We give sharp criteria for when a reductive group scheme satisfies Tannakian reconstruction. When the base is Noetherian, we explicitly identify its Tannaka scheme.

The set of strata a reductive group can be viewed as an enlargement the unipotent classes. In this paper notion distinguished class is extended to larger set. Weyl are introduced and studied.

Let G be a complex connected reductive group which is defined over R, let G be its Lie algebra, and T the variety of maximal tori of G. For ξ ∈ G(R), let Tξ be the variety of tori in T whose Lie algebra is orthogonal to ξ with respect to the Killing form. We show, using the Fourier–Sato transform of conical sheaves on real vector bundles, that the “weighted Euler characteristic” (see below) of ...

Introduction. Let k be a field of characteristic not two and G a connected linear reductive k-group. By a k-involution θ of G, we mean a k-automorphism θ of G of order two. For k = R, C or an algebraically closed field, such involutions have been extensively studied emerging from different interests. As manifested in [8, 18, 28], the interactions with the representation theory of reductive grou...

If G is a Lie group, H ⊂ G is a closed subgroup, and τ is a unitary representation of H, then the authors give a sufficient condition on ξ ∈ ig∗ to be in the wave front set of IndH τ . In the special case where τ is the trivial representation, this result was conjectured by Howe. If G is a reductive Lie group of Harish-Chandra class and π is a unitary representation of G that is weakly containe...

Each infinitesimally faithful representation of a reductive complex connected algebraic group G induces a dominant morphism Φ from the group to its Lie algebra g by orthogonal projection in the endomorphism ring of the representation space. The map Φ identifies the field Q(G) of rational functions on G with an algebraic extension of the field Q(g) of rational functions on g. For the spin repres...

Consider the character of an irreducible admissible representation of a p-adic reductive group. The Harish-Chandra-Howe local expansion expresses this character near a semisimple element as a linear combination of Fourier transforms of nilpotent orbital integrals. Under mild hypotheses, we describe an explicit region on which the local character expansion is valid. We assume neither that the gr...