The Baum-Connes Conjecture andParametrization of Representations

Nigel Higson

Department of MathematicsPennsylvania State University

Henrifest, June 2009

Nigel Higson Parametrization of Representations

General problem: Obtain a description of the dual (the set ofirreducible representations, up to equivalence) of a semisimplegroup, e.g. SL(n,R).

(Typically infinite-dimensional representations, either temperedor admissible, up to infinitesimal equivalence.)

Various solutions: Gelfand, Harish-Chandra, Langlands,Knapp, Zuckermann, Vogan, Beilinson, Bernstein, ...

The Baum-Connes conjecture suggests an approach that isvery naiveapparently unnoticed beforeapparently viable

At the moment, the approach strays a bit far from theBaum-Connes point of view. Can it be framed as a refinedversion of Baum-Connes?

Nigel Higson Parametrization of Representations

Baum-Connes Assembly Map

Baum-Connes map for a locally compact group:

µ : K top∗ (G) −→ K∗

(C∗λ(G)

)Inspired by:

foliations (Connes)discrete series (Parthasarathy, Atiyah-Schmid,Connes-Moscovici)geometric K-homology (Atiyah, Baum-Douglas)

We should think of the RHS as the Atiyah-Hirzebruch K-theoryof the (reduced) unitary dual of G. In our cases this is (almost)exactly what it is.

Nigel Higson Parametrization of Representations

The Equivariant Index

Let D be an equivariant Dirac-type operator on a G-compactproper G-manifold M, acting on sections of some SSS.

Its index in K (C∗λ(G)) is obtained as follows:

Manufacture from C∞c (M,SSS) (by a completion operation) a(Hilbert) C∗λ(G)-module.Manufacture from D (by completing its graph) anunbounded self-adjoint operator on this Hilbert modulewith compact resolvent.Take the index of this Fredholm operator.

Nigel Higson Parametrization of Representations

Baum-Connes for Lie Groups

Here the conjecture is proved . . .

G = Connected semisimple groupK = Maximal compact subgroup

Assume K\G has a G-equivariant Spinc-structure, given by aK -equivariant Clifford algebra representationc : Cliff(p) −→ End(S) (where g = k⊕ p). Then

K top∗ (G) = R(G)

and to a representation τ : K → U(V ) corresponds the Diracoperator on K\G with coefficients in V .

Nigel Higson Parametrization of Representations

Index of the Dirac Operator on K\GG = Semisimple groupK = Maximal compact subgroup

Let D = Dirac with coefficients in an (irreducible) K -module V .

C∞c (K\G,SSS ⊗VVV ) is(C∞c (G)⊗ S ⊗ V

)K , and the HilbertC∗λ(G)-module completion is

H =(C∗λ(G)⊗ S ⊗ V

)K

This we can localize at any [π] ∈ Gλ:

H⊗C∗λ(G) Hπ =

(Hπ ⊗ S ⊗ V

)K

We obtain a finite-dimensional Hilbert space.

Nigel Higson Parametrization of Representations

Index of the Dirac Operator on K\G

The Dirac operator is

D =∑

o.n.b. for p

Xi ⊗ c(Xi)⊗ I

It too localizes to each (Hπ ⊗ S ⊗ V)K (where [π] ∈ Gλ).

We obtain

Dπ =

∑π(Xi)⊗c(Xi)⊗I : (Hπ⊗S⊗V

)K −→ (Hπ⊗S⊗V)K

Compare Parthasarathy-Atiyah-Schmid, except at issuehere is the topology of the dual, not measure theory.

Nigel Higson Parametrization of Representations

Reduced C*-Algebra of a Semisimple Group

Consider for example G = SL(n,C), after Gelfand & Naimark(or any complex semisimple group).

Let B = MAN = Borel subgroup of upper-triangular matrices.

For (σ, ϕ) ∈ M × A, π(σ,ϕ) := IndGB (σ ⊗ ϕ⊗ 1) is irreducible.

C∗λ(G) is represented as compact operators in each H(σ,ϕ).

Corresponding to permutations w ∈W are intertwiners

Iw : H(σ,ϕ) → H(wσ,wϕ).

There results an isomorphism

C∗λ(G)∼=−−−−→ C0

(M × A,K(H)

)W

and a Morita equivalence with C0(M × A)W .

