Quasi-simple representation

A continuous linear representation of a connected semi-simple real Lie group in a Banach space such that: 1) the operator is a scalar multiple of the identity operator on for any in the centre of ; and 2) if is the space of analytic vectors (cf. Analytic vector) in with respect to and if is the representation of the universal enveloping Lie algebra of in (cf. Universal enveloping algebra), then is a scalar multiple of the identity operator on for all in the centre of . These scalar multiples determine a character of the centre of , called the infinitesimal character of the quasi-simple representation. Two quasi-simple representations are said to be infinitesimally equivalent if they determine equivalent representations in the respective vector spaces of analytic vectors of the universal enveloping algebras. Every completely-irreducible representation of a group in a Banach space is a quasi-simple representation, and any irreducible quasi-simple representation of a group in a Banach space is infinitesimally equivalent to a completely-irreducible representation; the latter is the restriction to the invariant subspace of some quotient representation of the representation (generally non-unitary) in the fundamental series of representations of .

References

[1a]	Harish-Chandra, "Representations of a semisimple Lie group on a Banach space I" Trans. Amer. Math. Soc. , 75 (1953) pp. 185–243
[1b]	Harish-Chandra, "Representations of a semisimple Lie groups II" Trans. Amer. Math. Soc. , 76 (1954) pp. 26–65
[2]	J. Lepowsky, "Algebraic results on representations of semisimple Lie groups" Trans. Amer. Math. Soc. , 176 (1973) pp. 1–44
[3]	A.I. Fomin, "Characters of irreducible representations of real semisimple Lie groups" Funct. Anal. Appl. , 10 : 3 (1976) pp. 246–247 Funktsional. Anal. Prilozhen. , 10 : 3 (1976) pp. 95–96

References

[a1]	N.R. Wallach, "Real reductive groups" , Acad. Press (1988)