A new proof of the Borel-de Siebenthal Theorem, the classification of equi-rank groups, and some compact and semi-compact dual pairs.

Hubert Rubenthaler (1)

1 IRMA - Institut de Recherche Mathématique Avancée

We give a new classification of the simple real Lie groups $G$ which have a discrete series of representations. In fact our characterization is purely algebraic and we work only on the complexified Lie algebra $ {\go{g}} $ of $G$.If ${\go k}$ is the complexified Lie algebra of the maximal compact subgroup of $G$ then there are only two possibilities. Either $ {\go{g}} = {\go{g}} _{-1}\oplus {\go{g}} _0\oplus {\go{g}} _{1}$ and ${\go k}={\go g}_0$ (hermitian case, ${\go k}$ has a center) or $ {\go{g}} = {\go{g}} _{-2}\oplus {\go{g}} _{-1}\oplus {\go{g}} _0\oplus {\go{g}} _1 \oplus {\go{g}} _2$ and ${\go k}= {\go{g}} _{-2}\oplus {\go{g}} _0 \oplus {\go{g}} _2$ (${\go k}$ is semi-simple). Moreover in each case the parabolic subalgebra $ {\go{P}}=\sum_{p\geq 0} {\go{g}} _i $ is maximal. This allows an easy classification. Our results also give three families of compact or semi-compact dual pairs in the real Lie algebra $ {\go{g}} _{\bf R}$ of $G$.

Mots clés: "discrete series; dual pairs; graded Lie algebra"; discrete series; graded Lie algebra

Domaines: Anneaux et algèbres [math.RA]

Hubert Rubenthaler. A new proof of the Borel-de Siebenthal Theorem, the classification of equi-rank groups, and some compact and semi-compact dual pairs.. 1998. ⟨hal-00129720⟩