,fin as the scalar Re(inf. ch.(π)) 2 − Im(inf. ch.(π)) 2 .
Proof. Extend B to a complex bilinear form on the complexification g C . The restriction to any h C is nondegenerate, and so it determines a nondegenerate bilinear form on h * C in the usual way. Using the Harish-Chandra homomorphism (5.17), the formula in the lemma is then: This formula holds generally, when inf. ch.(π) is replaced by any element of h * C ; see for instance [KV95,Prop 4.120]. Let us now apply this to the unitary principal series representations of G. Let [P, σ] be an associate class, and let P = M P A P N P be the Langlands decomposition of P, so that σ is an irreducible square-integrable representation σ. Harish-Chandra showed that: 5.20. Theorem. Whenever M P carries an irreducible square-integrable representation, the Lie algebra t P of any maximal torus in K ∩ M P is a Cartan subalgebra of m P . 5.21. Theorem. Every irreducible, square-integrable representation of M has real infinitesimal character. Moreover, for every N > 0 the set of equivalence classes of irreducible, square-integrable representations of M with inf. ch.(σ) < N is finite.
For an exposition of these results, see for instance [Kna86]. We shall use the second statement in the second theorem in the next subsection. As a result of the first theorem, the Lie algebra t P ⊕ a P is a Cartan subalgebra of g, and we may compute the infinitesimal characters for the (P, σ)-principal series as follows: 5.22. Lemma (See for example [Kna86,Prop. 8.22]). The infinitesimal character of the unitary (P, σ)-principal series representation π σ,ϕ is inf. ch.(σ) ⊕ ϕ ∈ Hom R (t P , iR) ⊕ Hom R (a P , iR).
Note that the two summands in the infinitesimal character above are its real and imaginary parts, respectively.
Dirac Operator from the Representation Theory Point of View. Now form the Hilbert space Ind G P H σ as in Definition 2.4, and given a spin-module S, form the fixed space The same space is obtained if we replace Ind G P H σ by its subspace of K-finite vectors, and as a result [Ind G P H σ ⊗ S] K carries an action of g. So if we regard Ind G P H σ as carrying the principal series representation π σ,ϕ , then we may form the operator The following is an immediate consequence of Lemmas 5.19 and 5.22: 5.24. Lemma. The operator π σ,ϕ (Ω G ) + ρ G 2 acts on [Ind G P H σ ⊗ S] K as the scalar inf. ch.(σ)) 2 − ϕ 2 .
Putting this together with Theorems 5.13 and 5.16, we arrive at the following result: 5.25. Theorem. If π σ,ϕ is any (P, σ)-principal series representation, and if S is any irreducible spin module, then / D 2 σ,ϕ,S = S 2 − inf. ch.(σ) 2 + ϕ 2 . 5.26. Remark. Strictly speaking, to reach this conclusion we need the formula shows that / D 2 σ,0,S will be negative whenever inf. ch.(σ) 2 > S 2 , assuming that [Ind G P H σ ⊗ S] K is nonzero. But the Dirac operator is self-adjoint, so its square is positive semidefinite. So necessarily [Ind G P H σ ⊗ S] K is zero in these cases. The corollary now follows from Theorem 5.21.

