On the decomposition of tensor products of principal series representations for real-rank one semisimple groups

Author:
Robert Paul Martin

Journal:
Trans. Amer. Math. Soc. **201** (1975), 177-211

MSC:
Primary 22E45

DOI:
https://doi.org/10.1090/S0002-9947-1975-0374341-0

MathSciNet review:
0374341

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a connected semisimple real-rank one Lie group with finite center. It is shown that the decomposition of the tensor product of two representations from the principal series of consists of two pieces, and , where is a continuous direct sum with respect to Plancherel measure on of representations from the principal series only, occurring with explicitly determined multiplicities, and is a discrete sum of representations from the discrete series of , occurring with multiplicities which are, for the present, undetermined.

**[1]**N. Anh,*Restriction of the principal series of*SL*to some reductive subgroups*, Pacific J. Math.**38**(1971), 295-313. MR**0327980 (48:6322)****[2]**C. Chevalley,*Theory of Lie groups*. Vol. I, Princeton Math. Ser., vol. 8, Princeton Univ. Press, Princeton, N. J., 1946. MR**7**, 412.**[3]**E. G. Effros,*Transformation groups and -algebras*, Ann. of Math. (2)**81**(1965), 38-55. MR**30**#5175. MR**0174987 (30:5175)****[4]**J. Ernest,*A decomposition theory for unitary representations of locally compact groups*, Trans. Amer. Math. Soc.**104**(1962), 252-277. MR**25**#3383. MR**0139959 (25:3383)****[5]**S. Helgason,*Differential geometry and symmetric spaces*, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR**26**#2986. MR**0145455 (26:2986)****[6]**K. Johnson,*Composition series and intertwining operators for the spherical principal series*. II (to appear). MR**0385012 (52:5882)****[7]**A. Kleppner and R. L. Lipsman,*The Plancherel formula for group extensions*, Ann. Sci. École Norm. Sup.**5**(1972), 459-516. MR**0342641 (49:7387)****[8]**A. W. Knapp and E. M. Stein,*Intertwining operators for semi-simple groups*, Ann. of Math. (2)**93**(1971), 489-578. MR**0460543 (57:536)****[9]**B. Kostant,*On the existence and irreducibility of certain series of representations*, Bull. Amer. Math. Soc.**75**(1969), 627-642. MR**39**#7031. MR**0245725 (39:7031)****[10]**R. A. Kunze and E. M. Stein,*Uniformly bounded representations*. III:*Intertwining operators for the principal series on semi-simple groups*, Amer. J. Math.**89**(1967), 385-442. MR**38**#269. MR**0231943 (38:269)****[11]**R. L. Lipsman,*On the characters and equivalence of continuous series representations*, J. Math. Soc. Japan**23**(1971), 452-480. MR**44**#6906. MR**0289718 (44:6906)****[12]**G. Mackey,*Induced representations of locally compact groups*. I, Ann. of Math. (2)**55**(1952), 101-139. MR**13**, 434. MR**0044536 (13:434a)****[13]**-,*Induced representations of locally compact groups*. II.*The Frobenius reciprocity theorem*, Ann. of Math. (2)**58**(1953), 193-221. MR**15**, 101. MR**0056611 (15:101a)****[14]**M. A. Naĭmark,*Decomposition of the tensor product of irreducible representations of the proper Lorentz group into irreducible representations*, Trudy Moskov. Mat. Obšč.**8**(1959), 121-153; English transl., Amer. Math. Soc. Transl. (2)**36**(1964), 101-136. MR**22**#4966. MR**0114139 (22:4966)****[15]**L. Pukánszky,*On the Kronecker products of irreducible representations of the**real unimodular group*. I, Trans. Amer. Math. Soc.**100**(1961), 116-152. MR**30**#3177. MR**0172962 (30:3177)****[16]**P. J. Sally, Jr.,*Analytic continuation of the irreducible unitary representations of the universal covering group of*SL, Mem. Amer. Math. Soc. No. 69 (1967). MR**38**#3380. MR**0235068 (38:3380)****[17]**G. Warner,*Harmonic analysis on semi-simple Lie groups*. Vols. 1, 2, Die Grundlehren der math. Wissenschaften, Bände 188, 189, Springer-Verlag, Berlin and New York, 1972.**[18]**F. Williams,*Tensor products of principal series representations*, Lecture Notes in Math., vol. 358, Springer-Verlag, Berlin and New York, 1973. MR**0349903 (50:2396)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
22E45

Retrieve articles in all journals with MSC: 22E45

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1975-0374341-0

Keywords:
Semisimple Lie groups,
irreducible unitary representations,
cuspidal parabolic subgroups,
principal series,
discrete series,
tensor products

Article copyright:
© Copyright 1975
American Mathematical Society