Table Of Contentde Gruyter Expositions in Mathematics 26
Editors
Ο. H. Kegel, Albert-Ludwigs-Universität, Freiburg
V. P. Maslov, Academy of Sciences, Moscow
W. D. Neumann, The University of Melbourne, Parkville
R.O.Wells, Jr., Rice University, Houston
de Gruyter Expositions in Mathematics
1 The Analytical and Topological Theory of Semigroups, Κ. H. Hofmann,
J. D. Lawson, J. S. Pym (Eds.)
2 Combinatorial Homotopy and 4-Dimensional Complexes, H. J. Baues
3 The Stefan Problem, A. M. Meirmanov
4 Finite Soluble Groups, K. Doerk, T. O. Hawkes
5 The Riemann Zeta-Function, A. A. Karatsuba, S. M. Voronin
6 Contact Geometry and Linear Differential Equations, V. E. Nazaikinskii,
V. E. Shatalov, B. Yu. Stemin
7 Infinite Dimensional Lie Superalgebras, Yu. A. Bahturin, A. A. Mikhalev,
V. M. Petrogradsky, Μ. V. Zaicev
8 Nilpotent Groups and their Automorphisms, Ε. I. Khukhro
9 Invariant Distances and Metrics in Complex Analysis, M. Jarnicki, P. Pflug
10 The Link Invariants of the Chern-Simons Field Theory, E. Guadagnini
11 Global Affine Differential Geometry of Hypersurfaces, A.-M. Li, U. Simon,
G. Zhao
12 Moduli Spaces of Abelian Surfaces: Compactification, Degenerations, and
Theta Functions, K. Hulek, C. Kahn, S. H. Weintraub
13 Elliptic Problems in Domains with Piecewise Smooth Boundaries, S. A.
Nazarov, B. A. Plamenevsky
14 Subgroup Lattices of Groups, R. Schmidt
15 Orthogonal Decompositions and Integral Lattices, A. I. Kostrikin, PH.
Tiep
16 The Adjunction Theory of Complex Projective Varieties, M. C. Beltrametti,
A. J. Sommese
17 The Restricted 3-Body Problem: Plane Periodic Orbits, A. D. Bruno
18 Unitary Representation Theory of Exponential Lie Groups, H. Leptin,
J. Ludwig
19 Blow-up in Quasilinear Parabolic Equations, A.A. Samarskii, V.A.
Galaktionov, S. P. Kurdyumov, A. P. Mikhailov
20 Semigroups in Algebra, Geometry and Analysis, Κ. H. Hofmann, J. D.
Lawson, Ε. B. Vinberg (Eds.)
21 Compact Projective Planes, H. Salzmann, D. Betten, Τ. Grundhöf er, Η. Hühl,
R. Löwen, M. Stroppel
22 An Introduction to Lorentz Surfaces, Τ. Weinstein
23 Lectures in Real Geometry, F. Broglia (Ed.)
24 Evolution Equations and Lagrangian Coordinates, A. M. Meirmanov, V. V.
Pukhnachov, S. I. Shmarev
25 Character Theory of Finite Groups, B. Huppert
Positivity in Lie Theory:
Open Problems
Editors
Joachim Hilgert
Jimmie D. Lawson
Karl-Hermann Neeb
Ernest B.Vinberg
W
DE
_G
Walter de Gruyter · Berlin · New York 1998
Editors
Joachim Hilgert Jimmie D. Lawson Karl-Hermann Neeb Ernest Β. Vinberg
Institut für Mathematik Dept. of Mathematics Mathematisches Institut Chair of Algebra
Technische Universität Louisiana State University Universität Erlangen- Dept. of Mechanics
Clausthal Baton Rouge, LA 70803 Nürnberg & Mathematics
38678 Clausthal-Zellerfeld USA 91054 Erlangen Moscow State University
Germany Germany 119899 Moscow, Russia
1991 Mathematics Subject Classification:
17Bxx, 17Cxx, 20Gxx, 20Mxx, 22Exx, 32Exx, 43-XX, 52-XX, 60Bxx, 93-XX
Keywords:
Causality, compression semigroups, conformal geometry, control systems, discrete series represen-
tations, exponential function, harmonic analysis, holomorphic representations, invariant cones,
Jordan algebras, Lie algebras, Lie semigroups, linear monoids, one-parameter groups of probabi-
lity measures, reductive groups, reproducing kernels, singular representations, spherical functions,
Stein manifolds, symmetric spaces, total positivity, tube domains
© Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.
