SINGULAR VECTORS OF QUANTUM GROUP-REPRESENTATIONS FOR STRAIGHT LIE-ALGEBRA ROOTS

We give explicit formulae for singular vectors of Verma modules over U(q)(G), where G is any complex simple Lie algebra. The vectors we present correspond exhaustively to a class of positive roots of G which we call straight roots. In some special cases, we give singular vectors corresponding to arbitrary positive roots. For our vectors we use a special basis of U(q)(G-), where G- is the negative roots subalgebra of G, which was introducted in our earlier work in the case q = 1. This basis seems more economical than the Poincare-Birkhoff-Witt type of basis used by Malikov, Feigin, and Fuchs for the construction of singular vectors of Verma modules in the case q = 1. Furthermore, this basis turns out to be part of a general basis recently introduced for other reasons by Lusztig for U(q)(B-), where B- is a Borel subalgebra of G.