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 Mosc. Math. J., 2012, Volume 12, Number 3, Pages 605–620 (Mi mmj460)

The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group

Bertram Kostant

Department of Mathematics, M.I.T., Cambridge, MA 02139

Abstract: Let $G$ be a semisimple Lie group and let $\mathfrak{g}= \mathfrak{n}_- + \mathfrak{h} +\mathfrak{n}$ be a triangular decomposition of $\mathfrak{g}= Lie G$. Let $\mathfrak{b} = \mathfrak{h} +\mathfrak{n}$ and let $H,N,B$ be Lie subgroups of $G$ corresponding respectively to $\mathfrak{h}$, $\mathfrak{n}$ and $\mathfrak{b}$. We may identify $\mathfrak{n}_-$ with the dual space to $\mathfrak{n}$. The coadjoint action of $N$ on $\mathfrak{n}_-$ extends to an action of $B$ on $\mathfrak{n}_-$. There exists a unique nonempty Zariski open orbit $X$ of $B$ on $\mathfrak{n}_-$. Any $N$-orbit in $X$ is a maximal coadjoint orbit of $N$ in $\mathfrak{n}_-$. The cascade of orthogonal roots defines a cross-section $\mathfrak{r}_-^{\times}$ of the set of such orbits leading to a decomposition
$$X = N/R\times \mathfrak{r}_-^{\times}.$$
This decomposition, among other things, establishes the structure of $S(\mathfrak{n})^{\mathfrak{n}}$ as a polynomial ring generated by the prime polynomials of $H$-weight vectors in $S(\mathfrak{n})^{\mathfrak{n}}$. It also leads to the multiplicity 1 of $H$ weights in $S(\mathfrak{n})^{\mathfrak{n}}$.

Key words and phrases: Cascade of orthogonal roots, Borel subgroups, nilpotent coadjoint action.

DOI: https://doi.org/10.17323/1609-4514-2012-12-3-605-620

Full text: http://www.ams.org/.../abst12-3-2012.html
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MSC: 20C, 14L24
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Citation: Bertram Kostant, “The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group”, Mosc. Math. J., 12:3 (2012), 605–620

Citation in format AMSBIB
\Bibitem{Kos12} \by Bertram~Kostant \paper The cascade of orthogonal roots and the coadjoint structure of the nilradical of a Borel subgroup of a semisimple Lie group \jour Mosc. Math.~J. \yr 2012 \vol 12 \issue 3 \pages 605--620 \mathnet{http://mi.mathnet.ru/mmj460} \crossref{https://doi.org/10.17323/1609-4514-2012-12-3-605-620} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3024825} \zmath{https://zbmath.org/?q=an:1260.14058} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000309366400008} 

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