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Register a new userCoadjoint orbits of reductive type of seaweed Lie algebras
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A connected algebraic group Q defined over a field of characteristic zero is quasi-reductive if there is an element of its dual of reductive type, that is such that the quotient of its stabiliser by the centre of Q is a reductive subgroup of GL(q), where q=Lie(Q). Due to results of M. Duflo, coadjoint representation of a quasi-reductive Q possesses a so called maximal reductive stabiliser and knowing this subgroup, defined up to a conjugation in Q, one can describe all coadjoint orbits of reductive type. In this paper, we consider quasi-reductive parabolic subalgebras of simple complex Lie algebras as well as all seaweed subalgebras of gl(n) and describe the classes of their maximal reductive stabilisers.
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We study the coadjoint orbits of a Lie algebra in terms of Cartan class. In fact, the tangent space to a coadjoint orbit $mathcal{O}(alpha)$ at the point $alpha$ corresponds to the characteristic space associated to the left invariant form;$alpha$ and its dimension is the even part of the Cartan class of $alpha$. We apply this remark to determine Lie algebras such that all the nontrivial orbits (nonreduced to a point) have the same dimension, in particular when this dimension is 2 or 4. We determine also the Lie algebras of dimension $2n$ or $2n+1$ having an orbit of dimension $2n$.
We provide a micro-local necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let $bf G$ be a complex algebraic reductive group, and $bf Hsubset G$ be a spherical algebraic subgroup. Let $mathfrak{g},mathfrak{h}$ denote the Lie algebras of $bf G$ and $bf H$, and let $mathfrak{h}^{bot}$ denote the annihilator of $mathfrak{h}$ in $mathfrak{g}^*$. A $mathfrak{g}$-module is called $mathfrak{h}$-distinguished if it admits a non-zero $mathfrak{h}$-invariant functional. We show that the maximal $bf G$-orbit in the annihilator variety of any irreducible $mathfrak{h}$-distinguished $mathfrak{g}$-module intersects $mathfrak{h}^{bot}$. This generalizes a result of Vogan. We apply this to Casselman-Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that as suggested by Prasad, if $H$ is a symmetric subgroup of a real reductive group $G$, the existence of a tempered $H$-distinguished representation of $G$ implies the existence of a generic $H$-distinguished representation of $G$. Many models studied in the theory of automorphic forms involve an additive character on the unipotent radical of $bf H$, and we devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of W-algebras. As an application we derive necessary conditions for the existence of Rankin-Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan-Gross-Prasad conjectures for non-generic representations. We also prove more general results that ease the sphericity assumption on the subgroup, and apply them to local theta correspondence in type II and to degenerate Whittaker models.
same as arXiv:0904.1778
For a finite dimensional complex Lie algebra, its index is the minimal dimension of stabilizers for the coadjoint action. A famous conjecture due to Elashvili says that the index of the centralizer of an element of a reductive Lie algebra is equal to the rank. That conjecture caught attention of several Lie theorists for years. In this paper we give an almost general proof of that conjecture.
Let $G$ be a connected reductive algebraic group defined over an algebraically closed field $mathbbm k$ of characteristic zero. We consider the commuting variety $mathcal C(mathfrak u)$ of the nilradical $mathfrak u$ of the Lie algebra $mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $mathfrak u$ with only a finite number of orbits, we verify that $mathcal C(mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {em distinguished} $B$-orbits in $mathfrak u$. We observe that in general $mathcal C(mathfrak u)$ is not equidimensional, and determine the irreducible components of $mathcal C(mathfrak u)$ in the minimal cases where there are infinitely many $B$-orbits in $mathfrak u$.
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