Library of Congress Cataloging-in-Publication Data
Positivity in Lie theory : open problems / editors, J. Hilgert . .. [et al.].
p. cm. — (De Gruyter expositions in mathematics,
ISSN 0938-6572 ; 26)
Includes index.
ISBN 3-11-016112-5 (alk. paper)
1. Lie groups. I. Hilgert, Joachim. II. Series.
QA387.P68 1998
512'.55-dc21 98-6433
CIP
Die Deutsche Bibliothek — Cataloging-in-Publication Data
Positivity in Lie theory : open problems / ed. J. Hilgert ...
- Berlin ; New York : de Gruyter, 1998
(De Gruyter expositions in mathematics ; 26)
ISBN 3-11-016112-5
© Copyright 1998 by Walter de Gruyter GmbH & Co., D-10785 Berlin.
All rights reserved, including those of translation into foreign languages. No part of this book
may be reproduced or transmitted in any form or by any means, electronic or mechanical,
including photocopy, recording, or any information storage or retrieval system, without permission
in writing from the publisher.
Typeset using the authors' TßX files: I. Zimmermann, Freiburg
Printing: Arthur Collignon GmbH, Berlin. Binding: Lüderitz & Bauer GmbH, Berlin.
Cover design: Thomas Bonnie, Hamburg.
Preface
In December 1996 the Mathematical Research Institute at Oberwolfach hosted a con-
ference on Positivity in Lie Theory in which many of the main contributors to this
rapidly developing area discussed recent results and interesting open problems. A
characteristic feature of the field is that notions of positivity in Lie theory occur in
quite diverse settings, are motivated by a wide variety of problems and applications,
and are approached from quite varying mathematical viewpoints. While this diversity
is attractive for the specialists who work on these problems, it is often difficult for
newcomers to the field to see the relation between the various aspects and to pick the
right problems. The participants of the Oberwolfach meeting decided to put together a
collection of problems with commentary that would serve as an invitation and a guide
to the field. The result of this joint effort is the present book.
In each chapter the reader is introduced to a specific open problem or circle of
problems that the author considers important for further development. The level of
presentation is chosen in such a way that a graduate student with a sound knowledge of
basic Lie theory should be able to grasp the gist of the problem. The main definitions
are explained, preliminary results are quoted, and the relevant references are listed.
Moreover, the authors often tried to formulate smaller problems that might serve as
guidelines to the main problems. To make sure that this book does not loose its value
too quickly if certain problems get solved, a website has been set up under
http://lie.math.tu-clausthal.de/~Hilgert/Problembook
where known solutions will be announced and preprints can be made available.
The scope of the field. We briefly describe the scope of the field we call "Positivity
in Lie Theory". In chronogeometry one considers causal structures which consist of a
continuous choice of one half of the double cone of timelike vectors on a Lorentzian
manifold. The chosen cone is the positive or forward light cone. Then a central
problem of chronogeometry is to decide which pairs of points can be connected by
causal curves, i.e., curves with derivatives in the forward light cone.
In geometric control theory one models the states of a system by the points of
a differentiable manifold and the controls by a set of (control) vector fields on the
manifold. A typical problem for the theory is the problem of controllability, where
one wants to describe the set of states of a system reachable in positive time from
some fixed state with a given set of controls. Such questions lead to a manifold of
states and a set of (control) vector fields on the manifold. Many important features of
the control vector fields are encoded in the convex cones spanned by the vector fields
in the tangent spaces. In this way one again arrives at the concept of a general cone
field or "causal structure" on a manifold and the associated causal order.
vi Preface
If a group acts differentiably on a manifold one asks for invariant causal structures
and considers the subset of the group leaving invariant the causal future. In this way
one finds the "future" semigroup, which can be viewed as a "dynamical polysystem,"
a control theoretic analog of the one-parameter semigroups describing a dynamical
system. Thus semigroups play an important role in positivity questions related to
causal and control structures.
