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This article is cited in 21 scientific papers (total in 21 papers)
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 $\mathfrak{g}= \hbox{Lie}\,G$. 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.
Received: February 1, 2011
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
Linking options:
https://www.mathnet.ru/eng/mmj460 https://www.mathnet.ru/eng/mmj/v12/i3/p605
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Abstract page: | 201 | References: | 56 |
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