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% {Spherical Orbits and Abelian Ideals}
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\begin{document}
\title[Spherical Orbits and Abelian Ideals]
{Spherical Orbits and Abelian Ideals}
\author[G. R\"ohrle]{Gerhard R\"ohrle}
\address{Fakult\"{a}t f\"{u}r Mathematik,
Universit\"{a}t Bielefeld, Postfach 100131, 33501 Bielefeld, Germany.}
\email{roehrle@mathematik.uni-bielefeld.de}
\makeatletter
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\maketitle
This is a report on recent joint work with Dmitri Panyushev \cite{PR}.
Let $G$ be a reductive complex Lie group with Lie algebra
$\Lie G = \frakg$.
Let $B$ be a Borel subgroup with Lie algebra $\bb$.
Let $P$ be a parabolic subgroup of $G$ containing $B$
with unipotent radical $P_u$.
We denote the Lie algebra of $P$ and $P_u$ by
$\mathfrak p$ and $\mathfrak p_u$, respectively.
The group $P$ acts on any ideal of $\pp$ in $\pp_u$
by means of the adjoint representation.
Let $\mathfrak a$ be an abelian ideal of $\pp$ in $\pp_u$.
It was shown in \cite[Thm.\ 1.1]{Ro3} that $P$ operates on $\aaa$
with a finite number of orbits.
The proof of this result in \cite{Ro3} involved long and tedious case
by case considerations. The original proof of this theorem went as follows:
it readily reduces to the case of a Borel subalgebra $\bb$ of $\frakg$.
It then suffices to only consider the maximal abelian ideals of $\bb$.
These were classified in \cite{Ro3} and then it was shown that the number of
$B$--orbits is finite in each of these instances.
Using the structure theory for spherical nilpotent orbits \cite{Pa4}
we found a short conceptual proof of this fact.
The finiteness result for the number of $P$--orbits on such an
abelian ideal $\aaa$
is a consequence of one of the main results in \cite{PR}:
namely that for $\aaa$ an
abelian ideal in $\bb$
and $\OO$ any nilpotent orbit in $\frakg$ meeting $\aaa$
the orbit $\OO$ is a spherical $G$--variety.
A $G$--variety is called {\em spherical},
whenever $B$ acts on it with an open dense orbit.
By a fundamental theorem,
due to M.\ Brion \cite{Br1} and E.B.\ Vinberg \cite{Vi2}
independently, $B$
acts on a spherical $G$--variety with a finite number of orbits.
Besides presenting this conceptual proof of the finiteness result,
I shall exhibit a natural connection between abelian ideals of $\bb$ and
$\ZZ$--gradings of $\frakg$. In particular, all the maximal abelian
ideals of $\bb$ stem from certain $\ZZ$--gradings of $\frakg$.
Further, I shall also address the set of
maximal abelian ideals $\AAA_{max}$ of $\bb$.
I shall present the classification of
$\AAA_{max}$ from \cite{Ro3} and will explain the existence of a
canonical bijection between $\AAA_{max}$ and the set of long simple roots
of $\frakg$ and discuss some properties of this map.
\begin{thebibliography}{99}
\bibitem{Br1}
{\sc M.~Brion},
{\em Quelques propri\'et\'es des espaces homog\'enes sph\'eriques},
Man.\ Math.\ {\bf 99} (1986), 191--198.
\bibitem{Pa4}
D.~Panyushev,
{\em On spherical nilpotent orbits and beyond},
Annales de l'Institut Fourier, {\bf 49}(5) (1999), 1453--1476.
\bibitem{PR}
{\sc D.~Panyushev, G.R\"ohrle},
{\em Spherical orbits and abelian Ideals},
Preprint 00--052 SFB 343, Diskrete Strukturen in der Mathematik,
Universit\"at Bielefeld (2000).
\bibitem{Ro3}
{\sc G.~R\"ohrle},
{\em On Normal Abelian Subgroups of Parabolic groups},
Annales de l'Institut Fourier, {\bf 48}(5), (1998), 1455--1482.
\bibitem{Vi2}
{\sc E.\,B.~Vinberg},
{\em Complexity of actions of reductive groups},
Funkt. Analiz i Prilozhen. {\bf 20}(1986), {\rus N0}\,1, 1--13 (Russian).
English translation: Funct.\ Anal.\ Appl.\ {\bf 20}, (1986), 1--11.
\end{thebibliography}
\end{document}