Date:	Fri, November 21, 2014
Time:	11:15
Place:	Seminar Room (120), Research I

Abstract: Let \(\mathfrak{g}\) be the Lie algebra of a classical complex Lie group \(G\), \(\mathfrak{n}\) be a maximal nilpotent subalgebra of \(\mathfrak{g}\), and \(U(\mathfrak{n})\) be the universal enveloping algebra of \(\mathfrak{n}\). J. Dixmier (for \(\mathfrak{sl}_n(\mathbb{C})\)), A. Joseph and B. Kostant (for arbitrary \(\mathfrak{g}\)) described the center \(Z(\mathfrak{n})\) of \(U(\mathfrak{n})\). In particular, they proved that \(Z(\mathfrak{n})\) is a polynomial ring. A. Panov provided a description of the set of Kostant's generators of \(Z(\mathfrak{n})\) in terms of the representation theory of \(G\). We will explain Panov's approach for arbitrary \(\mathfrak{g}\) and consider the example \(\mathfrak{sl}_n(\mathbb{C})\)) in details.
The orbit method provide a description of primitive ideals of \(U(\mathfrak{n})\) in terms of coadjoint orbits. Namely, to each linear form \(f\) on \(\mathfrak{n}\) one can assign the primitive ideal \(J(f)\) of \(U(\mathfrak{n})\). The induced Dixmier map from the set of coadjoint orbits on \(\mathfrak{n}^\ast\) to the set of primitive ideals of the enveloping algebra is a bijection. It turns out that centrally generated ideals correspond to orbits of maximal dimension. We will show how to prove this fact using a modification of Panov's construction.
The talk is based on the joint work with Mikhail Ignatyev.