Antonio J. Di Scala, Thomas Leistner
Connected subgroups of SO(2,n) acting irreducibly on R$^{2,n}$
Preprint series: ESI preprints

MSC:

22E46 Semisimple Lie groups and their representations

53C35 Symmetric spaces, See also {32M15, 57T15}

53C40 Global submanifolds, See also {53B25}

53C05 Connections, general theory

53A30 Conformal differential geometry

Abstract: We classify all connected subgroups of $SO(2,n)$ that act irreducibly on $\rr^{2,n}$. Apart from
$SO_0(2,n)$ itself these are $U(1,n/2)$, $SU(1,n/2)$, if $n$ even, $S^1\cdot SO(1,n/2)$
if $n$ even and $n\ge 2$, and $SO_0(1,2)$ for $n=3$. Our proof is based on the Karpelevich Theorem
and uses the classification of totally geodesic submanifolds of complex hyperbolic space and of the
Lie ball. As an application we obtain a list of possible irreducible holonomy groups of Lorentzian
conformal structures, namely $SO_0(2,n)$, $SU(1,n)$, and $SO_0(1,2)$.


Keywords: irreducible orthogonal representations, Lie ball, complex hyperbolic space, totally geodesic submanifolds of symmetric spaces, conformal holonomy, Berger list