Nigel Higson Parametrization of Representations

Computation of the Dirac Index

G = Complex semisimple, so that Gλ = (M × A)/WD(σ,ϕ) : (H(σ,ϕ) ⊗ S ⊗ V

)K −→ (H(σ,ϕ) ⊗ S ⊗ V)K

This is a cycle for K ∗((M × A)/W

). One can calculate:

D2 = ΩG ⊗ I ⊗ I − I ⊗ ΩK ⊗ I + I ⊗ I ⊗ ΩK ,

where Ω = Casimir. After some more calculation,

D2 = ‖ϕ2‖ − ‖σ‖2 + ‖τ + ρ‖2.Still further calculation shows that:

Theorem (Penington and Plymen)Index(D) is supported on the component of (M × A)/W labelledby ρ plus the highest weight τ of V and is the push-forward ofthe base representation into this euclidean space.

Nigel Higson Parametrization of Representations

Some Questions

It would be interesting to calculate D more carefully on otherthan the ρ+ τ component. (Connections to the Dirac family ofFreed-Hopkins-Teleman and the orbit method?)

What about the push-forwards of other representations, asidefrom those with real infinitesimal character? Is there a cyclegroup more refined than R(K ) that more fully accounts for thedual as a geometric space?

Nigel Higson Parametrization of Representations

Contraction of G to a subgroup K

Examined by Wigner et al in the 1950’s, 1960’s . . .

G = Lie groupK = Lie subgroup

The contraction of G along K is

G0 = K n(Lie(G)/Lie(K )

)Notice that: G0 = normal bundle for the inclusion of K into G.

If G acts isometrically on X , and if Y ⊆ X is K -invariant, thenG0 acts on the normal bundle of Y in X (affine-isometrically onthe fibers).

Nigel Higson Parametrization of Representations

Cartan Motion Group

G = Semisimple groupK = Maximal compact subgroup

If g = k⊕ p, then

G0 = K n p

and one can form Gt with Lie algebra gt :

[X ,Y ]t =

[X ,Y ] if X or Y ∈ k

t [X ,Y ] if X and Y ∈ p.

Nigel Higson Parametrization of Representations

Baum-Connes Conjecture, Again

The family Gt determines a specific map

K∗(C∗λ(G0)) −→ K∗(C∗λ(G))

and the Baum-Connes conjecture is equivalent to the assertionthat this map is an isomorphism.

(Why? Roughly speaking because the Baum-Connes left-handsides for G and G0 are the same.)

Equivalent (slightly loose) formulation of the assertion: thecontinuous field C∗λ(Gt ) is K -theoretically indistinguishablefrom a constant field.

What does this version of Baum-Connes look like, from arepresentation-theoretic point of view?

Nigel Higson Parametrization of Representations

Mackey’s Analogy

Mackey (early 1970’s): “. . . there ought to exist a ‘natural’one-to-one correspondence between almost all of the unitaryrepresentations of G0 and almost all the unitary representationsof G . . . ”

By almost all he probably meant with respect to Plancherelmeasure.

Given Mackey’s suggestion (and supporting computations),there is an interesting tension between measure theory (in theform of his analogy) and cohomology (in the form ofBaum-Connes) . . .

Nigel Higson Parametrization of Representations

The Reduced Duals of G0 and G

G0 = Compact n Vector Group = K n V

By Fourier and Peter-Weyl,

C∗(G0) ∼= K n C∗(V ) ∼= K n C0(V ) ∼= C0

(V ,K

(L2(K )

))K.

This gives

G0 =⊔φ∈bV

Kφ/

K ,

while on the other hand

Gλ =(M × A

)/W

for complex semisimple G.

Nigel Higson Parametrization of Representations

A Mackey Bijection

⊔ϕ∈V Kϕ

/K ∼= M × A

/W

V/K = A/W

For all ϕ ∈ A, Kϕ is connected with maximal torus M.

⇒ Kϕ = M/Wϕ by Cartan-Weyl.

⇒ G0 =⊔ϕ∈V Kϕ

/K =

⊔ϕ∈bA Kϕ

/W

=⊔ϕ∈bA M/Wϕ

/W

= (M × A)/W= Gλ

Nigel Higson Parametrization of Representations

More Calculations

Some work of students . . .

Chris George: Similar bijection for SL(n,R) (and similarstructure theorem for the C∗-algebra—next slide—in progress).

John Skukalek: Similar bijection and structure theorem foralmost-connected groups with connected component of identitycomplex semisimple.

