C^{∗}-ALGEBRAS AND GROUPS

P. Julg 22. 5. 2002

Sommerschule Göttingen

General idea.

Klein  Hilbert
groups (geometry), acting on manifolds  Hilbert space

Find representations of groups in Hilbert spaces.

1. C^{∗}-algebras and groups

1.1. Definition. A unitary representation of a group G in a Hilbert space H is a homomorphism π: G → U(H). If G has a locally compact topology, then π is strongly continuous. 1.a. Full and reduced C^{∗}-algebras of G (G discrete). Consider four algebras: ℂG, ℓ^{1}G, C^{∗}G, C_{r}^{∗}G.

ℂG is the algebra of G of G over ℂ, which is a vector space with (algebraic) basis (u_{g})_{g},

ℂG = { a = ∑_{g∈G} a_{g}u_{g} (finite sum) } .

Multiplication extends u_{g} · u_{g′} = u_{gg′} (convolution product).

a = ∑ a_{g}u_{g} ,  b = ∑ b_{g}u_{g} ,  then  a · b = ∑ c_{g}u_{g} ,  c_{g} = ∑_{h} a_{gh^{−1}}b_{h} .

Involution extends g ↦ g^{−1} antilinearly.

a^{∗} = ∑ \overline{a}_{g}u_{g^{−1}} .

Define ℓ^{1}G by

ℓ^{1}G = { a = ∑_{g} a_{g}u_{g} | ∑ |a_{g}| < ∞ } .

Convolution and the involution extend to ℓ^{1}G.

Typeset by AMS-TEX

Given a unitary representation π: G → U(H_{π}), extend π to a ∗-algebra homomorphism ℂG → B(H_{π}),

π: a = ∑ a_{g}u_{g} ⟼ ∑_{g∈G} a_{g}π(g) .

Because |π_{g}| = 1, this homomorphism extends to ℓ^{1}G.

We think of C^{∗}-algebras as subalgebras of B(H) for some Hilbert space which is closed under involution and under the operator norm. To any unitary representation of π, we associate a C^{∗}- subalgebra C_{π}^{∗}G ⊂ B(H_{π}), which is the closure in B(H_{π}) of π(ℂG) (or equivalently, π(ℓ^{1}G)). Abstractly, take the completion of ℂG (or ℓ^{1}G) with respect to the norm ∥π(a)∥_{B(H_{π})}.

1.2. Example. If π is the trivial representation, then C_{π}^{∗}G ≅ ℂ.

Take π = λ to be the left regular representation

λ: G ⟶ U(ℓ^{2}G) , g ⟼ (ε_{x} ⟼ ε_{g^{−1}x}) .

By definition, C_{r}^{∗}G = C_{λ}^{∗}G ⊂ B(ℓ^{2}G) is the reduced C^{∗}-algebra of G.

Take H_{u} = ⨁_{π} H_{π} to be the Hilbert sum over all isomorphism classes of representations of G, then C^{∗}G = C_{u}^{∗}G is the full C^{∗}-algebra of G. Equivalently, C^{∗}G is the completion of ℂG for the norm

(1.3)  ∥a∥_{C^{∗}G} = sup_{π unitary} ∥π(a)∥ .

Universal property. For all unitary representations π, there exists a unique extension of C_{π}^{∗}G → B(H_{π}) to C^{∗}G.

Note: already for the free group F_{2} of two generators, the map C^{∗}G → G_{r}^{∗}G is not injective (F_{2} is not amenable).

1.b. Locally compact groups.

In the definitions above, replace ℂG by the space of compactly supported continuous functions C_{c}G on G, replace ℓ_{1}G by L^{1}G with respect to left Haar measure. The product is still given by convolution,

(f_{1} ∗ f_{2})(g) = ∫_{G} f_{1}(gh^{−1}) f_{2}(h) dh ,
f^{∗}(g) = \overline{f(g^{−1})} Δ(g)^{−1} ,

where Δ(g) is the Radon-Nikodym derivative of left Haar measure with respect to right Haar measure (or vice versa). Define C_{π}^{∗}G ⊂ B(H_{π}) by

π: f ⟼ ∫_{G} f(g) π(g) dg

with respect to left Haar measure. Similarly, C_{r}^{∗}G = C_{λ}^{∗}G ⊂ B(L^{2}(G)), and C^{∗}G is the completion of ℂG with respect to (1.3).

