The orbit method and primitive ideals for semisimple Lie algebras.

*(English)*Zbl 0585.17008
Lie algebras and related topics, Proc. Semin., Windsor/Ont. 1984, CMS Conf. Proc. 5, 281-316 (1986).

[For the entire collection see Zbl 0574.00001.]

The orbit method – the correspondence between the orbits of coadjoint representations of the Lie group \(G\) and the unitary irreducible representations of \(G\) – has its algebraic aspect, where the natural bijection is supposed between the orbits of coadjoint representations and the completely prime and (or) primitive ideals of \(U(\mathfrak g)\), \(\mathfrak g\) being the Lie algebra of the group \(G\).

The purpose of this article is to discuss the conjectures, results and examples on this theme. The central place in the article is occupied by the conjecture of the author on the bijection between the completely prime \(\mathbb C\)-algebras \(A\) (with the algebra homomorphism \(\phi : U(\mathfrak g)\to A\) such that \(A\) is the finitely generated \(U(\mathfrak g)\)-module, and \(G\) acts on \(A\) algebraically by the automorphisms, extending Ad-action of \(G\) on \(U(\mathfrak g)\)) and the irreducible affine varieties \(X\) (with the algebraic action of \(G\) on \(X\) and the finite \(G\)-morphism \(\pi: X\to\mathfrak g^*\)). When \(G\) is any connected complex reductive Lie group, the examples of algebras \(A\) are the algebras of twisted differential operators on the flag varieties from the Bernstein-Beilinson construction.

The orbit method – the correspondence between the orbits of coadjoint representations of the Lie group \(G\) and the unitary irreducible representations of \(G\) – has its algebraic aspect, where the natural bijection is supposed between the orbits of coadjoint representations and the completely prime and (or) primitive ideals of \(U(\mathfrak g)\), \(\mathfrak g\) being the Lie algebra of the group \(G\).

The purpose of this article is to discuss the conjectures, results and examples on this theme. The central place in the article is occupied by the conjecture of the author on the bijection between the completely prime \(\mathbb C\)-algebras \(A\) (with the algebra homomorphism \(\phi : U(\mathfrak g)\to A\) such that \(A\) is the finitely generated \(U(\mathfrak g)\)-module, and \(G\) acts on \(A\) algebraically by the automorphisms, extending Ad-action of \(G\) on \(U(\mathfrak g)\)) and the irreducible affine varieties \(X\) (with the algebraic action of \(G\) on \(X\) and the finite \(G\)-morphism \(\pi: X\to\mathfrak g^*\)). When \(G\) is any connected complex reductive Lie group, the examples of algebras \(A\) are the algebras of twisted differential operators on the flag varieties from the Bernstein-Beilinson construction.

Reviewer: S. Prishchepionok

##### MSC:

17B35 | Universal enveloping (super)algebras |

17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |