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

Martin, Robert Paul

doi:10.1090/S0002-9947-1975-0374341-0

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

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

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

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Abstract:

Let $G$ 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 $G$ consists of two pieces, ${T_c}$ and ${T_d}$, where ${T_c}$ is a continuous direct sum with respect to Plancherel measure on $\hat G$ of representations from the principal series only, occurring with explicitly determined multiplicities, and ${T_d}$ is a discrete sum of representations from the discrete series of $G$, occurring with multiplicities which are, for the present, undetermined.

References

Nguyen Huu Anh, Restriction of the principal series of $\textrm {SL}(n,\,\textbf {C})$ to some reductive subgroups, Pacific J. Math. 38 (1971), 295–314. MR 327980, DOI 10.2140/pjm.1971.38.295

Theory of Lie groups

7

Edward G. Effros, Transformation groups and $C^{\ast }$-algebras, Ann. of Math. (2) 81 (1965), 38–55. MR 174987, DOI 10.2307/1970381
John A. Ernest, A decomposition theory for unitary representations of locally compact groups, Trans. Amer. Math. Soc. 104 (1962), 252–277. MR 139959, DOI 10.1090/S0002-9947-1962-0139959-X
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455
Kenneth D. Johnson, Composition series and intertwining operators for the spherical principal series. II, Trans. Amer. Math. Soc. 215 (1976), 269–283. MR 385012, DOI 10.1090/S0002-9947-1976-0385012-X
Adam Kleppner and Ronald L. Lipsman, The Plancherel formula for group extensions. I, II, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459–516; ibid. (4) 6 (1973), 103–132. MR 342641, DOI 10.24033/asens.1235
A. W. Knapp and E. M. Stein, Intertwining operators for semisimple groups, Ann. of Math. (2) 93 (1971), 489–578. MR 460543, DOI 10.2307/1970887
Bertram Kostant, On the existence and irreducibility of certain series of representations, Bull. Amer. Math. Soc. 75 (1969), 627–642. MR 245725, DOI 10.1090/S0002-9904-1969-12235-4
R. A. Kunze and E. M. Stein, Uniformly bounded representations. III. Intertwining operators for the principal series on semisimple groups, Amer. J. Math. 89 (1967), 385–442. MR 231943, DOI 10.2307/2373128
Ronald L. Lipsman, On the characters and equivalence of continuous series representations, J. Math. Soc. Japan 23 (1971), 452–480. MR 289718, DOI 10.2969/jmsj/02330452
George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 44536, DOI 10.2307/1969423
George W. Mackey, Induced representations of locally compact groups. II. The Frobenius reciprocity theorem, Ann. of Math. (2) 58 (1953), 193–221. MR 56611, DOI 10.2307/1969786
M. A. Naĭmark, Decomposition of a tensor product of irreducible representations of the proper Lorentz group into irreducible representations. I. The case of a tensor product of representations of the fundamental series, Trudy Moskov. Mat. Obšč. 8 (1959), 121–153 (Russian). MR 0114139
Lajos Pukánszky, On the Kronecker products of irreducible representations of the $2\times 2$ real unimodular group. I, Trans. Amer. Math. Soc. 100 (1961), 116–152. MR 172962, DOI 10.1090/S0002-9947-1961-0172962-1
Paul J. Sally Jr., Analytic continuation of the irreducible unitary representations of the universal covering group of $\textrm {SL}(2,\,R)$, Memoirs of the American Mathematical Society, No. 69, American Mathematical Society, Providence, R.I., 1967. MR 0235068

Harmonic analysis on semi-simple Lie groups

Floyd L. Williams, Tensor products of principal series representations, Lecture Notes in Mathematics, Vol. 358, Springer-Verlag, Berlin-New York, 1973. Reduction of tensor products of principal series. Representations of complex semisimple Lie groups. MR 0349903, DOI 10.1007/BFb0067484