Suppose $G$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$F,$ $P=MN$ is a maximal parabolic subgroup of $G,$ $frak{n}$ is‎ ‎the Lie algebra of the unipotent radical $N.$ Under the adjoint‎ ‎action of its stabilizer in $M,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

The aim of this note is to prove that the parabolic closure of any subset of a Coxeter group is a parabolic subgroup. To obtain that, several technical lemmas on the root system of a parabolic subgroup are established. MSC 2000 Subject Classifications: Primary 20F55; Secondary 20H15

Let (W,S) be a Coxeter system, and let X be a subset of S. The subgroup of W generated by X is denoted by WX and is called a parabolic subgroup. We give the precise definition of the commensurator of a subgroup in a group. In particular, the commensurator of WX in W is the subgroup of w in W such that wWXw ∩WX has finite index in both WX and wWXw . The subgroup WX can be decomposed in the form ...

The unitary group of the hyperbolic hermitian space of dimension two over a quaternion division algebra over a number field is a non-quasisplit inner form of Sp(4), and does not have a parabolic subgroup corresponding to the Klingen parabolic subgroup. However, it has CAP representations with respect to the Klingen parabolic subgroup. We construct them by using the theta lifting from the unitar...

In this paper, we show that the center of every Coxeter group is finite and isomorphic to (Z2) n for some n ≥ 0. Moreover, for a Coxeter system (W, S), we prove that Z(W ) = Z(W S\S̃) and Z(W S̃ ) = 1, where Z(W ) is the center of the Coxeter group W and S̃ is the subset of S such that the parabolic subgroup W S̃ is the essential parabolic subgroup of (W, S) (i.e. W S̃ is the minimum parabolic subgr...

for a composition $lambda$ of $n$ our aim is to obtain reduced forms ‎for all the elements in the kazhdan-lusztig (right) cell containing ‎$w_{j(lambda)}$‎, ‎the longest element of the standard parabolic‎ ‎subgroup of $s_n$ corresponding to $lambda$‎. ‎we investigate how far this is possible to achieve by looking at‎ ‎elements of the form $w_{j(lambda)}d$‎, ‎where $d$ is a prefix of ‎an element...

A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion...

suppose $g$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$f,$ $p=mn$ is a maximal parabolic subgroup of $g,$ $frak{n}$ is‎ ‎the lie algebra of the unipotent radical $n.$ under the adjoint‎ ‎action of its stabilizer in $m,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

group GC defined over Q whose connected component G 0 Q has no rational character. It is also necessary to suppose that the centralizer of a maximal Q split torus of G0C meets every component of GC. The reduction theory of Borel applies, with trivial modifications, to G; it will be convenient to assume that Γ has a fundamental set with only one cusp. Fix a minimal parabolic subgroup P 0 C defin...