 
Summary: Twisted Verma modules and their quantized analogues
Henning Haahr Andersen
1. Introduction
In [AL] we studied twisted Verma modules for a nite dimensional semisim
ple complex Lie algebra g. In fact, we gave three rather dierent constructions
which we showed lead to the same modules. Here we shall brie
y recall one of
these approaches  the one based on Arkhipov's twisting functors [Ar]. We then
demonstrate that this construction can also be used for the quantized enveloping
algebra U q (g):
In analogy with their classical counterparts the quantized twisted Verma mod
ules belong to the category O q for U q (g) and have the same composition factors as
the ordinary Verma modules for U q (g): They also possess Jantzen type ltrations
with corresponding sum formulae.
I would like to thank Catharina Stroppel and Niels Lauritzen for some very
helpful comments.
2. The classical case
2.1. Let h denote a Cartan subalgebra of g and choose a set R + of positive
roots in the root system R attached to (g; h): Then we have the usual triangular
decomposition g = n hn + of g with n
+ (respectively n ) denoting the nilpotent
