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
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