G. I. Lehrer

Reflection Subquotients of Unitary Reflection Groups

Let G be a finite group generated by (pseudo-) reflections in a complex vector space and let g be any linear transformation which normalises G. In an earlier paper, the authors showed how to associate with any maximal eigenspace of an element of the coset gG, a subquotient of G which acts as a reflection group on the eigenspace. In this work, we address the questions of irreducibility and the coexponents of this subquotient, as well as centralisers in G of certain elements of the coset. A criterion is also given in terms of the invariant degrees of G for an integer to be regular for G. A key tool is the investigation of extensions of invariant vector fields on the eigenspace, which leads to some results and questions concerning the geometry of intersections of invariant hypersurfaces.

The Brauer category and invariant theory

A category of Brauer diagrams, analogous to Turaevs tangle category, is introduced, and a presentation of the category is given; specifically, we prove that seven relations among its four generating homomorphisms suffice to deduce all equations among the morphisms. Full tensor functors are constructed from this category to the category of tensor representations of the orthogonal group

Reflection subgroups of finite and affine Weyl groups

O(V)

Hodge-Deligne equivariant polynomials and monodromy of hyperplane arrangements

or the symplectic group

A toral configuration space and regular semisimple conjugacy classes

Sp(V)

The Two-Step Nilpotent Representations of the Extended Affine Hecke Algebra of Type A

over any field of characteristic zero. The first and second fundamental theorems of invariant theory for these classical groups are generalised to the category theoretic setting. The major outcome is that we obtain new presentations for the endomorphism algebras of the module

On Gel'fand-Graev characters of reductive groups with disconnected centre.

V^{\otimes r}

The spherical building and regular semisimple elements

. These are obtained by appending to the standard presentation of the Brauer algebra of degree

A Quantum Analogue of the First Fundamental Theorem of Classical Invariant Theory

r

Homology stability for classical regular semisimple varieties

one additional relation. This relation stipulates the vanishing of an element of the Brauer algebra which is quasi-idempotent, and which we describe explicitly both in terms of diagrams and algebraically. In the symplectic case, if