Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over finite fields. We generalize Lusztig’s results to reductive groups over arbitrary finite local rings. This generalization uses the Greenberg functor and the theor...

We compute the characters of many supercuspidal representations of reductive p-adic groups. Specifically, we deal with representations that arise via Yu’s construction from data satisfying a certain compactness condition. Each character is expressed in terms of a depth-zero character of a smaller group, the (linear) characters appearing in Yu’s construction, Fourier transforms of orbital integr...

For a fairly general reductive group G/Qp , we explicitly compute the space of locally algebraic vectors in the Breuil–Herzig construction (ρ)ord , for a potentially semistableBorel-valued representationρ ofGal(Q̄p/Qp). The point beingwedealwith thewhole representation, not just its socle—and we go beyond GLn(Qp). In the case of GL2(Qp), this relation is one of the key properties of the p-adic l...

Let F be any field. Let G be any reductive linear algebraic group and consider a finite dimensional rational representation V of G. Then the Falgebra F[V ] of polynomial invariants for G acting on V is finitely generated. The Noether Number β(G,V ) is the highest degree of an element of a minimal homogeneous generating set for F[V ]. We survey what is known about Noether Numbers, in particular ...

In this paper we generalize some constructions and results due to Cayley and Hilbert. We define the concept of Ω–process for an arbitrary algebraic monoid with zero and unit group G. Then we show how to produce from the process and for a linear rational representation of G, a number of elements of the ring of G-invariants, that is large enough as to guarantee its finite generation. Moreover, we...

The definition of index goes back to Dixmier [3, 11.1.6]. This notion is important in Representation Theory and also in Invariant Theory. By Rosenlicht’s theorem [12], generic orbits of an arbitrary action of a linear algebraic group on an irreducible algebraic variety are separated by rational invariants; in particular, ind g = tr.degK(g). The index of a reductive algebra equals its rank. Comp...

The action of a reductive algebraic group G on G/P−, where P− is a parabolic subgroup, differentiates to a representation of the Lie algebra g of G by vector fields on U+, the unipotent radical of a parabolic opposite to P−. We show that both the regular functions on U+ and the polynomial vector fields on U+ form g-modules that are dual to parabolically induced modules, construct an explicit co...

Let Z = G/Q be a complex flag manifold and let Go be a real form of G. Then the representation theory of the real reductive Lie group Go is intimately connected with the geometry of Go-orbits on Z. The open orbits correspond to the discrete series representations and their analytic continuations, closed orbits correspond to the principal series, and certain other orbits give the other series of...

Let π be an irreducible representation of a real reductive group G. The global character θπ of π may be considered as a function on the regular semisimple elements of G. The global character determines π, and it is of great interest to compute it. For example Harish-Chandra found the discrete series representations of G by computing their characters. Fix a Cartan subgroup H of G and let D be th...

Vitamin B(12), reduced by titanium (III) citrate to vitamin B(12s), catalyzes the reductive dechlorination of chlorophenols. Reductive dechlorination of pentachlorophenol and of all tetrachlorophenol and trichlorophenol isomers was observed. Reaction of various chlorophenols with vitamin B(12) favored reductive dechlorination at positions adjacent to another chlorinated carbon, but chlorines or...