Buildings are highly structured, geometric objects, primarily used in the finer study of the groups that act upon them. In Buildings and Classical Groups, the author develops the basic theory of buildings and BN-pairs, with a focus on the results needed to apply it to the representation theory of p-adic groups. In particular, he addresses spherical and affine buildings, and the "spherical building at infinity" attached to an affine building. He also covers in detail many otherwise apocryphal results. Classical matrix groups play a prominent role in this study, not only as vehicles to illustrate general results but as primary objects of interest. The author introduces and completely develops terminology and results relevant to classical groups. He also emphasizes the importance of the reflection, or Coxeter groups and develops from scratch everything about reflection groups needed for this study of buildings. In addressing the more elementary spherical constructions, the background pertaining to classical groups includes basic results about quadratic forms, alternating forms, and hermitian forms on vector spaces, plus a description of parabolic subgroups as stabilizers of flags of subspaces. The text then moves on to a detailed study of the subtler, less commonly treated affine case, where the background concerns p-adic numbers, more general discrete valuation rings, and lattices in vector spaces over ultrametric fields. Buildings and Classical Groups provides essential background material for specialists in several fields, particularly mathematicians interested in automorphic forms, representation theory, p-adic groups, number theory, algebraic groups, and Lie theory. No other available source provides such a complete and detailed treatment.

This book is an elementary self-contained introduction to some constructions of representation theory and related topics of differential geometry and analysis. Topics covered include the theory of various Fourier-like integral operators such as Segal-Bargmann transforms, Gaussian integral operators in $L^2$ and in the Fock space, integral operators with theta-kernels, the geometry of real and $p$-adic classical groups and symmetric spaces. The heart of the book is the Weil representation of the symplectic group (real and complex realizations, relations with theta-functions and modular forms, $p$-adic and adelic constructions) and representations in Hilbert spaces of holomorphic functions of several complex variables. This book is addressed to graduate students and researchers in representation theory, differential geometry, and operator theory. Prerequisites are standard university courses in linear algebra, functional analysis, and complex analysis.

Introduction to buildings and their geometries with emphasis on model theoretic constructions, covering recent developments.

These notes are a slightly revised and extended version of mim- graphed notes written on the occasion of a seminar on buildings and BN-pairs held at Oberwolfach in April 1968. Their main purpose is to present the solution of the following two problems: (A) Determination of the buildings of rank >; and irreducible, spherical type, other than ~ and H ("of spherical type" means "with finite Weyl 4 group", about the excluded types H, cf. the addenda on p. 274). Roughly speaking, those buildings all turn out to be associated to simple algebraic or classical groups (cf. 6. ;, 6. 1;, 8. 4. ;, 8. 22, 9. 1, 10. 2). An easy application provides the enumeration of all finite groups with BN-pairs of irreducible type and rank >;, up to normal subgroups contained in B (cf. 11. 7). (B) Determination of all isomorphisms between buildings of rank > 2 and spherical type associated to algebraic or classical simple groups and, in parti cular, description of the full automorphism groups of such buildings (cf. 5. 8, 5. 9, 5. 10, 6. 6, 6. 1;, 8. 6, 9. ;, 10. 4). Except for the appendices, the notes are rather strictly oriented - ward these goals.

This book is addressed to mathematicians and advanced students interested in buildings, groups and their interplay. Its first part introduces - presupposing good knowledge of ordinary buildings - the theory of twin buildings, discusses its group-theoretic background (twin BN-pairs), investigates geometric aspects of twin buildings and applies them to determine finiteness properties of certain S-arithmetic groups. This application depends on topological properties of some subcomplexes of spherical buildings. The background of this problem, some examples and the complete solution for all "sufficiently large" classical buildings are covered in detail in the second part of the book.

Buildings are combinatorial constructions successfully exploited to study groups of various types. The vertex set of a building can be naturally decomposed into subsets called Grassmannians. The book contains both classical and more recent results on Grassmannians of buildings of classical types. It gives a modern interpretation of some classical results from the geometry of linear groups. The presented methods are applied to some geometric constructions non-related to buildings Grassmannians of infinite-dimensional vector spaces and the sets of conjugate linear involutions. The book is self-contained and the requirement for the reader is a knowledge of basic algebra and graph theory. This makes it very suitable for use in a course for graduate students.