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