Dirac Operator from a Functional Analytic Point of View. The Dirac operator / D S can be viewed as an unbounded operator on the Hilbert space
The Dirac operator so viewed is essentially self-adjoint (see [Che73]), and there is therefore an associated one-parameter group of unitary operators exp(it / D S ). These restrict to operators [Che73] again); this is the finite propagation property of the Dirac operator.
But we are more interested in viewing / D S as an unbounded operator on the space [C * r (G) ⊗ S] K , which becomes a Hilbert C * -module over C * r (G) when equipped with the right action of C * r (G) on the first factor and the C * r (G)-valued inner product f 1 ⊗ s 1 , f 2 ⊗ s 2 = f * 1 f 2 s 1 , s 2 . See [Lan95] for general information about Hilbert C * -modules. The operators exp(it / D S ) in (5.28) extend to a one-parameter group of unitary operators As in the previous section, given a (P, σ)-principal series representation π σ,ϕ : G −→ U Ind G P H σ we may form the operator Of course, the operator / D σ,ϕ,S is acting on a finite-dimensional Hilbert space, and is therefore already bounded itself.
associated with the C * -algebra isomorphism of Theorem 2.14, the operator for all f ∈ C 0 (a * P , K(Ind G P H σ )) Wσ and all s ∈ S, where the product · on the right hand side is composition of linear operators on the finite-dimensional space r (G) ⊗ S] K . We shall prove this using the representation theory calculations from the previous section, since those results are at hand. The original proof, due to [Kas83], uses the basic elliptic estimates for the Dirac operator and the Rellich lemma. See [BCH94] for a general account of these these matters.
Proof. The formula for / F S in Lemma 5.30 and the formula for / D S,σ,ν in Theorem 5.25 combine to give a formula for 1− / F 2 S as an operator on The direct sum here is actually a finite direct sum, in view of Corollary 5.27, and in each summand 1− / F 2 S acts as multiplication by a C 0 -scalar-valued function. Each such operator is compact, thanks to the finite-dimensionality of the spaces [Ind G P H σ ⊗ S] K . Now the compact operators on any Hilbert C * -module form an ideal in the C * -algebra of all bounded, adjointable operators, and by definition a bounded adjointable operator is Fredholm if it is invertible modulo this ideal. In the present case, we see from the theorem above that / F S is its own inverse, modulo compact operators. Therefore: 5.32. Corollary. The operator is a bounded, self-adjoint, odd-graded, Fredholm operator on the Z 2 -graded Hilbert The Connes-Kasparov Index Homomorphism. In order to define the index homomorphism it is convenient to use Kasparov's approach of C *algebra K-theory [Kas81] (see [Hig90,Section 3] for an exposition).
Kasparov defines the K 0 -group of a C * -algebra A as the group of homotopy classes of bounded, self-adjoint, Fredholm operators F on Z 2 -graded Hilbert A-modules. In addition, he defines the K 1 -group in the same way, but using Hilbert A-modules that carry an additional odd-graded skewsymmetry which is required to anti-commute with F.

Remark.
We note for later use that, as a consequence of the way homotopy is defined, it is an elementary property of K-theory that if a Fredholm operator is actually invertible (not merely invertible modulo compact operators), then it determines the zero class in K-theory.
Kasparov's definitions are made with Dirac operators in mind, and it follows immediately from the definitions and the results we have summarized above that if S is any spin module, then the Fredholm operator / F S determines a class Index( / F S ) ∈ K dim(G/K) (C * r (G)) (in the case where dim(G/K) is odd, the skew-symmetry γ is Clifford multiplication by the generator in Cliff(s ⊕ R) associated to the R-summand).

5.35.
Definition. The Connes-Kasparov index homomorphism is the homomorphism of abelian groups R spin (K, s) −→ K dim(G/K) (C * r (G)) that maps the class of a spin module S to the index of / F S in K-theory.
Our aim is to prove that: 5.36. Theorem (Connes-Kasparov Isomorphism). If G is a connected, linear real reductive Lie group, then Connes-Kasparov index map ) is an isomorphism of abelian groups. Moreover K dim(G/K)+1 (C * r (G)) = 0. 5.37. Remarks. There is a Connes-Kasparov index homomorphism for any almost-connected Lie group (that is, any Lie group with finitely many connected components) and moreover it is an isomorphism in this generality [CEN03]. The definition of the index homomorphism for connected Lie groups is essentially the same as the one we have presented. But beyond connected groups, and even within the realm of real reductive groups, the definition of the index homomorphism needs to be adjusted [EP09]. Among other things it is possible that both K-theory groups for C * r (G) might be nonzero at the same time, as is the case for GL(2, R), for instance.

THE MATCHING THEOREM
In this section we shall state a purely representation-theoretic result that will lead quickly (in the next section) to a proof that the Connes-Kasparov index homomorphism is an isomorphism. The result relates representation theory from the perspective of Harish-Chandra to Dirac cohomology.

Statement of the
Moreover if µ is the highest weight of mod(S), then µ+ρ K (compare Definition 5.14) is the so-called Harish-Chandra parameter of σ. Compare [AS77, Theorem 9.3] or [Laf02a, §2]. 6.4. Example. If G = SL(2, R), the content of the matching theorem is described in [BCH94,Example 4.25] (see [CCH16,Example 6.10] for a description of the structure of the reduced C*-algebra). The essential associate classes to consider are of two types. If [P, σ] = [G, H n ] as in the previous example, with H n a discrete series with Harish-Chandra parameter labeled by the integer n = 0, the matching spin module S n such that [H n ⊗ S n ] SO(2) = 0 is S n = S irr ⊗ C 1−n where C 1−n denotes the one-dimensional representation of weight 1 − n of SO(2). One more essential associate class is given by [P, σ] = [P min , ε], where P min is the subgroup of upper-triangular matrices in SL(2, R) and ε is the non-trivial character of the subgroup M = {±I 2 }, so that Ind G P min H ε is the base of the non-spherical principal series. A direct calculation (see [BCH94,(4.2.5)]) shows that the matching spin module under Theorem 6.2 is S 0 = S irr ⊗ C 1 . 6.5. Example. If G is a complex reductive group, then all essential associate classes are attached to the minimal parabolic P min = MAN, for which M is a maximal torus in a maximal compact subgroup of G. The Connes-Kasparov isomorphism was established in the case in [PP83]. More generally, the Matching Theorem was established for semi-simple Lie groups having only one conjugacy class of Cartan subgroups by Alain Valette in [Val85] (see Theorem 3.12). Both references express the correspondence in terms of highest weights, the spin module S irr ⊗ V µ with V µ of highest weight µ being matched with the associate class [P min , σ] where σ ∈M is the weight µ + ρ K .

