View source for Analytic representation

''holomorphic representation''

A representation of a complex Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123901.png" /> in a topological vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123902.png" /> in which all matrix elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123903.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123904.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123905.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123906.png" /> is the dual topological vector space, are holomorphic on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123907.png" />. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123908.png" /> is called an anti-analytic representation if its matrix elements become holomorphic after complex conjugation. An analytic (anti-analytic) representation of a connected Lie group is uniquely determined by a corresponding Lie algebra representation of this group (cf. [[Representation of a Lie algebra|Representation of a Lie algebra]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012390/a0123909.png" /> is a semi-simple complex Lie group, then all its topologically irreducible analytic (anti-analytic) representations are finite-dimensional.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Naimark,   "Theory of group representations" , Springer  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  D.P. Zhelobenko,   "Compact Lie groups and their representation" , Amer. Math. Soc.  (1973)  (Translated from Russian)</TD></TR></table>