Some help from the experts . . .

With the ample assistance of David Vogan, the generalsemisimple (or reductive) case seems well within reach.

Nigel Higson Parametrization of Representations

Minimal K-Types

What is the connection to Baum-Connes?

Label each irrep of G by its minimal K -type(s) (for G complexsemisimple, the set K is partially ordered by highest weightsand there is a unique minimal K -type in each irrep).

Theorem: The bijection preserves minimal K -types (whichalways have multiplicity one).

Theorem: The labels determine an increasing filtration ofC∗λ(Gt ) by continuous fields of ideals. The correspondingfields of subquotients are Morita-equivalent to constant fields(of commutative C∗-algebras).

This is for complex G, so far.

Nigel Higson Parametrization of Representations

Infinitesimal Representations

The Hecke algebra of G is the convolution algebra H(G,K ) ofdistributions on G supported on K .

LemmaNondegenerate H(G,K )-modules ≡ (g,K )-modules

The Hecke algebra is filtered by distribution order.

LemmaThe associated graded algebra for H(G,K ) is the Heckealgebra H(G0,K ) for the contracted group.

There is therefore a natural deformation from H(G0,K ) intoH(G,K ) . . .

Nigel Higson Parametrization of Representations

Local Hecke Algebras

Fix τ ∈ K and form the local Hecke algebra

H(G, τ) = pτH(G)K pτ

Lemma

Irreducible H(G, τ)-modules ∼= Irreducible (g,K )-moduleswith nonzero τ -component

As before, H(G0, τ) is the associated graded algebra of H(G, τ)

Both are finite type algebras (Harish-Chandra).

Nigel Higson Parametrization of Representations

Spherical Representations

LemmaFor G complex semisimple

H(G,1) ∼= Z (U(g))

(consider g as a complex Lie algebra) and more generally

H(G, τ) ∼=(U(g)⊗ End(τ∗)

)K

So H(G0,1) ∼= S(g)K and H(G0, τ) ∼=(S(g)⊗ End(τ∗)

)K .

Harish-Chandra defined

Z (U(g)) −→ U(a)

and proved it to be an isomorphism onto U(a)W as follows:The image is in U(a)W (intertwiners)The associated graded map is bijective onto S(a)W

(Chevalley).Nigel Higson Parametrization of Representations

Theorem (Harish-Chandra/Chevalley Isomorphism)

H(G,1) ∼= S(a)W ∼= H(G0,1)

This identifies the spherical representations of G and G0.

There are now several proofs (including an amazing Diracoperator-based proof of Alekseev-Meinrenken . . . )

The theorem identifies the spherical duals of G and G0. Does itextend to other parts of the dual?

Nigel Higson Parametrization of Representations

Classification of Irreducible Representations

Complex G: Classification begun by Berezin in 1950’s (atGelfand’s suggestion), completed 20 years later (Zhelobenko,Duflo, . . . ).

Let’s continue to focus on these.

Harish-Chandra’s Subquotient Theorem: All irreps aresubquotients of the principal series.

Vogan’s Refinement: All irreps with minimal K -type σ occur incertain specific principal series.

By Vogan’s theorem, the lowest τ -K -type representations factorthrough the image of a generalized Harish-Chandrahomomorphism

H(G, τ) −→ U(a).

Nigel Higson Parametrization of Representations

Theorem (A Higher Harish-Chandra Isomorphism)The images of

H(G, τ) −→ U(a)

and the associated graded map

H(G0, τ) −→ S(a)

are equal.

Proof.One can use Harish-Chandra’s method: apply a Chevalley-typerestriction theorem and an intertwiner argument (Weylcharacter formula).

Nigel Higson Parametrization of Representations

Real Groups

(Work in progress with David Vogan.) The correctHarish-Chandra maps have noncommutative target, in general(related to non-uniqueness of minimal K -types). However . . .

Theorem (or “Theorem”)For each τ ∈ K there exists

H(G, τ) −→ U(aτ )

(defined by Vogan) that classifies τ -minimal K -type reps.

The associated graded map classifies τ -minimal K -typerepresentations of G0, and the images of these two maps areequal.

The correspondence between irreducible representations of Gand G0 so-determined is a well-defined bijection between theduals of G and G0.

Nigel Higson Parametrization of Representations

Nigel Higson Parametrization of Representations