Quasi-simple representation

A continuous linear representation $\pi$ of a connected semi-simple real Lie group $G$ in a Banach space $E$ such that: 1) the operator $\pi(x)$ is a scalar multiple of the identity operator on $E$ for any $x$ in the centre of $G$; and 2) if $F$ is the space of analytic vectors in $E$ with respect to $\pi$ and if $\pi_F$ is the representation of the universal enveloping Lie algebra $\mathfrak{G}$ of $G$ in $F$ (cf. Universal enveloping algebra), then $\pi_F(z)$ is a scalar multiple of the identity operator on $F$ for all $z$ in the centre of $\mathfrak{G}$. These scalar multiples determine a character of the centre of $\mathfrak{G}$, 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 $G$ 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 $G$.

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)