FIRST PROOF OF THE CONNES-KASPAROV ISOMORPHISM
In this section we shall use the Matching Theorem formulated in the previous section to prove that the Connes-Kasparov index homomorphism is an isomorphism. We shall follow the shortest route to do so, which uses the fact, proved by Kasparov, that the index homomorphism is a split-injective homomorphism of abelian groups. While this is certainly a significant new ingredient in the proof, injectivity is a considerably simpler and more accessible result than surjectivity. (In any case, in the next section we shall take a different approach to the proof of the Connes-Kasparov isomorphism that avoids Kasparov's result. ) Kasparov proved split injectivity in a much broader context than the one we are considering here-involving both continuous and discrete groupsin the course of proving groundbreaking results on the Novikov conjecture in differential topology. But let us record his result as it applies in our case: ) is a split injection of abelian groups.

Proof of the Connes-Kasparov Isomorphism Theorem Using Split Injectivity.
To begin Theorems 4.9 shows in particular that K dim(G/K)+1 C * r (G) = 0, which is one of the assertions in Theorem 5.36. The main task is to show that the index homomorphism is an isomorphism of abelian groups.
The C * -algebra isomorphism in Theorem 2.14 determines a K-theory direct sum decomposition If S is a spin-module for (K, s), then we shall denote by  Proof of Theorem 5.36. We shall use the Matching Theorem. Let S be an irreducible spin module. Lemma 7.2 implies that the image of S under the index homomorphism is concentrated in the summand in (7.1) associated to the unique [P, σ] to which S is matched. Since the index homomorphism is injective, the image there must be nonzero. In fact, because the index homomorphism is split injective, while the summand is isomorphic to Z, the image must be a generator. That is, the index homomorphism maps the basis of R spin (K, s) determined by the irreducible spin modules to the basis determined up to signs by Theorem 4.3.

SECOND PROOF OF THE CONNES-KASPAROV ISOMORPHISM
In this final section we shall study the Dirac operator / D S in more detail, and by doing so give proof of the Connes-Kasparov isomorphism that is independent of Kasparov's split-injectivity result. This is probably more in line with the approach that Wassermann intended to take, as sketched in the note [Was87].

