In December 1996 the Mathematical Research Institute at Oberwolfach hosted a conference 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 guidelines to the main problems.
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.
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. In this way 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 infinitesimal 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 non-compact 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.
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 the general linear group 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.
{ last modified : Oct 30 1997 }