Memoirs of the American Mathematical Society 2008; 90 pp; softcover Volume: 193 ISBN10: 0821841319 ISBN13: 9780821841310 List Price: US$64 Individual Members: US$38.40 Institutional Members: US$51.20 Order Code: MEMO/193/903
 This paper studies quantum invariant differential operators for quantum symmetric spaces in the maximally split case. The main results are quantum versions of theorems of HarishChandra and Helgason: There is a HarishChandra map which induces an isomorphism between the ring of quantum invariant differential operators and the ring of invariants of a certain Laurent polynomial ring under an action of the restricted Weyl group. Moreover, the image of the center under this map is the entire invariant ring if and only if the underlying irreducible symmetric pair is not of four exceptional types. In the process, the author finds a particularly nice basis for the quantum invariant differential operators that provides a new interpretation of difference operators associated to Macdonald polynomials. Table of Contents  Introduction
 Background and notation
 A comparison of two root systems
 Twisted Weyl group actions
 The HarishChandra map
 Quantum radial components
 The image of the center
 Finding invariant elements
 Symmetric pairs related to type AII
 Four exceptional cases
 Appendix: Commonly used notation
 Bibliography
