Frechet completions of moderate growth old and (somewhat) new results

This article has two objectives. The first is to give a guide to the proof of the (so-called) Casselman-Wallach theorem as it appears in Real Reductive Groups II. The emphasis will be on one aspect of the original proof that leads to the new result in this paper which is the second objective. We show how a theorem of van der Noort combined with a clarification of the original argument in my book lead to a theorem with parameters (an alternative is one announced by Berstein and Krotz). This result gives a new proof of the meromorphic continulation of the smooth Eisenstein series.

A quantum polylog algorithm for non-normal maximal cyclic hidden subgroups in the affine group of a finite field

We give an algorithm to solve the quantum hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group of a finite field (if the field has order $q$ then the group has order $q(q-1)$) with probability $1-varepsilon$ with (polylog) complexity $O(log(q)^{R}log(varepsilon)^{2})$ where $R<infty.$

Action of the conformal group on steady state solutions to Maxwells equations and background radiation

The representation of the conformal group (PSU(2,2)) on the space of solutions to Maxwells equations on the conformal compactification of Minkowski space is shown to break up into four irreducible unitarizable smooth Frechet representations of moderate growth. An explicit inner product is defined on each representation. The frequency spectrum of each of these representations is analyzed. These representations have notable properties; in particular they have positive or negative energy, they are of type $A_{frak q}(lambda)$ and are quaternionic. Physical implications of the results are explained.

Bessel Models for General Admissible Induced Representations: The Compact Stabilizer Case

A holomorphic continuation of Jacquet type integrals for parabolic subgroups with abelian nilradical is studied. Complete results are given for generic characters with compact stabilizer and arbitrary representations induced from admissible representations. A description of all of the pertinent examples is given. These results give a complete description of the Bessel models corresponding to compact stabilizer.

On the algebraic set of singular elements in a complex simple Lie algebra

Let $G$ be a complex simple Lie group and let $g = hbox{rm Lie},G$. Let $S(g)$ be the $G$-module of polynomial functions on $g$ and let $hbox{rm Sing},g$ be the closed algebraic cone of singular elements in $g$. Let ${cal L}s S(g)$ be the (graded) ideal defining $hbox{rm Sing},g$ and let $2r$ be the dimension of a $G$-orbit of a regular element in $g$. Then ${cal L}^k = 0$ for any $k<r$. On the other hand, there exists a remarkable $G$-module $Ms {cal L}^r$ which already defines $hbox{rm Sing},g$. The main results of this paper are a determination of the structure of $M$.