On the Characters of a Semisimple Lie Group

where dx is the Haar measure of G and C?(G) is the set of all (complexvalued) functions on G which are everywhere indefinitely differentia t e and which vanish outside a compact set. V is called the Gârding subspace of § . Let R and C be the fields of real and complex numbers respectively and g0 the Lie algebra of G. We complexify g0 to Q and denote by $8 the universal enveloping algebra of Q [2a]. Then there exists a (uniquely determined) representation TTV of S3 on V such that 7rF(X)^ = lim^o(lA){7r(exp tX)f-f} (XGöo, ^ G F , tGR). Let S denote the center of S3. We say that TT is quasi-simple if there exist homomorphisms rj and % of Z and 3 respectively into C such that 7r(r)0 = î?(r)0, x r ( s ) * = x ( * # for all f £ Z , * G 3 , 4>G£ and ^ £ F . 77 is then called the central character and x the infinitesimal character of 7T. An irreducible unitary representation is automatically quasi-simple [5]. Let A be a bounded linear operator on § . We say that A is of the trace class or A has a trace if for every complete orthonormal set (\{/j)j£j in § the series ^CiG^ W'i» ^ i ) converges absolutely and its sum is independent of the choice of the complete orthonormal set. We call this sum the trace of A and denote it by Sp A. Now suppose w is quasi-simple and irreducible. Then it can be shown (see [2e]) that for any /GGC°°(G) the operator ff(x)ir{x)dx is of the trace class. If we denote its trace by TT(J) we get a linear function Tr on CC°°(G) which is actually a distribution (see [4; and 2e]). We call this distribution the character of 7r. Our object is to try to determine TV.