1.4. Example. If G is abelian, then

C^{∗}G ≅ C_{0}(Ĝ) ,

where C_{0} denotes functions that vanish at infinity (arbitrarily small outside compact subsets), the isomorphism is given by Fourier transform.

Ĝ = { χ: G ⟶ 𝕋 = { z ∈ ℂ | |z| = 1 } } .

Each representation χ: G → U(ℂ) gives χ: C^{∗}G → ℂ = B(ℂ).

f ∈ C^{∗}G ⟶ ℂ f ⟼ χ(f) = ∫_{G} f(g)χ(g) dg .

For example,

G = ℤ,  Ĝ = 𝕋 ,  ∑_{n} a_{n}u_{n} ⟼ ∑_{n} a_{n}e^{inθ} .

Or,

G = 𝕋,  Ĝ = ℤ ,  and  G = ℝ,  Ĝ = ℝ .

1.5. Example. If G is compact, let Ĝ be the set of classes of irreducible unitary representations, which are always finite dimensional. For all π ∈ Ĝ, C^{∗}G → B(H_{π}) = End H_{π} (note that H_{π} is now finite-dimensional).

f ⟼ π(f) = f̂(π) = ∫ f(g) π(g) dg ∈ End(H_{π}) = M_{n_{π}}(ℂ) .

In other words, there is a bundle of matrix algebras End(H_{π}) over Ĝ.

C^{∗}G ≅ \overline{⨁_{π∈Ĝ} End H_{π}} ,

with respect to the norm (1.3).

1.6. Example. Let G = SL_{2}(ℝ), or, more generally, SL_{n}(ℝ), SL_{n}(ℂ) (more generally, semisimple Lie groups over ℝ or ℂ). Hope: define a good dual space Ĝ, the set of unitary isomorphism classes of irreducible unitary representations; over π ∈ Ĝ, let C_{π}^{∗}G be the algebra k(H_{π}) ⊂ B(H_{π}) of compact operators (inductive limit of matrix algebras); define a Fourier transform,

π ∈ Ĝ ⇝ π(f) ∈ k(H_{π}) ,

so f ∈ C^{∗}G becomes a section f̂ in the bundle of algebras k(H_{π}) over Ĝ. Note that in contrast to Example 1.4 and Example 1.5, we may have C^{∗}G ≠ C_{r}^{∗}G, but there exists a closed subset Ĝ_{r} ⊂ Ĝ, such that C^{∗}G → B(H_{π}) factors over C_{r}^{∗}G iff π ∈ Ĝ_{r}.

There is no natural topology on Ĝ, but there is a map from Ĝ to the two-sided ideals of C^{∗}G, π ↦ ker π ⊂ C^{∗}G, giving a surjective map

Ĝ ⟶ Prim C^{∗}G .

For semisimple Lie groups G with finite center, this map is bijective (in general, this is not the case). Jacobson topology: if T ⊂ Prim C^{∗}G, set

T̂ = { J ∈ Prim C^{∗}G | J ⊃ ⋂_{I∈T} I } ,

this gives a T_{1}-topology (points are closed), which is in general not Hausdorff.

For any f ∈ C^{∗}G, any π ∈ Ĝ, π(f) ∈ B(H_{π}) is a compact operator, moreover, C_{π}^{∗}G = k(H_{π}) (this is the analogon of finite-dimensionality in Example 1.5).

For G = SL_{2}(ℝ), the space Ĝ_{r} consists of a halfline with two points at the end, another halfline, and a family of pairs of points (discrete series). The space Ĝ \ Ĝ_{r} consists of a complementary series which extends the second halfline, and at the other end has three limit points: the first two discrete series representations, and the trivial representation (which does not belong to Ĝ_{r}). Then,

C_{r}^{∗}G = ⨁_{π discrete series} k(H_{π}) ⊕ C_{0}([0, ∞), k(H)) ⊕ \{ f ∈ C_{0}([0, ∞), k(H)) | f(0) = ( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} ) \} ,

where at zero, H is the direct sum of two irreducible representations, so f(0) has to respect this decomposition. The full C^{∗}-algebra is constructed similarly, where in the second summand, again f(0) decomposes into operators on a one-dimensional component and two Hilbert spaces.

The programme of Example 1.4–1.6 fails for general discrete groups.

