By Alessandro Figá-Talamanca, Claudio Nebbia

ISBN-10: 0521424445

ISBN-13: 9780521424448

The unitary irreducible representations are labeled in 3 sequence: a continuing sequence of round, distinct representations, and a countable sequence of cupsidal representations as outlined through G.I. Ol'shiankii.

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**Example text**

C n (12). It is also obvious that I z O_%O. Let d(x,o)=n+1, with n>-0. Observe that PZ(x,w)= r)1(w)+cixi2(x) where 711EXn and ci*O. Cn . 2 Therefore x (w)=Ef(y)PZ(y,w), with may be written as x f supported on {y: d(y,o)sn}v{x}. Since Izgx(w)=E'(y)P1-Z(y,w), it follows that Izgx r)3 (w) then L(y)9xdv=0. Therefore where 713EKn. 1-Z('xdv) identically zero on 3f n={y: d(o,y)5n}. So, for any yE3f Pz('Q3dv)(y)= = fQ(y)713(w)dv=0 for and therefore r)3 0. compute xdv)(y) P1-z(gxdv)-c3Rz(gxdv)(y)=0. This implies that to are n, c3.

Let oeX, and let z be a complex number µ=µ(z)=(qz+q 1-z such that )/(q+l), and z*kin/ln q, for keZ. Then there exists mEX' such that f(x) = P m(x) = IS2PZ(o,x,w) dm(w). z PROOF. For simpler notation, we shall write P(x,w) in the place of P(o,x,w). Let S be a finite subtree of 1, containing o. We say that a vertex of 3 is an interior point if each of its q+l A vertex of S which is not an nearest neighbors lies in S. interior point is said to belong to the boundary 8S of St, or to be a boundary point.

Therefore the first application of a power of t' and all successive applications of powers of t and t' map z into geodesics which do not meet T. We have proved that w(T)n'=r, if w contains a power of T'. On the other hand if w is a nonzero power of t, then w(x) is not the identity on T. We have thus proved that any reduced word in t and t' is not the identity as an element of Aut(X), and therefore that t and t' generate a free group. In addition the intersection of this group with any stabilizer of a point of ' is the identity, and therefore this free group is discrete.

### Harmonic Analysis and Representation Theory for Groups Acting on Homogenous Trees by Alessandro Figá-Talamanca, Claudio Nebbia

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