A NOTE ON WEYL GROUPS AND THE QUANTUM TRANSLATION PRINCIPLE FOR ROOT SYSTEMS OF CLASSICAL TYPE

ABSTRACT

Let U be the quantum analog of the universal enveloping algebra of a Lie algebra. In [4] Joseph and Letzter used a quantum translation principle to prove that certain categories of highest weight modules and Harish-Chandra bimodules are equivalent. The proof depends on certain properties of strongly dominant weights under the action of the Weyl group. In this note we investigate what happens when the weight under consideration is just dominant, not strongly dominant. When the root system is of classical type, it turns out that the translation principle no longer holds, but the equivalence of categories is nevertheless still true.

ACKNOWLEDGMENT

I thank the referee for the careful reading of the manuscript.