By Alessandro Figá-Talamanca, Claudio Nebbia

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

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With kEZ, 39 We conclude that co c1=0, and the set B is linearly independent. We will prove now that the space of µ-eigenfunctions on S has dimension n, where n is the number of elements of 8S={yl...... yy}. Again we use induction on the size of S. If S' is obtained by adding one then t must be adjacent to some point point, say t, ylEB3. It follows that either yl remains a boundary point in S', or it to 3, becomes an interior point. In the first case a µ-eigenfunction on S may be extended to a ii-eigenf unction on 8S', by assigning an arbitrary value at t, because the condition of being a µ-eigenfunction applies only to interior points and t is not adjacent to an interior point.

Observe that Aut(3f)+nG is a subgroup invariant of index character of 2. G. Therefore which is (-1)d(go,o) is a K-bi- of course a positive- definite spherical function. Likewise the function identically 1 is a positive-definite spherical function. 54 Ch. I I We shall prove that a spherical function associated with a real eigenvalue -1

It remains to prove that the representation it is in other words that every nonzero vector irreducible, cyclic vector. then g=Ecio(gig), If is a PKg $ ciK¢(gIkg)dk= J (Ecl0(gi))n. Since linear combinations of left translates of 0 are dense, we conclude that, for every PKC is a constant multiple of fin. On the other hand, if f;*O and PKn(g)F;=O for then n(g)etgn for every every gEG, because gn is cyclic. g, which implies f;=O, Therefore, for some gEG, PKn(g)g*O. In order to show that f; is cyclic it is enough to prove that n(g)g is cyclic for some g.