The purpose of this article is to prove an explicit invariant trace formula. In the preceding paper [1 (j)], we studied two families of invariant distributions. Now we shall exhibit these distributions as terms on the two sides of the invariant trace formula. We refer the reader to the introduction of [l (j)], which contains a general discussion of the problem. In this introduction, we shall de...

The Helly dimension of a group is a purely group theoretic invariant (the name is motivated by the loose analogy with a well known theorem in convex geometry). Its version for algebraic groups is defined in terms of Zariski closed cosets. This quantity comes up naturally in the study of actions on product varieties. We prove that the Helly dimension of an algebraic group is finite, and give som...

An introduction of nonstandard analysis in purely algebraic terms is presented. As an application, we give a nonstandard proof of a characterization theorem for compact subsets of Sym(N).

We extend one exact sequence of Colliot-Thélène and Sansuc for tori over number fields to one for arbitrary connected groups. Introduction. Let G be a connected linear algebraic group defined over a number field k. Denote by A(G) = ∏

This is an expository article on the theory of algebraic stacks. After introducing the general theory, we concentrate in the example of the moduli stack of vector budles, giving a detailed comparison with the moduli scheme obtained via geometric invariant theory.

The Dirac quantization ‘procedure’ for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular interpretation of the Dirac procedure known as refined algebraic quantization. We find technical conditions under which refined algebraic quantization can reproduce the general implementation of the Dirac s...

Roughly, this class will be about studying one fundamental object in two different guises. Let X be an algebraic curve over a field k, and let G be some algebraic group over X . We assume for this lecture that G splits trivially as G0 ×X , but this is not necessary for what we will do in the course. The fundamental object we will study is the collection of G-bundles on X . We will consider two ...

We show that the algebraic intersection number of Scott and Swarup for splittings of free groups coincides with the geometric intersection number for the sphere complex of the connected sum of copies of S × S.

We start by introducing Kac-Moody algebras and completing the classification of finite dimensional semisimple Lie algebras. We then discuss the classification of finite dimensional representations of semisimple Lie algebras (and, more generally, integrable highest weight representations of Kac-Moody algebras). We finish by discussing the structure and representation theory of reductive algebrai...

We provide a general, unified, framework for external zonotopal algebra. The approach is critically based on employing simultaneously the two dual algebraic constructs and invokes the underlying matroidal and geometric structures in an essential way. This general theory makes zonotopal algebra an applicable tool for a larger class of polytopes.