Difference between revisions of "Quasi-simple representation"
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− | A continuous [[ | + | 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 vector]]s 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==== | ====References==== | ||
− | <table><TR><TD valign="top">[1a]</TD> <TD valign="top"> Harish-Chandra, "Representations of a semisimple Lie group on a Banach space I" ''Trans. Amer. Math. Soc.'' , '''75''' (1953) pp. 185–243</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top"> Harish-Chandra, "Representations of a semisimple Lie groups II" ''Trans. Amer. Math. Soc.'' , '''76''' (1954) pp. 26–65</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J. Lepowsky, "Algebraic results on representations of semisimple Lie groups" ''Trans. Amer. Math. Soc.'' , '''176''' (1973) pp. 1–44</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[1a]</TD> <TD valign="top"> Harish-Chandra, "Representations of a semisimple Lie group on a Banach space I" ''Trans. Amer. Math. Soc.'' , '''75''' (1953) pp. 185–243</TD></TR> | ||
+ | <TR><TD valign="top">[1b]</TD> <TD valign="top"> Harish-Chandra, "Representations of a semisimple Lie groups II" ''Trans. Amer. Math. Soc.'' , '''76''' (1954) pp. 26–65</TD></TR> | ||
+ | <TR><TD valign="top">[2]</TD> <TD valign="top"> J. Lepowsky, "Algebraic results on representations of semisimple Lie groups" ''Trans. Amer. Math. Soc.'' , '''176''' (1973) pp. 1–44</TD></TR> | ||
+ | <TR><TD valign="top">[3]</TD> <TD valign="top"> 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</TD></TR> | ||
+ | </table> | ||
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====References==== | ====References==== | ||
− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> N.R. Wallach, "Real reductive groups" , Acad. Press (1988)</TD></TR></table> | + | <table> |
+ | <TR><TD valign="top">[a1]</TD> <TD valign="top"> N.R. Wallach, "Real reductive groups" , Acad. Press (1988)</TD></TR> | ||
+ | </table> | ||
+ | |||
+ | {{TEX|done}} |
Latest revision as of 17:00, 9 October 2016
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 |
Comments
References
[a1] | N.R. Wallach, "Real reductive groups" , Acad. Press (1988) |
Quasi-simple representation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quasi-simple_representation&oldid=14861