PERFECTING GROUP SCHEMES

Algebraic groups play much the same role for algebraists as Lie groups play for analysts. This book is the first comprehensive introduction to the theory of algebraic group schemes over fields that includes the structure theory of semisimple algebraic groups, and is written in the language of modern algebraic geometry. The first eight chapters study general algebraic group schemes over a field and culminate in a proof of the Barsotti–Chevalley theorem, realizing every algebraic group as an extension of an abelian variety by an affine group. After a review of the Tannakian philosophy, the author provides short accounts of Lie algebras and finite group schemes. The later chapters treat reductive algebraic groups over arbitrary fields, including the Borel–Chevalley structure theory. Solvable algebraic groups are studied in detail. Prerequisites have also been kept to a minimum so that the book is accessible to non-specialists in algebraic geometry.

It is obvious that the topological closure of an orbit of an algebraic group acting algebraically on a projective manifold over tE is an algebraic set containing the orbit as a Zariski open set. This article treats the above situation when the group is a connected reductive complex Lie group acting holomorphically on a compact Kaehler manifold. Recall (cf. § II below) that a connected complex reductive Lie group, G, has the structure of a linear algebraic group, and this algebraic structure is compatible with the underlying analytic structure. Let G be any projective manifold in which G is Zariski open and which induces the above algebraic structure on G. A complex connected Lie group G is said to act projeetively on a compact Kaehler manifold X if G acts holomorphically on X and the Lie algebra of holomorphic vector-fields that G generates on X is annihilated by every holomorphic one form on X. This definition is justified in § II. Note (cf. §III) that G acts projectively on X if either H~(X, Q)=0, or G is semi-simple, or if every generator of the solvable radical of G has a fixed point on X, or if G is linear algebraic acting algebraically on a projective X. The main result of the paper (cf. § II) where G is as above for reduct i~ G is: Proposition. Let G be a complex connected reduetive Lie group acting projeetively on a compact Kaehler X. Let q~: Y ~ X be a hotomorphic map where Y is a normal reduced complex space. Consider the equivariant map q~:G× Y ~ X , ~ extends meromorphically (in the sense of Remmert) to CJ × Y. Taking Y to be a point one gets the analog of the result mentioned in the opening sentence. Another simple corollary is the classical result that the linear algebraic structure chosen on G which makes it algebraic is unique; and in fact that any reductive connected subgroup of a linear algebraic group over ~ is an algebraic subgroup. As a further application of the techniques used a new proof of an improved form of a fixed point theorem (cf. [20]) of the author is given: Proposition. Let S be a complex solvable Lie group acting holomorphically on a compact Kaehler manifold X. The following are equivalent:

A combinatorial classification of 2-regular simple modules for Nakayama algebras

Rational Cherednik algebras of G(ℓ,p,n) from the Coulomb perspective

Introduction .220 Chapter I. A fixed point theorem for finite diffeomorphism groups generated by reflections 221 Chapter II. Geometric weight system and a fixed point theorem for compact connected Lie groups.... 231 1. Actions of compact simple Lie groups of rank 1.232 2. Basic properties of the geometric weight system.237 3. A fixed point theorem for compact connected Lie groups.240 Chapter III. Classification of the types of principal isotropy subgroups for actions of compact connected Lie groups on acyclic manifolds .242 1. Classification of principal orbit types for actions of simple compact connected Lie groups on acyclic manifolds .243 2. Classification of principal orbit types for actions of general compact connected Lie groups .247 References ...... 255

For the centrally extended Heisenberg double of SL2, its center is determined, the central factor algebras are described, and classifications of simple Harish-Chandra modules, simple Whittaker modules, and simple quasi-Whittaker modules are obtained. Two classes of simple weight modules with infinite-dimensional weight spaces are given. We also give a classification of simple modules that decompose into a direct sum of simple finite-dimensional sl2-modules with finite multiplicities.

This is a semi–expository update and rewrite of my 1974 AMS AMS Memoir describing Plancherel formulae and partial Dolbeault cohomology realizations for standard tempered representations for general real reductive Lie groups. Even after so many years, much of that Memoir is up to date, but of course there have been a number of refinements, advances and new developments, most of which have applied to smaller classes of real reductive Lie groups. Here we rewrite that AMS Memoir in in view of these advances and indicate the ties with some of the more recent (or at least less classical) approaches to geometric realization of unitary representations.

Discrete Series Representations for the Orbit Spaces Arising from Two Involutions of Real Reductive Lie Groups

Multiplicity One Theorems for Generalized Gelfand-Graev Representations of Semisimple Lie Groups and Whittaker Models for the Discrete Series

This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω‎ on M, by averaging all the left translates of ω‎ over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.

Classification of irreducible weight modules over higher rank Virasoro algebras

We give a definition of polynomial identities for affine group schemes; over a field F of characteristic 0, the identities of an affine group scheme turn out to coincide with the identities of a certain matrix algebra over F. We prove the identities of an algebraic affine group scheme are those of a matrix algebra whose size is related to the rank of the representative algebra of the group scheme. Moreover, we relate the existence of a polynomial identity of an affine group scheme with the existence of an abelian subgroupscheme of finite index, giving a Passman type result for affine group schemes.

Let G be a Lie group acting on a manifold X and preserving a G-invariant measure. Abstract harmonic analysis seeks to understand the action of G on X by understanding the unitary representation of G on L 2 (X). This may be done by "decomposing" L2 (X) (in an appropriate sense) into irreducible unitary representations. In order to do this, it is useful to have in hand a family of irreducible unitary representations; not necessarily exhaustive, but large enough to solve a range of interesting harmonic analysis problems. The Kirillov-Kostant philosophy of coadjoint orbits seeks to provide such a family. The purpose of these notes is to describe what is known about implementing that philosophy, particularly for reductive groups.

We introduce the combinatorial notion of a q-factorization graph intended as a tool to study and express results related to the classification of prime simple modules for quantum affine algebras. These are directed graphs equipped with three decorations: a coloring and a weight map on vertices, and an exponent map on arrows (the exponent map can be seen as a weight map on arrows). Such graphs do not contain oriented cycles and, hence, the set of arrows induces a partial order on the set of vertices. In this first paper on the topic, beside setting the theoretical base of the concept, we establish several criteria for deciding whether or not a tensor product of two simple modules is a highest- $\ell $ -weight module and use such criteria to prove, for type A, that a simple module whose q-factorization graph has a totally ordered vertex set is prime.

We continue the study of prime simple modules for quantum affine algebras from the perspective of q-fatorization graphs. In this paper we establish several properties related to simple modules whose q-factorization graphs are afforded by trees. The two most important of them are proved for type A. The first completes the classification of the prime simple modules with three q-factors by giving a precise criterion for the primality of a 3-vertex line which is not totally ordered. Using a very special case of this criterion, we then show that a simple module whose q-factorization graph is afforded by an arbitrary tree is real. Indeed, the proof of the latter works for all types, provided the aforementioned special case is settled in general.