1.7. Example. Let G = F_{n} be the free group of n ≥ 2 generators. The set Ĝ is given by isomorphism classes of n-tupels of unitaries in U(H). The pull-back topology of Prim C^{∗}G or Prim C_{r}^{∗}G is not useful, for example, the pull-back topology from Prim C_{r}^{∗}G is trivial, because C_{r}^{∗}G has no (closed) two-sided ideals.

The goal of the Baum-Connes conjecture is to describe the space Ĝ_{r} in general. By the philosophy of noncommutative geometry, one tries to replace the (ill-behaved) space Ĝ_{r} by the C^{∗}- algebra C_{r}^{∗}(G).

2. K-theory of C^{∗}-algebras

Let A be a C^{∗}-algebra, then K_{0}(A) and K_{1}(A) are abelian groups. To G, we associate K_{i}(C_{r}^{∗}G) for i = 0, 1. It is quite easy to compute K_{i}(C_{r}^{∗}G) for the examples 1.4–1.6. There is a map from a topological group to K_{i}(C_{r}^{∗}G), which by the Baum-Connes conjecture should be an isomorphism.

An operator T: H_{0} → H_{1} is Fredholm iff dim ker T and codim im T are finite. Define the index of a Fredholm operator as ind T = dim ker T − codim im T. Thus, the index lives in K_{0}(ℂ) ≅ ℤ. Equivalently, T is Fredholm iff there exists an operator S: H_{1} → H_{0} such that ST − id and T S − id are compact. This interpretation gives (by a six-term exact sequence) an interpretation of the index in K_{0}(k(H)).

A Hilbert module over a C^{∗}-algebra A is a right module E over A with a “scalar product”

⟨ · , · ⟩: H^{2} ⟶ A ,

such that

⟨ξa, ηb⟩ = a^{∗}⟨ξ, η⟩b , ⟨η, xi⟩ = ⟨ξ, η⟩^{∗} ,
⟨ξ, ξ⟩ is positive in A ,  and  ⟨ξ, ξ⟩ = 0 ⟺ ξ = 0 .

Then ∥⟨ξ, ξ⟩∥^{\frac{1}{2}} defines a Banach norm on E, and we assume that E is complete.

A map T: E_{0} → E_{1} is an algebraic A-homomorphism, then T is bounded if it has an adjoint. An operator is of finite rank, if it factors over some A^{n}. A compact operator is an operator norm limit of a sequences of finite rank operators.

2.1. Example. E = A and E = A^{n} with ⟨ξ, η⟩ = ∑ ξ_{i}^{∗}η_{i} generalise Euclidean spaces. Hilbert spaces correspond to

E = H_{A} = { ξ = (ξ_{i})_{i} | ∑_{i} ξ_{i}^{∗}ξ_{i} converges in A } .

We assume that scalars act from the right, while matrices act from the left. For any idempotent matrix e ∈ M_{n}(A) (i.e., e = e^{∗} = e^{2}), ker e and im e ⊂ A^{n} are finitely generated projective modules. These modules generate K_{0}(A). One may understand the dimension of E to be its class in K_{0}(A).

The index of a (bounded) Fredholm operator (invertible modulo compact operators) is given as [ker T] − [ker T^{∗}] ∈ K_{0}(A) for a suitable perturbation of T by a compact operator (one has to show that for some perturbation, this difference is defined in K_{0}(A), and that the index does not depend on the perturbation).

2.a. Differential operators. Let M be a smooth compact manifold, let D be a differential operator on M,

D: C^{∞}(M, E_{0}) ⟶ C^{∞}(M, E_{1}) ,

E_{0}, E_{1} are bundles, and locally

Df(x) = ∑_{|α|≤n} a_{α}(x) , \frac{∂^{α}f}{∂x^{α}} .

The principal symbol is defined as

σ_{D}(x, ξ) = ∑_{|α|=n} a_{α}(x)(iξ)^{α} ∈ End(E_{0}, E_{1}) .

The operator D is called elliptic iff σ_{D}(x, ξ) is invertible for all ξ ∈ T^{∗}M \ M × {0}. Example: Laplace operator, σ_{Δ}(x, ξ) = |ξ|^{2}.

The Dirac operator on ℝ^{n} is an operator on a vector bundles whose square is the Laplacian,

D = ∑ c_{i} \frac{∂}{∂x_{i}} ,

with c_{i}c_{j} + c_{j}c_{i} = −2δ_{ij}. Then

σ_{D}(x, ξ) = i ∑ c_{i}ξ_{i} ,  and  σ_{D}(x, ξ)^{2} = |ξ|^{2} .

