Publications (5)0 Total impact
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ABSTRACT: . Let k be an algebraically closed field and V a finite dimensional k- space. Let GL(V ) be the general linear group of V and P a parabolic subgroup of GL(V ). Now P acts on its unipotent radical Pu and on pu = Lie Pu , the Lie algebra of Pu , via the adjoint action. More generally, we consider the action of P on the l-th member of the descending central series of pu denoted by p (l) u . All instances when P acts on p (l) u for l 0 with a finite number of orbits are known. In this note we give a complete description of the closure relations among the P -orbits on p (l) u in all these finite cases. There is a canonical bijection between the set of P -orbits on p (l) u and the set F (Delta)(e) of isomorphism classes of Delta--filtered modules of a particular dimension vector e of a certain quasi-hereditary algebra A(t; l). These isomorphism classes in turn are given by the orbits of the reductive group G(e) = Q GL(e i ) on the variety R(Delta)(e) of all representations of...
05/1999;
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G. R Ohrle
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ABSTRACT: . Let G be a reductive algebraic group, P a parabolic subgroup of G with unipotent radical Pu , and A a closed connected unipotent subgroup of Pu which is normalized by P . We show that P acts on A with finitely many orbits provided A is abelian. This generalizes a well-known finiteness result, namely the case when A is central in Pu . Also, we obtain an analogous result for the adjoint action of P on invariant linear subspaces which are abelian as subalgebras of the Lie algebra of Pu under some mild characteristic restrictions. 1. Introduction Throughout, G denotes a (connected) reductive algebraic group defined over an algebraically closed field k of characteristic p 0 and P is a parabolic subgroup of G with unipotent radical P u . The aim of this note is the following result. Theorem 1.1. Let G be a reductive algebraic group, P a parabolic subgroup of G, and A a closed connected normal subgroup of P in P u . If A is abelian, then P has finitely many orbits on A. The part...
05/1999;
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ABSTRACT: . For an algebraic group R acting morphically on an algebraic variety X the modality of the action, mod(R : X), is the maximal number of parameters upon which a family of R-orbits on X depends. Let G be a reductive algebraic group defined over an algebraically closed field K. Let P be a parabolic subgroup of G. Then P acts on its unipotent radical Pu via conjugation and on pu , the Lie algebra of Pu , via the adjoint action. The modality of P is defined as modP := mod(P : pu ). In this paper we discuss an algorithm which is used to compute upper bounds for modP along with some results obtained by this algorithm. One is a classification of parabolic subgroups P of simple algebraic groups G of semisimple rank 2 and modality 0. For parabolic subgroups of semisimple rank 3 we present some partial results. This extends results from [11] and [14] where the cases of semisimple rank 0 and 1 are handled. For exceptional groups G we show that P ae G has modality zero provided the cl...
05/1999;
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ABSTRACT: Let G be a classical algebraic group defined over an algebraically closed field of characteristic zero. In case the simple factors of G are of type Ar , Br or Cr , all parabolic subgroups P of G with a finite number of orbits on the unipotent radical Pu are determined: this is the case precisely when the class of nilpotency of Pu is at most four. For groups of type Dr we obtain partial results, depending on the structure of P .
08/1998;
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02/1970;
