By J. H., E. N. Nilson, and J. L. Walsh. AHLBERG

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_d, where R is a certain (unique) rational function on gv, such that (all! E £leG) n V(G)). ] This is substantially the Plancherel Theorem for G; for clarification and amplification, see Pukanszky [2], [4]; see also Dixmier [3], [5] and Kirillov [1], [2], [3], [4]. [The following point should be mentioned. It is tacitly supposed that the orbit space A is equipped with the quotient topology; now the points of A are in a natural one-to-one correspondence A f--t 0). with the points of G and so it is only natural to ask: Is this correspondence a homeomorphism?

J=1 = v and consider the corresponding standard representation UI' of G on V(K); reading the preceding argument backwards, we immediately see that CP. occurs as a 'coefficient' in UI' and hence is quasibounded. Summary Every zonal spherical function cP, on G occurs as a coefficient in some (not necessarily irreducible) continuous representation of G on a Hilbert space (which will, in general, depend on cP'). One may ask: What is the necessary and sufficient condition on v to ensure that the corresponding zonal spherical function CP.

The closure of + (respectively -) is G -£ - } (respectively G -£ +}); the points £+}, £- } are both open but not closed .... ] (5) Suppose that G = SL(2, C) - then, as is well-known, the irreducible unitary representations of G fall into three distinct classes, namely the trivial one dimensional representation 1 (say), the representations in the principal P-series (P a minimal parabolic subgroup of G), and the representations in the complementary series. The representations Urn, r of the principal P-series are indexed by a pair (m, r) with m an integer and r a real number; two representations in the principal P-series corresponding to distinct parameter pairs (mj, rj), (mz, r2) are unitarily equivalent iff mj = -mz, rj = -rz; in view of this, let us agree to index the principal P-series by the pairs (m, r) for m "> 0 (with r "> 0 when m = 0).