Following the basic strategy of Lie theory one studies causal structures via their in-
finitesimal objects. This leads to the study of convex cones with invariance conditions
in Lie algebras or more general representation spaces. It turns out that invariant cones
in Lie algebras also play an important role in representation theory where they show
up in dissipativity conditions on the spectrum of the operators representing the Lie
algebra. This connection also leads to complex analysis via the possibility of finding
holomorphic extensions of unitary representations and at the same time yields tools
to study certain Stein domains.
In the study of general homogeneous spaces another manifestation of positivity
appears in the form of "compression" semigroups associated to distinguished open
subsets associated with various classes of homogeneous manifolds. Here to any subset
of the manifold one considers the semigroup of transformations carrying this set into
itself. Important examples that occur naturally in questions of control theory, harmonic
analysis and complex geometry can be found in flag manifolds.
Important classes of compression semigroups arise in complex analysis where
one studies actions of real groups G on complex manifolds Μ by biholomorphic
mappings. In many important cases, such as the action of the automorphism group of
a bounded symmetric domain, the action of the real group extends to a holomorphic
action of a certain complex semigroup S by integrating the corresponding flows also
for positive imaginary time. The holomorphic action of the complex semigroup S and
its holomorphic representation theory now can be used to study the complex geometry
of Μ with respect to the action of G. This correspondence ties up complex geometry
and representation theory in a quite subtle way.
The classical positivity concepts for matrices have been generalized in various
ways. In harmonic analysis and representation theory one considers positive definite
functions and kernels. On the other hand the notion of a totally positive matrix has
been generalized from GL(n, M) to arbitrary real reductive groups.
Finally a generalization of a toric variety, the Zariski closures of a linear algebraic
group, which again turns out to be a semigroup, has been considered. Semigroups
of this type are of a quite different character, however, than those associated with an
infinitesimal cone of generators.
These examples show that the link between the topics treated under the heading
of "positivity" in Lie theory is the presence of orderings at the level of manifolds,
semigroups at the level of groups, and cones at the level of vector spaces and Lie
algebras.
Preface vii
Recent developments. The problems described in this book primarily reflect recent
developments. The intensive work during the eighties on the structure of subsemi-
groups of Lie groups as well as semigroup closures of linear algebraic groups laid the
foundation for more specialized and application-oriented research conducted in recent
years. New results on structure theory of Lie semigroups and causal spaces are now
usually motivated by and obtained in the context of either control theory, harmonic
analysis or representation theory.
Harmonic analysis. A major undertaking at the moment is the study of various
types of Hardy spaces associated to symmetric spaces with invariant causal struc-
tures. It has its origins in the "Gelfand-Gindikin Program" launched in 1977 with
the goal of developing a new approach to the Plancherel Theorem for semisimple Lie
groups by grouping irreducible representations together into geometrically constructed
"chunks." The way to construct these geometric pieces is to consider G-invariant do-
mains in the complexification of G and boundary values of holomorphic functions (or
more general objects). Hardy spaces for causal symmetric spaces live on (curved) tube
domains modelled on invariant cones, and yield the so called holomorphic discrete
series. They are a first successful attempt to give content to the Gelfand-Gindikin
Program for semisimple symmetric spaces. It is an open problem (even in the group
case) to construct cohomological Hardy spaces for the "non-convex" tube domains.
New light is shed on this problem through the use of Jordan algebras. It opens the
way to compare the "curved" Hardy spaces with classical families of Hardy spaces on
"linear" tube domains. The contributions of W. BERTRAM, J. FARAUT, S. GlNDIKIN
and G. 0LAFSSON all deal with different aspects of this circle of questions. Whereas
FARAUT and 0LAFSSON concentrate more on the harmonic analysis of causal sym-
metric spaces in which convex cones play a central role, W. BERTRAM explains the
setting in which one can transcend the causal situation using Jordan algebra techniques
involving also non-convex cones and generalized conformal structures. GlNDIKIN ex-
posits his philosophy of generalized conformal structures in the context of complex
analysis, in particular, Stein spaces.
Another very active field is the study of spherical functions on ordered symmetric
spaces. Originally motivated by the study of integral operators in scattering theory
respecting causality, it has developed into a mature theory parallel to Harish-Chandra's
treatment of Riemannian symmetric spaces. Very interesting new phenomena which
have no parallel in the Riemannian setting are emerging in the case of non-reductive
causal symmetric spaces. These aspects are also touched upon in the articles of
FARAUT and 0LAFSSON.