Dirac operators are operators with a symbol as above.

2.2. Theorem. If M is compact and D is elliptic, then D is Fredholm, and ind D ∈ Z.

Note: “Fredholm” only in a suitable sense, because D is not bounded.

Let V be a compact manifold, D an elliptic operator. Let Ṽ denote the universal cover of V, so V = Ṽ/G, where G = π_{1}(G). The lift D̃ of D to Ṽ is no longer Fredholm, but it is G-equivariant. However, regard the space of sections as an C^{∗}G-module (or C_{r}^{∗}G-module), then D̃ is Fredholm as an operator on C^{∗}-modules, and it has an index ind D̃ ∈ K_{0}(C^{∗}G) → K_{0}(C_{r}^{∗}G).

2.3. Conjecture (Baum-Connes). The index map

K_{∗}^{Γ}(\underline{E}Γ) ⟶^{μ} K_{∗}(C_{r}^{∗}Γ)

is an isomorphism.

Injectivity allows one to deduce topological consequences of analytic results, e.g., Novikov conjecture, Gromow-Lawson conjecture. Surjectivity allows one to construct representation theoretic data from geometric data.

Fermat’s theorem: x^{n} + y^{n} + z^{n} has no integer solutions for n ≥ 3. Currently, one attacks the Baum-Connes conjecture like solving Fermat’s problem by choosing only particular inetegers n each time. According to Gromov, no non-trivial statement about all groups can be true, in particular, the Baum-Connes conjecture should be false. Maybe, Gromov’s “random groups”, or variants thereof, will eventually give counterexamples (according to Baum, these groups will be counterexamples against any possible non-trivial conjecture).

2.4. Question. Does the Baum-Connes conjecture hold for SL(3, ℤ)?

Injectivity in Conjecture 2.3 is already proved. In fact, injectivity is true for most “reasonable groups”.

2.b. Injectivity results. Kasparov (1980). Assume that Γ is a discrete subgroup of an almost connected (locally compact, second countable) group, then the Baum-Connes map is injective. In particular, the Novikov conjecture holds for these groups. The proof uses the space G/K = \underline{E}Γ, where K is a maximal compact subgroup. Note that G/K has a nice contraction to a point. One uses a “dual Dirac construction”

K_{∗}^{Γ}(\underline{E}Γ)@ < ν << K_{∗}(C_{r}^{∗}Γ) .

2.5. Theorem (Kasparov). The map ν ∘ μ is the identity on K_{∗}^{Γ}(\underline{E}Γ).

A priori, the map μ ∘ ν is an idempotent on K_{∗}(C_{r}^{∗}Γ). The Baum-Connes conjecture is then equivalent in this case to γ = 1.

2.c. Surjectivity?. Let Γ ⊂ G be as above, so injectivity in Conjecture 2.3 holds.

2.6. Fact. The composition

(2.7)  K_{∗}^{Γ}(\underline{E}Γ) ⟶^{μ_{max}} K_{∗}(C^{∗}Γ) ⟶^{λ_{∗}} K_{∗}(C_{r}^{∗}Γ)

gives the Baum-Connes map μ.

Injectivity of μ_{max} follows from injectivity of μ.

2.8. Example. For Γ = SL(2, ℤ), one knows that the maps μ, μ_{max} are isomorphic, and one can prove (without using Baum-Connes) that λ_{∗} is an isomorphism. This situation is very special.

For Γ = SL(3, ℤ), one only knows injectivity of μ and μ_{max}. It is known that λ_{∗} is not injective, so μ_{max} cannot be surjective. One of the main difficulties consists in understanding the precise difference between C^{∗}(Γ) and ℂ_{r}^{∗}(Γ).

2.9. Definition. Γ has Kazhdan’s property T if it satisfies the following equivalent conditions: (1) all actions of Γ by affine isometries on a Hilbert space have a fixed point;

(2) for all unitary representations π of Γ, if Γ almost has invariant vectors, then Γ has an invariant vector.

Example: SL(3, ℤ).

2.10. Definition. Γ has the Haagerup property (Gromov’s T -amenability), if it satisfies the following equivalent conditions:

(1) The exists an action of Γ on a Hilbert space by affine isometries which is metrically proper;

(2) There exists a unitary representation of Γ which has coefficients vanishing at ∞ and almost has invariant vectors.

Example: SL(2, ℤ), other SL(2, ·), SO(n, 1), SU(n, 1).

