These volumes contain papers based on lectures presented at the conference, 'Algebraic Groups and Their Generalizations', held at Pennsylvania State University in July 1991. An outgrowth of the remarkable proliferation of Lie theory in the last fifteen years, this conference reflected both the diversification of technique in the classical theory and the beginnings of the study of new objects. These new objects include quantum groups and vertex operator algebras, as well as various kinds of infinite-dimensional groups and algebras inspired by new work in mathematical physics and quantum field theory. The first volume focuses on classical methods, while the second centers on quantum and infinite-dimensional methods. Each section begins with expositions and then turns to new results. This collection provides readers with an excellent introduction to these astonishing new mathematical worlds.
Part 1. Classical methods: Sur les decompositions cellulaires des espaces $G/B$ by C. Chevalley Introduction to middle intersection cohomology and perverse sheaves by A. Borel The Bruhat graph of a Coxeter group, a conjecture of Deodhar, and rational smoothness of Schubert varieties by J. B. Carrell Simulating perverse sheaves in modular representation theory by E. Cline, B. Parshall, and L. Scott A brief survey of Kazhdan-Lusztig theory and related topics by V. Deodhar Green theory for Hecke algebras and Harish-Chandra philosophy by R. Dipper Liftable deformations and Hecke algebras by M. Schaps A Hecke algebra of the symmetric group by E. A. Siegel Character sheaves: Applications to finite groups by B. Srinivasan Real algebraic quotients by R. J. Bremigan Frobenius splitting of spherical varieties by M. Brion and S. P. Inamdar Generalized Kloosterman sums by R. Dabrowski Symmetric $k$-varieties by A. G. Helminck Identities for prounipotent groups by A. R. Magid On the structure of nonreduced parabolic subgroup-schemes by C. Wenzel Weight modules without highest weight by A. J. Coleman Extremal composition factors for groups of Lie type by J. E. Humphreys Relative invariants of the polynomial rings over the finite and tame type quivers by K. Kazuhiko Hilbert series for modules of covariants by B. Broer The first fundamental theorem of invariant theory and spherical subgroups by R. E. Howe Algebraic families of $O(2)$-actions on affine space $\mathbf C^4$ by M. Masuda and T. Petrie Algebraic equivariant vector bundles and the linearity problem by L. Moser-Jauslin Equivariant matrix valued functions by G. F. Seelinger Constructive invariant theory by D. L. Wehlau Part 2. Quantum and infinite-dimensional methods: Finite dimensional representations of quantum groups by H. H. Andersen An introduction to quantum groups by P. Cartier Examples of compact matrix pseudogroups arising from Drinfeld's twisting operation by B. Enriquez Face algebras and their Drinfeld doubles by T. Hayashi Representation theory for quantized enveloping algebras by G. Letzter Rational representations of Hopf algebras by Z. Lin Filtrations of modules over the quantum algebra by J. Paradowski Quantum groups as invariance groups by A. Sudbery The quantum hyperalgebra of $SL q(2)$ by M. Takeuchi IC bases and quantum linear groups by J. Du Bases for quantum Demazure modules. II by V. Lakshmibai Problems on canonical bases by G. Lusztig $2$-categories and Zamolodchikov tetrahedra equations by M. M. Kapranov and V. A. Voevodsky Abelian intertwining algebras--A generalization of vertex operator algebras by C. Dong and J. Lepowsky Discrete series of the Virasoro algebra and the moonshine module by C. Dong, G. Mason, and Y. Zhu Constructions of vertex operator algebras by A. J. Feingold Binary trees and finite-dimensional Lie algebras by Y.-Z. Huang Holomorphic line bundles over Hilbert flag varieties by A. G. Helminck and G. F. Helminck New classes of infinite-dimensional Lie groups by L. Natarajan, E. Rodriguez-Carrington, and J. A. Wolf On forms of Kac-Moody algebras by G. Rousseau Semi-infinite cohomology of Lie algebras by A. A. Voronov.