A note on nonunitary principal series representations

Author:
Hrvoje Kraljević

Journal:
Proc. Amer. Math. Soc. **70** (1978), 213-216

MSC:
Primary 22E45

DOI:
https://doi.org/10.1090/S0002-9939-1978-0473107-9

MathSciNet review:
0473107

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Abstract: It is proven in a rather elementary way that any nonunitary principal series representation of a semisimple Lie group *G* is of finite length, having as a trivial consequence that the set of infinitesimal equivalence classes of quasisimple irreducible representations of *G* with a given infinitesimal character is finite.

**[1]**W. Casselman,*The differential equations satisfied by matrix coefficients*, 1975 (manuscript).**[2]**A. W. Knapp and N. R. Wallach,*Szego kernels associated with discrete series*, Invent. Math.**34**(1976), 163-200. MR**0419686 (54:7704)****[3]**B. Kostant,*On the tensor product of a finite and an infinite dimensional representation*, J. Functional Analysis**20**(1975), 257-285. MR**0414796 (54:2888)****[4]**J. I. Lepowsky and N. R. Wallach,*Finite- and infinite-dimensional representations of linear semisimple groups*, Trans. Amer. Math. Soc.**184**(1973), 223-246. MR**0327978 (48:6320)****[5]**D. Miličić,*Notes on asymptotics of admissible representations*, 1976 (unpublished).**[6]**-,*Asymptotic behaviour of matrix coefficients of the discrete series*, Duke Math. J.**44**(1977), 59-88. MR**0430164 (55:3171)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1978-0473107-9

Keywords:
Semisimple Lie groups,
nonunitary principal series

Article copyright:
© Copyright 1978
American Mathematical Society