Representation theory. A central question in the representation theory of semisimple
groups is the construction of the inner products for Harish-Chandra modules, known
to be unitarizable. Positivity methods are a powerful tool to attack this question for
the class of highest weight representations. In the special case of representations with
scalar lowest ÄT-type they already yield a construction of the Hilbert space structure
based on Kirillov's orbit method. It is to be expected that they also give new insights in
viii Preface
the general case. Problems related to the construction of Hilbert spaces carrying singu-
lar unitary representations are contained in the chapters written by V. F. MOLCHANOV
and Yu. NERETIN.
Representation theory of Lie semigroups plays an important role in the study of
Hardy spaces, spherical functions and unitarizable highest weight representations,
but these examples do not exhaust its potential. A key feature of the holomorphic
representation theory of complex semigroups is that it can be viewed as a technique
extending the method of Fourier expansion - a central tool in the realm of compact
groups - by more general desintegration methods available for representations of
complex semigroups which are based on highest weight representations. So far these
techniques are well developed for representations by bounded operators on Hilbert
spaces. The contribution of K.-H. NEEB describes a circle of ideas aiming at an
extension of this theory to certain unbounded representations on more general Banach
spaces. An important step in this direction is a better understanding of spaces of
holomorphic functions on complex semigroups and in particular a description of a
good convolution algebra of holomorphic functions.
The articles of S. GLNDIKIN and Yu. NERETIN deal with the converse problem of
using complex analytic methods to obtain interesting representations of real groups.
These two aspects of the theory illustrate quite well that there are many bridges between
complex analysis and representation theory that should be explored in the future.
These bridges are most clearly visible for group actions on complexifications of causal
symmetric spaces which often serve as model cases.
Convex cones in Lie algebras which are invariant under the adjoint action have
played an important role in the development of Lie semigroup theory and its appli-
cations to representation theory and harmonic analysis. Later one also considered
convex cones in the tangent space of a symmetric space invariant under the isotropy
representation of the stabilizer group. In both these cases one has a fairly complete
picture. Much less is known for the general case of convex cones left invariant by
an arbitrary linear group action. J. HILGERT and K.-H. NEEB give an overview what
is known in general and present a series of suggestions for attacking the problem of
classifying invariant cones in real representations other than the adjoint representation.
Control theory. Concepts and methods from Lie theory play an important role in
various aspects of geometric control theory. Typically in this approach to control
theory one models the states of a system as points on a smooth manifold and the
controls of the system as vector fields on the manifold. This set of controls may be
viewed as a "dynamical polysystem," and in studying such questions as reachability
and controllability, one is interested in the "forward semigroup" generated by finite
compositions of positive time phase transitions from the flows corresponding to the
given vector fields. This semigroup is in general infinite dimensional, but for a number
of interesting problems arising from mechanics or other problems demonstrating a
high degree of symmetry, the semigroup in question is both finite dimensional and
an infinitesimally generated subsemigroup of a Lie group. In this case the problem
at hand can often be pulled back to an equivalent problem on the Lie group, and
Preface ix
the methods of Lie group and Lie semigroup theory can be applied. The most typical
class of problems arising in this context concerns the so-called bilinear control systems,
which can be interpreted as control systems on matrix groups in which the vector fields
in question are right-invariant. These systems and related problems are discussed in
the contribution of F. COLONIUS, W. KLIEMANN and L. SAN MARTIN.
For studying reachable sets, investigators have found the notion of the Lie saturate
of a set of vector fields useful (where the Lie saturate is the largest collection of vector
fields having the same closed reachable sets). These Lie saturates are closely connected
to Lie wedges, the appropriate tangent objects of Lie semigroups in the Lie algebra in
the finite dimensional setting. The invariant cone fields on coset spaces of Lie groups,
which are a prevalent aspect of positivity in Lie groups, are an important special case
of the typical setting postulated in the non-smooth approach to control theory. The
Pontryagin Maximum Principle has sharpened and simplified versions for the setting
of Lie groups, which makes sharper results available and also sometimes facilitates
calculations. Conversely the concepts and methods of geometric control theory have
proved to be useful tools in the development of the Lie theory of semigroups.
