Unitarization of a singular representation of \(\text{SO}(p,q)\). (English) Zbl 0748.22009
From the abstract: A geometric construction of a certain singular unitary representation of \(\text{SO}_ e(p,q)\), with \(p+q\) even is given. The representation is realized geometrically as the kernel of an \(\text{SO}_ e(p,q)\)-invariant operator on a space of sections over a homogeneous space for \(\text{SO}_ e(p,q)\). The \(K\)-structure of these representations is elucidated and we demonstrate their unitarity by explicitly writing down an \(\mathfrak{so}(p,q)\)-invariant positive Hermitian form. Finally, we demonstrate that the annihilator in the universal enveloping algebra of this representation is the Joseph ideal, which is the maximal primitive ideal associated with the minimal coadjoint orbit.
Reviewer: P. Holod (Kiev)
MSC:
| 22E70 | Applications of Lie groups to the sciences; explicit representations |
| 22E46 | Semisimple Lie groups and their representations |
| 17B35 | Universal enveloping (super)algebras |
| 17B37 | Quantum groups (quantized enveloping algebras) and related deformations |
| 81R05 | Finite-dimensional groups and algebras motivated by physics and their representations |
Keywords:
singular unitary representation; homogeneous space; positive hermitian form; annihilator; universal enveloping algebra; Joseph ideal; maximal primitive ideal; minimal coadjoint orbitReferences:
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