An action is metrically proper if gB_{1} ∩ B_{2} = ∅ except for finitely many g ∈ Γ, whenever B_{1}, B_{2} are bounded sets.

2.11. Theorem (Higson-Kasparov, 1996). Let G be a locally compact group with the Haagerup property, then the maps μ, μ_{max}, λ_{∗} in (2.7) are isomorphisms.

Proof. The proof uses an infinite-dimensional generalisation of the dual Dirac construction.

2.12. Question (Connes, 1982–1998). Find a single infinite discrete group Γ of property T such that the Baum-Connes conjecture is true.

2.13. Remark. Property T implies that the trivial representation is an isolated point in Γ̂ in the topology induced from the ideals of C^{∗}(Γ), so C^{∗}Γ = ker 1_{r} ⊕ ℂ, so

K_{0}(C^{∗}Γ) = K_{0}(ker 1_{r}) ⊕ K_{0}(ℂ) .

and then K_{0}(ℂ) = ℤ ⊂ ker λ_{∗}.

V. Lafforgue answered Connes’ question by taking Γ to be a cocompact lattice in G, where G is either a real rank 1 simple group of property T (e.g. Sp(n, 1), F_{2} acting on the octonion hyperbolic plane), or SL(3, ℂ), SL(3, ℝ) (ℚ_{p} also works, but not by Lafforgue).

There are two steps in Lafforgues proof. First, formulate a Banach analogue of the Baum-Connes conjecture: Construct a factorisation of μ_{m}ax,

K_{∗}^{G}(\underline{E}G) ⟶^{μ_{romanBanach}} K_{∗}(L^{1}G) ⟶ K_{∗}(C^{∗}G) ⟶ K_{∗}(C_{r}^{∗}G) .

Kasparov’s dual Dirac construction gives an inverse ν_{Banach} of μ_{Banach} for “reasonable” groups. This uses an action on a non-positively curved manifold.

Second step. Instead of L^{1}G, take other Banach completions of the group algebras that depends only on the growth of the coefficients (this is not true for C_{r}^{∗}(G)). Suppose that one of these completions, say A(G), is contained in C_{r}^{∗}(G). Then K_{∗}(A(G)) = K_{∗}(C_{r}^{∗}(G)). This is possible for semi-simple Lie groups or semi-simple p-adic groups, and some discrete groups (those which satisfy Joliseaint’s rapid decay condition).

3. Baum-Connes conjecture for Lie groups

Consider an elliptic G-invariant operator on a G-proper, G-compact G-manifold X. These operators have an index in K_{0}(C^{∗}G) ⟶^{λ_{∗}} K_{0}(C_{r}^{∗}G).

If G acts properly on X and there is a G-equivariant complex Hermitian vector bundle E → X. Denote by C_{c}^{∞}(X, E) the compactly supported smooth sections of E. Then G acts on C_{c}^{∞}(X, E). Two natural completions of C_{c}^{∞}(X, E) are L^{2}(X, E) with respect to a G-invariant measure on X (exists because the action is proper). Note that the operator F = \frac{D}{\sqrt{1+D^{2}}} will be bounded, but in general not Fredholm.

The second completion will be a C^{∗}G- (or C_{r}^{∗}G-)module.

3.1. Definition. For ξ, η ∈ C_{c}^{∞}(X, E), define ⟨ξ, η, ⟩ ∈ C_{c}^{∞}(G) by

⟨ξ, η⟩(g) = (ξ, gη) .

Properness of the action guarantees that ⟨ξ, η⟩ is indeed compactly supported on G. Then this scalar product extends with values in C^{∗}G or C_{r}^{∗}G. Note that ⟨ξ, η⟩ satisfies the axioms for (pre- )Hilbert modules (positivity is most difficult). In particular, there exist completions E and E_{r} as C^{∗}G- and C_{r}^{∗}G-modules.

Let D: C_{c}^{∞}(X, E_{0}) → C_{c}^{∞}(X, E_{1}) be an elliptic differential operator, and set F = \frac{D}{\sqrt{1+D^{2}}} . Then F M_{f} − M_{f}F is compact for f ∈ C_{0}(X), and (1 − F^{∗}F) and (1 − F F^{∗}) are locally compact with respect to the L^{2} norms.

