Analytic representation

holomorphic representation

A representation of a complex Lie group $G$ in a topological vector space $E$ in which all matrix elements $(\phi(g)\xi,\eta)$, $\xi\in E$, $\eta\in E'$, where $E'$ is the dual topological vector space, are holomorphic on $G$. A representation $\phi$ 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). If $G$ is a semi-simple complex Lie group, then all its topologically irreducible analytic (anti-analytic) representations are finite-dimensional.

References

[1]	M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian)
[2]	D.P. Zhelobenko, "Compact Lie groups and their representation" , Amer. Math. Soc. (1973) (Translated from Russian)