Orbits, multiplicities and differential operators.

*(English)*Zbl 0799.14030
Adams, Jeffrey (ed.) et al., Representation theory of groups and algebras. Providence, RI: American Mathematical Society. Contemp. Math. 145, 199-227 (1993).

Section 1 contains five examples illustrating certain properties of actions of Lie groups on manifolds (orbits on the sphere, orbits in flag varieties, invariant differential operators on the cone, homogeneous varieties of \(SL (1000, \mathbb{C})\) without nonconstant regular functions).

Section 2 contains a generalization to real Lie groups of a result of M. Brion and E. Vinberg concerning actions of algebraic groups on the full flag varieties. Namely, it is shown there that if a subgroup \(H\) of a reductive real Lie group \(G\) acts with an open orbit on the maximal flag manifold \(Y\) of \(G\), then \(H\) has only finitely many orbits in \(Y\). In section 3 multiplicities of induced representations are used to detect the existence of open orbits. Linear algebraic groups of general type (i.e. not necessarily reductive) are considered. It is shown that spherical varieties are exactly those which are multiplicity free, thereby generalizing a result of M. KrĂ¤mer. In section 4 multiplicities are related to invariant differential operators, and it is shown that when the group is reductive, spherical spaces are characterized by the commutativity of all rings of invariant twisted differential operators. The last section contains solutions to the problems of section 1.

For the entire collection see [Zbl 0773.00011].

Section 2 contains a generalization to real Lie groups of a result of M. Brion and E. Vinberg concerning actions of algebraic groups on the full flag varieties. Namely, it is shown there that if a subgroup \(H\) of a reductive real Lie group \(G\) acts with an open orbit on the maximal flag manifold \(Y\) of \(G\), then \(H\) has only finitely many orbits in \(Y\). In section 3 multiplicities of induced representations are used to detect the existence of open orbits. Linear algebraic groups of general type (i.e. not necessarily reductive) are considered. It is shown that spherical varieties are exactly those which are multiplicity free, thereby generalizing a result of M. KrĂ¤mer. In section 4 multiplicities are related to invariant differential operators, and it is shown that when the group is reductive, spherical spaces are characterized by the commutativity of all rings of invariant twisted differential operators. The last section contains solutions to the problems of section 1.

For the entire collection see [Zbl 0773.00011].

Reviewer: V.L.Popov (Moskva)

##### MSC:

14M17 | Homogeneous spaces and generalizations |

16S32 | Rings of differential operators (associative algebraic aspects) |

17B45 | Lie algebras of linear algebraic groups |

20G20 | Linear algebraic groups over the reals, the complexes, the quaternions |

14M15 | Grassmannians, Schubert varieties, flag manifolds |

22E30 | Analysis on real and complex Lie groups |