On the other hands, F extends to a bounded, adjointable operator

F̃: E_{0} ⟶ E_{1} ,

with F̃^{∗} = F̃^{∗}. Because X is G-compact, the operators (1 − F̃^{∗}F̃) and (1 − F̃F̃^{∗}) are G-compact operators. Moreover, F̃ commutes with G.

Let G = SL(2, ℝ) or any semi-simple Lie group with finite center, and let K = SO(2) be a maximal compact subgroup, then G/K is a Riemannian symmetric space of non-conpact type, the G-action is proper and G-compact, and G/K is a model for \underline{E}G. One can compute

K_{∗}^{G}(G/K) = K_{∗}(C^{∗}(𝔤/𝔥 ⋊ K)) = K_{∗}(C_{0}((𝔤/𝔥)^{∗} ⋊ K)) ≃ K_{∗+dim G/K}(C^{∗}K)
= \{ \begin{matrix}R(K) & if ∗ + dim G/K is even \\ 0 & otherwise, \end{matrix}

where 𝔤/𝔥 = T_{o}(G/K), with o = [K] ∈ G/K. Note that “≃” is only true if G/K is G-spin.

Now assume that G/K is even-dimensional. Here is a special case of the Baum-Connes conjecture.

3.2. Conjecture (Connes-Kasparov). The Dirac induction map

R(K) ⟶ K_{0}(C_{r}^{∗}G)

is an isomorphism, and K_{1}(C_{r}^{∗}G) = 0.

Let χ ∈ K̂ be an irreducible representation of K on a finite-dimensional complex vector space V_{χ}. Consider the G-equivariant Hermitian bundle

E_{χ} = G ×_{K} V_{χ} ⟶ G/K

with fibre V_{χ}. Then

C_{c}^{∞}(G/K, E_{χ}) = C_{c}^{∞}(G, V_{χ})^{K} = (V_{χ} ⊗ C_{c}^{∞}(G))^{K} ,

and

E = (V_{χ} ⊗ C^{∗}G)^{K} = e_{K}(V_{χ} ⊗ C^{∗}G) ,

where

e_{K} = ∫_{K} χ(k) ⊗ u_{k} dk .

Because G/K is G-spin, there exists a G-equivariant spinor bundle S = S^{+} ⊕ S^{−} → G/K. Now define the Dirac induction map for χ ∈ K̂ by

[χ] ⟼ ind D_{χ} ,

where

D_{χ}: C_{c}^{∞}(S^{+} ⊗ E_{χ} ⟶ G/K) ⟶ C_{c}^{∞}(S^{−} ⊗ E_{χ} ⟶ G/K)

is the Dirac operator.

For K = SO(2), the irreducible representations are characters χ_{n}: ϑ → e^{inϑ}. We know that Ĝ_{r} consists of a halfline with a double-point at the end (representations ρ_{λ}, λ ≥ 0, ρ_{0} = ρ_{0}^{+} ⊕ ρ_{0}^{−}), a halfline (σ_{λ}, λ ≥ 0), and a countable set of points π_{n}. We have

C_{r}^{∗}G = \{ ϕ ∈ C_{0}([0, ∞), k(H)) | φ(0) = ( \begin{matrix}∗ & 0 \\ 0 & ∗ \end{matrix} ) \} ⊕ C_{0}([0, ∞), k(H)) ⊕ ⨁_{n≠0} k(H) .

The generators of K_{0}(C_{r}^{∗}G) correspond to the double point ρ(0) and the discrete series representations π_{n}, and K_{1}(C_{r}^{∗}G) = 0.

Exercise: Let A be the algebra of functions on [0, ∞) with values in M_{2}(ℂ) which are diagonal at 0, then K_{0}(A) = ℤ and K_{1}(A) = 0. Note that there are two evaluations ev_{1}, ev_{2}: A → ℂ at 0, which both induce opposite isomorphisms in K_{0}.

3.3. Theorem. Let G = SL(2, ℝ), K = SO(2). The map

μ: R(K) ⟶ K_{0}(C_{r}^{∗}G) , χ ⟼ ind D_{χ}

is given by

μ(χ) = ⟼ \{ \begin{matrix}±[ρ_{0}^{±}] & for χ = χ_{0} = 1, and \\ ±[π_{n}] & for χ = χ_{n}, n ≠ 0. \end{matrix}

There is a natural choice of signs, but it is difficult.

