International Congress of Mathematicians

Banach KK-theory and the Baum-Connes Conjecture 807K,(H) is a finite rank projector, whose rank is(H,x).Lemma 2.1.1 We have (H, [dy]) = dim(F* ® S* ® H) K .2.2. Dual-Dirac operatorsFrom now on we assume that G is a connected semi-simple Lie group with finitecenter and we still assume that G/K has a G-invariant spin structure. Kasparovhasconstructed an element n £ Hom(Ki(C* ed (Gj),R(Kj) (coming from an elementof KKi(C* ed (G), C* ed (Kj), itself coming from an element of KKQ,ì(C, CO (G/K))).Kasparov has shown that n o p ied = Id#(jq [29, 31].Here is the detail of the construction. The G-invariant riemannian structureon G/K given by the chosen If-invariant euclidian metric on p has non-positive curvature.Let p be the distance to the origin and £ = d(\/l + p 2 )- Yet V be a finitedimensional complex representation of If, endowed with an invariant hermitian metric.Let Hy be the space of L 2 sections of the hermitian G-equivariant fibre bundleon G/K associated to the representation of K on S ® V and let c^y be the Cliffordmultiplication by £. In other words Hy is the subspace of If-invariant vectors inL 2 (G) ® S® V, where K acts by right translations L 2 (G), and c^y is the restrictionto this subspace of the tensor product of the Clifford multiplication by £ on L 2 (G)®Swith Id y. Left translation by G on G/K or on L 2 (G) gives rise to a (G*-)morphismn v : C* ed (G) -• Cc(Hy) and (H v ,ny,c^v) defines n v £ KK bAn (C* ed (G),C) (infact in KKi(C* ed (G),Cj). We denote by [ny] £ Hom(lfj(G* ed (G)),Z) the associatedmap, and n = EvbAAp 7 ] € Hom(Ki(C* ed (Gj),R(Kj), where the sum is overthe irreducible representations of K.Since the Connes-Kasparov conjecture is true, p re d '• R(K) —¥ Ki(C* ed (Gj)and n : Ki(C* ed (Gj) —¥ R(K) are inverse of each other and K i+ i(C* ed (Gj) = 0.Let 17 be a discrete series representation of G, i.e. an irreducible unitaryrepresentationwith a positive mass in the Plancherel measure. We recall that thisis equivalent to the fact that some (whence all) matrix coefficient c x (g) = (x, n(g)x),x £ H, ||x|| = 1, is square-integrable. Then ||c œ ||^2(G) is indépendant of x, and itsinverse is the formal degree dn of 17, which is also the mass of 17 in the Plancherelmeasure. We introduce a first ingredient.Ingredient 1. All discrete series representations of G are isolated in thetempered dual.In other words, all matrix coefficients belong to C* ed (G). In fact a standardasymptotic expansion argument shows that for any lf-finite vector x £ H, c x belongsto the Schwartz algebra ([17], II, corollary 1 page 77).Therefore there exists an idempotent p £ C* ed (G) such that the image inL 2 (G) of the image of p by the left regular representation is H* as a representationof G on the right. In fact we can take p = dnci for any x £ H, \\x\\ = 1, wherec^(g) = c x (g). The class of p in K 0 (C* ed (Gj) only depends on 17 and we denote itby [H]. It is easy to see that i : (DH^ —t K 0 (C* ed (Gj), (UH)H ^ E if ïI H[H], wherethe sums are over the discrete series representations of G, is an injection. Indeed,if 17 and H' are discrete series representations of G, (H', [H]) = 1 if 17 = 17' and 0otherwise.