All the papers or books I read so far on quantum groups about Lusztig’s
restricted specialization consider only primitive roots of unity of odd order
and additional conditions. In most cases, it is claimed
that these restrictions could be removed without too much harm but details
are not given. So I have tried to do the calculation myself. Below is what
I find for the finite dimensional representations of .
Note that most authors consider the case odd. Then
and we can even restrict the study
to the representations for which is always 1. Indeed, all these
assumptions make the expression below much simpler.
We consider a simple Lie algebra of rank and
use standard notations for roots, for
coroots and for the weight space.
Let be an integer and a primitive -root of the
unity. We let if is even and otherwise. For
all , define ,
, and . We assume that
no is a multiple of , which is obviously true for large enough.
We denote the restricted specialization as defined in
Chari and Pressley’s guide to quantum groups and keep their notations for
, , and
of . Finally, we let
be a finite -module.