Orbits of parabolic subgroups on metabelian ideals

We consider the action of a parabolic subgroup of the General Linear Group on a metabelian ideal. For those actions, we classify actions with finitely many orbits using methods from representation theory.Comment: 10 pages, 6 eps figure

Inductive freeness of Ziegler’s canonical multiderivations for reflection arrangements

Let A be a free hyperplane arrangement. In 1989, Ziegler showed that the restriction A 00 of A to any hyperplane endowed with the natural multiplicity is then a free multiarrangement. We initiate a study of the stronger freeness property of inductive freeness for these canonical free multiarrangements and investigate them for the underlying class of re ection arrangements. More precisely, let A = A (W) be the re ection arrangement of a complex re ection group W. By work of Terao, each such re ection arrangement is free. Thus so is Ziegler's canonical multiplicity on the restriction A 00 of A to a hyperplane. We show that the latter is inductively free as a multiarrangement if and only if A 00 itself is inductively free

On the K(π,1)-problem for restrictions of complex reflection arrangements

Let W⊂GL(V) be a complex reflection group and A(W) the set of the mirrors of the complex reflections in W. It is known that the complement X(A(W)) of the reflection arrangement A(W) is a K(π,1) space. For Y an intersection of hyperplanes in A(W), let X(A(W)Y) be the complement in Y of the hyperplanes in A(W) not containing Y. We hope that X(A(W)Y) is always a K(π,1). We prove it in case of the monomial groups W=G(r,p,ℓ). Using known results, we then show that there remain only three irreducible complex reflection groups, leading to just eight such induced arrangements for which this K(π,1) property remains to be proved

Edifices : Building-like spaces associated to linear algebraic groups

Acknowledgements: We are grateful to Bernhard Mühlherr for his encouragement and for helpful conversations. We thank the editors of this special volume in honour of Jacques Tits for inviting us to contribute, and for their forbearance during the manuscript’s slow gestation. The second author was supported by a VIP grant from the Ruhr-Universität Bochum. Some of this work was completed during visits to the Mathematisches Forschungsinstitut Oberwolfach: we thank them for their support. We are also indebted to the referees for their careful reading of the paper and for many suggestions making various arguments more transparent.Peer reviewe

On reflection subgroups of finite Coxeter groups

Let W be a finite Coxeter group. We classify the reflection subgroups of W up to conjugacy and give necessary and sufficient conditions for the map that assigns to a reflection subgroup R of W the conjugacy class of its Coxeter elements to be injective, up to conjugacy

Cocharacter-closure and spherical buildings

Let k be a field, let G be a reductive k-group and V an affine k-variety on which G acts. In this note we continue our study of the notion of cocharacter-closed G(k)-orbits in V . In earlier work we used a rationality condition on the point stabilizer of a G-orbit to prove Galois ascent/descent and Levi ascent/descent results concerning cocharacter-closure for the corresponding G(k)-orbit in V . In the present paper we employ building-theoretic techniques to derive analogous results

Coxeter arrangements and Solomon's descent algebra

In our recent paper [3], we claimed that both the group algebra of a finite Coxeter group W as well as the Orlik-Solomon algebra of W can be decomposed into a sum of induced one-dimensional representations of centralizers, one for each conjugacy class of elements of W, and gave a uniform proof of this claim for symmetric groups. In this note we outline an inductive approach to our conjecture. As an application of this method, we prove the inductive version of the conjecture for nite Coxeter groups of rank up to 2