3.4. Corollary. The masp μ is an isomorphism.

Proof of Theorem 3.3. It is enough to compute π_{∗}(ind D_{χ}) ∈ ℤ for any χ = χ_{n}, and π = π_{n} or π = ρ_{0}^{+}. But π_{∗}(ind D_{χ}) = ind(π_{∗}D_{χ}), where π_{∗}D_{χ} is a Fredholm operator on E_{χ}^{±} ⊗_{C_{r}^{∗}G} H_{π}. Recall that E_{χ}^{±} = (V_{χ} ⊗ S^{±} ⊗ C_{r}^{∗}G)^{K}, and thus

E_{χ}^{±} ⊗_{C_{r}^{∗}G} H_{π} = (V_{χ} ⊗ S^{±} ⊗ H_{π})^{K} = Hom_{K}(V_{χ}^{∗} ⊗ S^{±∗}, H_{π}) ,

which is a finite dimensional vector space. In particular

ind(π_{∗}Dχ) = dim(V_{χ} ⊗ (S^{+} − S^{−}) ⊗ H_{π})^{K} .

For ρ_{0}^{±} one obtains ±1, but the sum will give zero. □

This approach has been carried out for all semi-ssimple Lie groups by A. Wassermann in 1983. This approach is not satisfactory, because one has to know already the complete representation theory, but finally, one gets much less information.

So, try to find a proof of this without using representation theory. This has been done by V. Lafforgue in 1998. Lafforgue uses the dual Dirac induction mamp ν. The question is, does this approach tell you anything about representation theory?

Consider for n ≠ 0 the map

R(K) ⟶ K_{0}(C_{r}^{∗}G) , [χ_{n}] ⟶ [π_{n}] .

So you obtain a bijection on bases, not only an isomorphism of ℤ-modules. Note that this is not implied by the Baum-Connes conjecture, but it follows from Kasparov’s dual Dirac construction. In other words, μ is given by a unitary ℤ-matrix.

3.5. Proposition (V. Lafforgue). For all χ ∈ K̂ and for all π ∈ Ĝ_{d} (discrete series representations), we have

π_{∗}(μ([χ])) = χ_{∗}(ν([π])) ∈ ℤ .

Take π ∈ Ĝ_{d}. One has to show that [π] = ±μ([χ]) for some χ ∈ K̂, and ν[π] = ±[χ] for some χ. Now,

ν([π]) = ∑ n_{π,χ}[χ] ,  with  n_{χ,π} = χ_{∗}(ν[π]) = π_{∗}(μ[χ]) .

So

[π] = μ(ν[pi]) = ∑ n_{π,χ}μ[χ] ,

and

π_{∗}[pi] = ∑ n_{π,χ}^{2} = 1 ,

so only one of the integers n_{π,χ} can be one, the others vanish. This gives a new proof of the Atiyah-Schmid construction. One only needs that the matrix coefficients of discrete series representations give K-theory elements.

To see connection with Atiyah-Schmid, one proceeds as above, but replaces π by L^{2}(G). This gives

λ_{∗}E_{χ}^{±} = (V_{χ} ⊗ S^{±} ⊗ L^{2}(G))^{K} = (ker D_{χ}^{+}) ⊗_{C_{r}^{∗}G} L^{2}(G) ,

which is precisely the L_{2}-kernel of the operator D_{χ}^{+} (a Hilbert module over the scalars, i.e., a Hilbert space). Note that λ_{∗} is pushing C_{r}^{∗}G-modules, it is not the map C^{∗}G → C_{r}^{∗}G.

What about the map λ_{∗}C^{∗}G → C_{r}^{∗}G? Because G = SL(2, ℝ) has the Haagerup property, we know that λ_{∗}: K_{0}(C^{∗}G) → K_{0}(C_{r}^{∗}G) is an isomorphism. For groups with Kazhdan’s property T, the corresponding map λ_{∗} in K-theory is only surjective, and the kernel contains a copy of Z generated by [1_{G}]. In general, what is the K-theory of C^{∗}G? For Lie groups, one knows at least

K_{∗}(C^{∗}G) = im μ_{max} ⊕ ker λ_{∗} .

3.6. Example. By F. Pierrot, one knows that ker λ_{∗} is generated by the trivial representation for SL(3, ℂ), but for SL(n, ℂ) it is more complicated for n ≥ 4.

For Sp(n, 1), there are more “extra” representationss, i.e. isolated representations (but not discrete series) discovered by Bardoni-Silva, and complementary series representations with different “ends”. Here,

ker λ_{∗} = ⨁_{n} ℤ .