For a connected reductive group G and a finite–dimensional G–module V , we study the invariant Hilbert scheme that parameterizes closed G–stable subschemes of V affording a fixed, multiplicity–finite representation of G in their coordinate ring. We construct an action on this invariant Hilbert scheme of a maximal torus T of G, together with an open T–stable subscheme admitting a good quotient. ...

For any finite, real reflection group W , we construct a geometric basis for the homology of the corresponding non-crossing partition lattice. We relate this to the basis for the homology of the corresponding intersection lattice introduced by Björner and Wachs in [4] using a general construction of a generic affine hyperplane for the central hyperplane arrangement defined by W .

We evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements, and mention results for products of skew-primitive elements. Examples include quantum groups associated to semisimple and abelian Lie algebras, and degenerate affine Hecke algebras of reductive type with trivial parameter.

This paper finishes the classification of primitive affine distance-transitive graphs by dealing with the only case left open, namely where the generalized Fitting subgroup of the stabilizer of a vertex is a finite quasisimple group of classical Lie type defined over the characteristic dividing the number of vertices of the graph. All graphs that are found to occur are known.

This paper introduces calibrated representations for affine Hecke algebras and classifies and constructs all finite-dimensional irreducible calibrated representations. The primary technique is to provide indexing sets for controlling the weight space structure of finite-dimensional modules for the affine Hecke algebra. Using these indexing sets we show that (1) irreducible calibrated representa...

Many different divisible designs are already known. Some of them possess remarkable automorphism groups, so called dual translation groups. The existence of such an automorphism group enables us to characterize its associated divisible design as being isomorphic to a substructure of a finite affine space. AMS Classification: 05B05, 05B30, 20B25, 51N10

Let G be a compact definable C group, X a compact affine definable CG manifold and 2 ≤ r < ∞. We prove that the set of equivariant definable Morse functions on X whose loci are finite unions of nondegenerate critical orbits is open and dense in the set of G invariant definable C functions with respect to the definable C topology.

We introduce two-parameter quantum toroidal algebras of simply laced types and provide their group theoretic realization using finite subgroups of SL2(C) via McKay correspondence. In particular our construction contains a realization of the vertex representation of the two-parameter quantum affine algebras of ADE types.

We provide simple characterizations of short-braid avoiding and fully commutative elements in an affine Weyl group W , generalizing results of Fan and Stembridge for finite Weyl groups. Our results rely on the combinatorics of the compatible subsets of the root system of W .