  These are the notes from a series of lectures the author gave at Harvard
University in the Fall of 1994. The goal of these lectures is to give a
self-contained exposition of recent result of Cherednik (\cite{C6}), who proved
Macdonald's inner product conjectures (see \cite{M2}) for arbitrary root
systems, using the algebraic structure he called ``double affine Hecke
algebra''.

Quantum Algebra;Classical Analysis and ODEs