- Algebraische Lie Theorie
- Algebraische Gruppen
- Endliche Gruppen vom Lie-Typ
- DFG project: Serre's notion of Complete Reducibility and Geometric Invariant Theory (within the
DFG Priority Programme in Representation Theory).
- DFG project: Computational aspects of the Cohomology of Coxeter arrangements: On Conjectures of Lehrer-Solomon and Felder-Veselov (within the
DFG Priority Programme "Algorithmic and experimental methods in Algebra,
Geometry and Number Theory").
- DFG project: Arrangements of complex reflection groups: Geometry and combinatorics (joint with M. Cuntz (Kaiserslautern) within the
DFG Priority Programme "Algorithmic and experimental methods in Algebra, Geometry and Number Theory").
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Supersolvable restrictions of reflection arrangements
with N. Amend and T. Hoge
Let A = (A,V) be a complex hyperplane arrangement and let L(A) denote its intersection lattice. The arrangement A is called supersolvable, provided its lattice L(A) is supersolvable. For X in L(A), it is known that the restriction A^X is supersolvable provided A is. Suppose that W is a finite, unitary reflection group acting on the complex vector space V. Let A(W) = (A(W), V) be its associated hyperplane arrangement. In earlier work by the last two authors, we classified all supersolvable reflection arrangements. Extending this work, the aim of this note is to determine all supersolvable restrictions of reflection arrangements. It turns out that apart from the obvious restrictions of supersolvable reflection arrangements there are only a few additional instances. Moreover, in our previous work, we classified all inductively free restrictions A(W)^X of reflection arrangements A(W). Since every supersolvable arrangement is inductively free, the supersolvable restrictions A(W)^X of reflection arrangements A(W) form a natural subclass of the class of inductively free restrictions A(W)^X. Finally, we characterize the irreducible supersolvable restrictions of reflection arrangements by the presence of modular elements of dimension 1 in their intersection lattice. This in turn shows that reflection arrangements as well as their restrictions are of fiber type if and only if they are strictly linearly fibered.
On inductively free Restrictions of Reflection Arrangements
with N. Amend and T. Hoge
Let W be a finite complex reflection group acting on the complex vector space V and let A(W) = (A(W), V) be the associated reflection arrangement. In an earlier paper by the last two authors, we classified all inductively free reflection arrangements A(W). The aim of this note is to extend this work by determining all inductively free restrictions of reflection arrangements.
Spherical subgroups in simple algebraic groups
with F. Knop
Let G be a simple algebraic group. A closed subgroup H of G is called spherical provided it has a dense orbit on the flag variety G/B of G. Reductive spherical subgroups of simple Lie groups were classified by Kr\"amer in 1979. In 1997, Brundan showed that each example from Kr\"amer's list also gives rise to a spherical subgroup in the corresponding simple algebraic group in any positive characteristic. Nevertheless, there is no classification of all such instances in positive characteristic to date. The goal of this paper is to complete this classification. It turns out that there is only one additional instance (up to isogeny) in characteristic 2 which has no counterpart in Kr\"amer's classification.
Equivariant K-theory of generalized Steinberg varieties
with J. M. Douglass
We describe the equivariant K-groups of a family of generalized Steinberg varieties that interpolates between the Steinberg variety of a reductive, complex algebraic group and its nilpotent cone in terms of the extended affine Hecke algebra and double cosets in the extended affine Weyl group. As an application, we use this description to define Kazhdan-Lusztig "bar" involutions and Kazhdan-Lusztig bases for these equivariant K-groups.
Former and current PhD students:
Simon Goodwin (University of Birmingham)
Michael Bate (University of York)
Russell Fowler (Npower)
Glenn Ubly (NHS)
Sebastian Herpel (University of Kaiserslautern)