to Group Actions in Symplectic and Complex Geometry.- I. Finite-dimensional manifolds.- II. Elements of Lie groups and their actions.- III. Manifolds with additional structure.- IV. Symplectic manifolds with symmetry.- V. Kählerian structures on coadjoint orbits of compact groups and associated representations.- Literature.- Infinite-dimensional Groups and their Representations.- I. Calculus in locally convex spaces.- II. Dual spaces of locally convex spaces.- III. Topologies on function spaces.- IV. Representations of infinite-dimensional groups.- V. Generalized coherent state representations.- References.- Borel-Weil Theory for Loop Groups.- I. Compact groups.- II. Loop groups and their central extensions.- III. Root decompositions.- IV. Representations of loop groups.- V. Representations of involutive semigroups.- VI. Borel-Weil theory.- VII. Consequences for general representations.- References.- Coadjoint Representation of Virasoro-type Lie Algebras and Differential Operators on Tensor-densities.- I. Coadjoint representation of Virasoro group and Sturm-Liouville operators; Schwarzian derivative as a 1-cocycle.- II. Projectively invariant version of the Gelfand-Fuchs cocycle and of the Schwarzian derivative.- III. Kirillov's method of Lie superalgebras.- IV. Invariants of coadjoint representation of the Virasoro group.- V. Extension of the Lie algebra of first order linear differential operators on S1 and matrix analogue of the Sturm-Liouville operator.- VI. Geometrical definition of the Gelfand-Dickey bracket and the relation to the Moyal-Weil star-product.- References.- From Group Actions to Determinant Bundles Using (Heat-kernel) Renormalization Techniques.- I. Renormalization techniques.- II. The first Chern form on a class of hermitian vector bundles.- III.The geometry of gauge orbits.- IV. The geometry of determinant bundles.- V. An example: the action of diffeomorphisms on complex structures.- References.- Fermionic Second Quantization and the Geometry of the Restricted Grassmannian.- I. Fermionic second quantization.- II. Bogoliubov transformations and the Schwinger term.- III. The restricted Grassmannian of a polarized Hilbert space.- IV. The non-equivariant moment map of the restricted Grassmannian.- V. The determinant line bundle on the restricted Grassmannian.- References.

Infinite dimensional manifolds, Lie groups and algebras arise naturally in many areas of mathematics and physics. Having been used mainly as a tool for the study of finite dimensional objects, the emphasis has changed and they are now frequently studied for their own independent interest. On the one hand this is a collection of closely related articles on infinite dimensional Kähler manifolds and associated group actions which grew out of a DMV-Seminar on the same subject. On the other hand it covers significantly more ground than was possible during the seminar in Oberwolfach and is in a certain sense intended as a systematic approach which ranges from the foundations of the subject to recent developments. It should be accessible to doctoral students and as well researchers coming from a wide range of areas. The initial chapters are devoted to a rather selfcontained introduction to group actions on complex and symplectic manifolds and to Borel-Weil theory in finite dimensions. These are followed by a treatment of the basics of infinite dimensional Lie groups, their actions and their representations. Finally, a number of more specialized and advanced topics are discussed, e.g., Borel-Weil theory for loop groups, aspects of the Virasoro algebra, (gauge) group actions and determinant bundles, and second quantization and the geometry of the infinite dimensional Grassmann manifold.