We mention some important examples of this interaction between geometric con-
trol theory and Lie semigroup theory. Questions of controllability on a Lie group are
closely related to a knowledge of the maximal subsemigroups of the Lie group, since
a subset of the Lie algebra of a group G viewed as a dynamical polysystem of right
invariant vector fields on G is controllable if and only if it is not contained in the
Lie tangent wedge (the appropriate tangential object in the Lie algebra) of a maximal
subsemigroup. Substantial progress has been made in understanding the maximal
subsemigroups, although they have only been characterized in the solvable case. In
the case of semisimple Lie groups with finite center the maximal subsemigroups arise
as compression semigroups of certain open domains in the flag manifolds of the Lie
group. These open domains can be typically located in a "big cell" associated to a
Bruhat decomposition of the Lie group and hence pulled back to an open domain in
one of the nilpotent algebras associated with the Bruhat decomposition. The article of
D. MlTTENHUBER considers the construction of compression semigroups from such
a viewpoint.
The asymptotic behavior of a constant linear dynamical system is determined by
the spectrum of the coefficient matrix. As one moves to the study of analogous prob-
lems in time-varying and control systems, generalizations of these ideas are needed.
Recent developments and open problems in this direction are discussed in the arti-
cle of F. COLONIUS, W. KLIEMANN, and L. SAN ΜΑΚΉΝ. Instead of one-parameter
semigroups, one now has multi-parameter semigroups to consider. Varying notions of
spectrum and the theory of Lyapunov exponents replace the classical matrix spectrum.
The eigenspaces are replaced by control sets, maximal sets on which the semigroup
(almost) acts transitively.
Various other aspects. A classical problem in probability theory is the embedding
of a probability measure on a locally compact group into a one-parameter semigroup
of probability measures. As is explained in M. McCRUDDEN's contribution the last
χ Preface
years have seen impressive progress on this problem, but the problem is also a source
of rather subtle Lie theoretic questions. A related question is the identification of those
elements in a connected Lie group which lie on a one-parameter group. These are
precisely the elements in the image of the exponential mapping. Major results and open
problems related to the image of the exponential map are presented by D. ©OKOVIC
and Κ. H. HOFMANN.
The theory of algebraic semigroups has been developed since 1980. It weds the
theories of abstract semigroups and algebraic groups and contributes to both of them.
Reductive algebraic semigroups have been studied most extensively. Analogues of
Cartan subgroups, the Weyl group, the Bruhat decomposition, and the Tits building
have been introduced for them. Recently a complete classification of reductive alge-
braic semigroups was obtained. At the same time some important questions about
their structure and representations remain to be answered. Making use of reductive
algebraic semigroups, the so-called wonderful completion of a semisimple algebraic
group was obtained in a most natural way. The asymptotic semigroup of a semisimple
algebraic group was defined. In a sense it reflects the behavior of the group at infinity
in a manner analogous to the way its Lie algebra reflects its infinitesimal behavior. The
semigroup analogue of the Weyl group gives rise to a generalization of the Hecke al-
gebra, an interesting combinatorial object. The article of M. PUTCHA and L. RENNER
discusses some open problems in this area.
One of the earliest examples of a Lie semigroup was the semigroup of totally
positive matrices in GL(n, K). In his contribution G. LUSZTIG gives a generalization of
total positivity from GL(n, R) to arbitrary real reductive groups; the resulting elements
again form a Lie semigroup. An unexpected connection arising in the treatment is the
use of canonical bases and perverse sheaves in the study of these semigroups. The
contribution explains this connection and points out a number of problems arising in
this context.
In spite of the extensive work on foundational questions there remain a number
of important questions unanswered. J. D. LAWSON's article describes the problems
arising from attempts to find the right framework for Lie semigroups which are not
necessarily subsemigroups of Lie groups and the efforts to understand semigroups
which are freely generated by a local Lie semigroup.
Acknowlegdments. We thank the Mathematische Forschungsinstitut Oberwolfach
for providing the logistics for the conference on which the content of this book draws.
To the editors of the series de Gruyter Expositions in Mathematics and to the math-
ematics editor of the de Gruyter Verlag, MANFRED KARBE, we express our gratitude
for publishing this collection in the usual superb form.
The editors are grateful to KARL HEINRICH HOFMANN to whom the field owes so
much and who is always willing to share his experience.
