BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 82, Number 6, November 1976 TENSOR PRODUCTS OF UNITARY REPRESENTATIONS OF SL 2 (R) BY JOE REPKA Communicated by I. M. Singer, April 18, 1976 1. Introduction. We consider the tensor product of two irreducible unitary representations of G = SL 2 (R); in particular, we obtain its reduction as a direct integral of irreducible representations. This question has been solved in certain cases by Pukanszky [4] and Martin [3]. We restate their results and also do the remaining cases. 2. Notation. Let M = {± ƒ}; K = S0 2 (R); and let A (resp. N) be the sub- group consisting of all positive diagonal matrices (resp. upper triangular unipotent matrices). Let (e f 0 \ / ;os0 sin0\ * \0 e-*/ 6 \-sin0 COS0/ For s G /R, e G Af, let 17 be the one-dimensional representation of MAN given by n Sf€ : mh f n H* e(m) • e 5 *, m EM, n EN Let ir s 6 = Indjjj^î?^, a principal series representation. For — 1 < a < 0, let n c 0 be the (unitary) complementary series representation which is inflnitesimally isomorphic to the "nonunitary principal series" represen- tation induced from the representation of MAN given by mh f n |—• exp(a/)- The representations TI C 0 are all irreducible, as are all the n s € , except when 5 = 0 and e E M is nontrivial. In this case, n 0 € is the direct sum of two irre- ducible representations, denoted it J € and 7TQ e . For n G Z, define x n G K by X n (k d ) = e ind . For n > 2, we let T n (resp. T~ n ) be the discrete series representation with lowest weight n (resp. highest weight - ri). We also let T x = 7rJ e , T_ x = Tt^ e , the so-called "mock discrete series representations" with extreme weights 1 and — 1 respectively. The representations we have described exhaust the irreducible unitary repre- sentations of G. For details, see, e.g., Lang [2] . 3. A preliminary result. Before proceeding, we state a very easy but useful fact, for any separable locally compact group. AMS (MOS) subject classifications (1970). Primary 22E45; Secondary 22E43. Key words and phrases. Tensor product, unimodular group, unitary representation, holomorphic discrete series. Copyright © 1976, American Mathematical Society 930 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use