K-Theoretic Preliminaries.
We have already seen that the Connes-Kasparov index homomorphism carries the natural basis for R spin (K, s) to the natural basis 5 for the K-theory of C * r (G). A striking feature of the Connes-Kasparov index is that in fact it carries natural basis elements to natural basis elements at the level of cycles, and not merely at the level of K-theory classes. In this section we shall describe those cycles. 8.1. Definition. Let V be a finite-dimensional Euclidean vector space of dimension d. A Bott element for V consists of a finite-dimensional Z/2-graded Hilbert space S with dim(S) = 2 d/2 d even 2 (d−1)/2 d odd and linear family of odd-graded, self-adjoint operators In the odd case we also require that S carry a symmetry γ as in (5.33) that anti-commutes with all D v .
It follows from the elementary theory of Clifford algebras that Bott elements are unique up to isomorphism. Each Bott element may be regarded as a Fredholm operator on the Hilbert C 0 (V)-module C 0 (V, S) of the sort considered by Kasparov (D is unbounded, but one can take the bounded transform to obtain a bounded Fredholm operator F if preferred). There is therefore an index Index(D) ∈ K d (C 0 (V)). Here is one form of the Bott periodicity theorem: 8.2. Theorem. Let V be a finite-dimensional Euclidean vector space of dimension d. The K d -group of C 0 (V) is freely generated by the index of any Bott element, and the K d+1 -group is zero.
Representation-Theoretic Preliminaries. Now let [P, σ] be an essential associate class. As noted earlier, there is a decomposition of the parabolically induced representation π σ,0 into finitely many irreducible subrepresentations, Ind G P H σ = µ X µ , and the index set in the direct sum is the set R σ of characters of the finite abelian group R σ . But we can index the sum in a different way using Vogan's theory of minimal K-types [Vog81], and it will be very useful to do so in what follows. It will not be important to present the precise definition of minimal Ktype here. It will suffice to recall that the K-types of a representation π of G are the irreducible representations of K that occur upon restriction of π from G to K, and that every representation has a finite number of minimal K-types among these, which depend only on the set of all K-types in π.
The deeper properties of minimal K-types that we shall use below are as follows: 8.4. Theorem. Let [P, σ] be an essential associate class, and let S be the irreducible spin module to which it is matched.
(i) Each minimal K-type of Ind G P H σ has multiplicity one, and each irreducible direct summand X µ of Ind G P H σ , as in (8.3), includes precisely one of these minimal K-types.
(ii) If X µ is any irreducible summand of Ind G P H σ , then where the brackets [ ] in the exponent denote the integer part. (iii) If X µ is any irreducible summand of Ind G P H σ , and if V µ ⊆ X µ is its minimal K-type, then the inclusion is a vector space isomorphism.
We shall prove this theorem in [CHS22, Sec. 8] (mostly by collecting results from elsewhere in the representation theory literature).
It follows from parts (i) and (iii) of the theorem, together with the direct sum decomposition (8.3), that if {V µ } is the set of minimal K-types in Ind G P H σ , then the inclusion is a vector space isomorphism. This gives a very concrete and convenient description of the space [Ind G P H * σ ⊗ S] K . The following lemmas examine the Dirac operators that act on this space. 8.6. Lemma. Let [P, σ] be an essential associate class and let S be the irreducible spin module to which it is matched. The operators are linear functions of ϕ ∈ a * P . Proof. The action of g on the smooth vectors in any principal series representation space such as Ind G P H σ ⊗C ϕ is affine-linear in ϕ (compare [KV95,Prop. 11.47]), and so D σ,ϕ,S is affine-linear in ϕ. But since S is matched to (P, σ), the operator D σ,0,S is zero. So D σ,ϕ,S is actually linear in ϕ. Proof. The representations X µ,ν that appear in the direct sum decomposition Ind G P H σ ⊗C ν = µ X µ,ν , compare (4.4), have the same K-isotypic decompositions as the representations X µ . Therefore for every ν ∈ a * ,Rσ P the K-type V µ appears in X µ,ν as a minimal K-type, and the inclusion is a vector space isomorphism, since [X µ,ν ⊗ S] K depends only on the Kstructure of X µ,ν , and not on ν. The Dirac operator D σ,ν,S certainly maps [X µ,ν ⊗ S] K to itself, and so it maps [V µ ⊗ S] K to itself, as claimed.
8.8. Theorem. If [P, σ] is any essential associate class, and if S is the irreducible spin module to which it is matched, then for any µ the family of Dirac operators / D σ,η,S : X µ ⊗ S K −→ X µ ⊗ S K (η ∈ a * ,Rσ P ).
is a Bott element for a * ,Rσ P .
Proof. This follows from the preceding two lemmas and Theorem 5.25.

Completion of the Second Proof of the Connes-Kasparov Isomorphism.
The Matching Theorem shows that Connes-Kasparov index morphism maps an irreducible spin module for (K, s) to the index of the family of Dirac operators / D σ,η,S : (Ind G P H σ ) ℓ ⊗ S K −→ (Ind G P H σ ) ℓ ⊗ S K , in K dim(s) (C 0 (a * P , K(Ind G P H σ )) Wσ ), where [P, σ] is the unique essential associate class matched to S. By Theorem 8.8, for any summand X µ of Ind G P H σ the family of Dirac operators / D σ,η,S : X µ ⊗ S K −→ X µ ⊗ S K is a Bott element for a * ,Rσ P and therefore its index is a generator of the Ktheory group K dim(s) (C 0 (a * ,Rσ P )). But the above is precisely the image of the cycle that defines the Connes-Kasparov index Index( / D S ) ∈ K dim(s) C 0 a * P , K(Ind G P H σ ) Wσ under the K-theory isomorphism in Theorem 4.7. We have shown, therefore, that the natural generators of R spin (K, s), namely the irreducible spin modules, are mapped to the natural generators of the free abelian group K dim(s) (C * r (G)), namely to classes that map to Bott generators of a * ,Rσ P under the K-theory isomorphism in Theorem 4.7. The proof is complete.