Content deleted Content added
TakuyaMurata (talk | contribs) m →Compact Lie group case: lk |
TakuyaMurata (talk | contribs) →References: add Pukánszky condition |
||
Line 13:
Complex irreducible representations of [[compact Lie group]]s have been completely classified. They are always finite-dimensional, unitarizable (i.e. admit an invariant positive definite [[Hermitian form]]) and are parametrized by their [[highest weight]]s, which are precisely the dominant integral weights for the group. If ''G'' is a compact [[semisimple Lie group]] with a [[Cartan subalgebra]] ''h'' then its coadjoint orbits are [[closed set|closed]] and each of them intersects the positive Weyl chamber ''h''<sup>*</sup><sub>+</sub> in a single point. An orbit is '''integral''' if this point belongs to the weight lattice of ''G''.
The highest weight theory can be restated in the form of a bijection between the set of integral coadjoint orbits and the set of equivalence classes of irreducible unitary representations of ''G'': the highest weight representation ''L''(''λ'') with highest weight ''λ''∈''h''<sup>*</sup><sub>+</sub> corresponds to the integral coadjoint orbit ''G''·''λ''. The [[Kirillov character formula]] amounts to the character formula earlier proved by [[Harish-Chandra]].
== See also ==
*[[Pukánszky condition]]
== References ==
| |||