16 ALESSANDRO FIGA-TALAMANCA

Proof. Let

g

E

Boo be such that

g~

cJ,.

~.and

let v(h)

=

(1r(h)7r(g)~,~).

Observe that

fork

E K(~),

v(kh)

=

(1r(h)7r(g)~, 1r{k- 1 )~)

=

v(h), and fork

E gK(~)g- 1

=

K(g~),

v(hk)

=

v(h). Let J

=

~

n

g~.

Then, by the lemma

Therefore v(h) is right and left K(J)-invariant.

It

follows that v(h) is constant

on K(J). But J is a proper subtree

of~.

and consequently

P7r(J)~

=

0. Therefore

v(k)

=

0 fork E K(J). We conclude that v{lc)

=

0 and u(g)

=

0.

REFERENCES

1. J. Faraut, Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques, Analyse

Harmonique Nancy, CIMPA, 1980, pp. 315-446.

2. A. Figil.-Talamanca and C. Nebbia, Harmonic Analysis and Representation Theory for Groups

Acting on Homogeneous Trees, Cambridge University Press, 1991.

3. S. Lang, S£(2, R), Addison-Wesley, Reading Mass., 1975.

4. G. Mauceri, Square integrable representations and the Fourier algebra of a unimodular group,

Pacific Journ. of Math. 73 (1977), 143-154.

5. C. Nebbia, Classification of all irreducible unitary representations of the stabilizer of the

horicycles of a tree, Israel J. of Math. 70 (1990), 343-351.

6. G.l. Ol'shianskii, Classification of irreducible representations of groups of automorphisms of

Bruhat-Tits trees, Functional Anal. Appl. 11 (1977), 26-34.

7. T. Steger, Local Fields and Buildings, this volume.

DIPARTIMENTO DI MATEMATICA, UNIVERSITA D1 ROMA "LA SAPIENZA", P. A.MORO

7,

00185

RoMA, ITALY