0. Introduction 1. Affine Weyl groups (Reduction modulo W ) 2. Double Hecke algebras (Automorphisms, Demazure-Lusztig operators) 3. Macdonald polynomials (Intertwining operators) 4. Fourier transform on polynomials (Basic transforms) 5. Jackson integrals (Macdonald’s η-identities) 6. Semisimple representations (Main Theorem, GLn and other applications) 7. Spherical representations (Semisimple s...

We prove that every Lie algebra can be decomposed into a solvable Lie algebra and a semisimple Lie algebra. Then we show that every complex semisimple Lie algebra is a direct sum of simple Lie algebras. Finally, we give a complete classification of simple complex Lie algebras.

We prove an explicit formula for the invariant μ(g) for finite-dimensional semisimple, and reductive Lie algebras g over C. Here μ(g) is the minimal dimension of a faithful linear representation of g. The result can be used to study Dynkin’s classification of maximal reductive subalgebras of semisimple Lie algebras.

Suppose $G$ is a connected complex semisimple group and $W$ its Weyl group. The lifting of an element to semisimple. This induces well-defined map from the set elliptic conjugacy classes $G$. In this paper, we give uniform algorithm compute map. We also consider twisted case.

We classify semisimple rigid monoidal categories with two iso-morphism classes of simple objects over the field of complex numbers. In the appendix written by P. Etingof it is proved that the number of semisimple Hopf algebras with a given finite number of irreducible representations is finite.

In this note the notion of kernel of a representation of a semisimple Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H

1.1. Superrigidity. In the early seventies, Margulis proved his celebrated superrigidity theorem for irreducible lattices in semisimple Lie and algebraic groups of higher rank. One of the motivations for this result is that it implies arithmeticity : a complete classification of higher rank lattices. In the case where the semisimple group is not almost simple, superrigidity reads as follows (se...

We complete the quasi-isometric classification of irreducible lattices in semisimple Lie groups over nondiscrete locally compact fields of characteristic zero by showing that any quasi-isometry of a rank one S-arithmetic lattice in a semisimple Lie group over nondiscrete locally compact fields of characteristic zero is a finite distance in the sup-norm from a commensurator.