Contemporary Mathematics
Volume 325, 2003
Twisted Verma modules and their quantized analogues
Henning Haahr Andersen
1.
Introduction
In
[AL]
we studied twisted Verma modules for a finite dimensional semisim-
ple complex Lie algebra g. In fact, we gave three rather different constructions
which we showed lead to the same modules. Here we shall briefly 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 Uq(g).
In analogy with their classical counterparts the quantized twisted Verma mod-
ules belong to the category Oq for Uq(g) and have the same composition factors as
the ordinary Verma modules for Uq(g). They also possess Jantzen type filtrations
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
~
denote a Cartan subalgebra of g and choose a set R+ of positive
roots in the root system R attached to (g,
~).
Then we have the usual triangular
decomposition g
=
n-
EB~EBn+
of g with n+ (respectively n-) denoting the nilpotent
subalgebra corresponding to the positive (respectively negative) roots.
We set b
=
~
EB n+ and write U
=
U(g) and B
=
U(b) for the enveloping
algebras of g and
b.
Then the Verma module corresponding to A
E
~*
is defined as
where C. is the !-dimensional B-module obtained by composing A with the projec-
tion
b-+

Supported in part by the TMR programme "Algebraic Lie Representations" (ECM Network Con-
tract No. ERB FMRX-CT 97 /0100)
©
2003 American Mathematical Society
1
http://dx.doi.org/10.1090/conm/325/05661
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