for nonnegative integers nβα. Each root β corresponds to a unipotent subgroup Uβ of G, whose Lie algebra is gβ. There is an isomorphism of the additive Lie groupGa ∼= C with Uβ; this is T -equivariant, with multiplication by β(t) on C corresponding to conjugation by t on Uβ (u 7→ tut −1). The product of the groups Uβ for β ∈ R+ forms a unipotent group U , isomorphic to C , with N = #R+, and B =...

Using a class sum and a collection of related Radon transforms, we present a proof G. James’s Kernel Intersection Theorem for the complex unipotent representations of the finite general linear groups. The approach is analogous to that used by F. Scarabotti for a proof of James’s Kernel Intersection Theorem for the symmetric group. In the process, we also show that a single class sum may be used...

My research focuses on topics related to unipotent classes and Springer theory for algebraic groups. These concepts are intimately related to the representation theory of Weyl groups. I have recently also become interested in complex reflection groups, which can be viewed as generalizations of Weyl groups. Combinatorial methods and algorithms have a long history as a means for studying all of t...

1.1. Let V be a complete discrete valuation ring with perfect residue field of characteristic p > 0 and fraction field K of characteristic 0. Fix an algebraic closure K ⊂ K. Let X/V be a smooth proper scheme, and let D ⊂ X be a divisor with normal crossings over V . Assume that XK is connected. Let X ◦ denote X −D and assume given a point x ∈ X◦(V ). We can then consider the pro-unipotent funda...

This paper provides new, relatively simple, proofs of some important results about unipotent classes in simple linear algebraic groups. We derive the formula for the Jordan blocks of the Richardson class of a parabolic subgroup of a classical group. This result was originally due to Spaltenstein. Secondly, we derive the description of the natural partial order of unipotent classes in the genera...

Suppose $G$ is a split connected‎ ‎reductive orthogonal or symplectic group over an infinite field‎ ‎$F,$ $P=MN$ is a maximal parabolic subgroup of $G,$ $frak{n}$ is‎ ‎the Lie algebra of the unipotent radical $N.$ Under the adjoint‎ ‎action of its stabilizer in $M,$ every maximal prehomogeneous‎ ‎subspaces of $frak{n}$ is determined‎.

0.1. For any finite group Γ, a “nonabelian Fourier transform matrix” was introduced in [L1]. This is a square matrix whose rows and columns are indexed by pairs formed by an element of Γ and an irreducible representation of the centralizer of that element (both defined up to conjugation). As shown in [L2], this matrix, which is unitary with square 1, enters (for suitable Γ) in the character for...

Abstract In this paper, we study the generalized order- Jacobsthal sequences modulo for and the generalized order-k Jacobsthal-Padovan sequence modulo for . Also, we define the generalized order-k Jacobsthal orbit of a k-generator group for and the generalized order-k Jacobsthal-Padovan orbit a k-generator group for . Furthermore, we obtain the lengths of the periods of the generalized order-3 ...