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Invariant Differential Operators for Quantum Symmetric Spaces
Gail Letzter, Viriginia Polytechnic Institute and State University, Blacksburg, VA

Memoirs of the American Mathematical Society
2008; 90 pp; softcover
Volume: 193
ISBN-10: 0-8218-4131-9
ISBN-13: 978-0-8218-4131-0
List Price: US$64
Individual Members: US$38.40
Institutional Members: US$51.20
Order Code: MEMO/193/903
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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 Harish-Chandra and Helgason: There is a Harish-Chandra 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 Harish-Chandra 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
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