The
Kac module is obtained from the bosonic representation
as a so-called
parabolic induction, a particular case of induced
representation. The induced representation of
is denoted
and defined as follows:
(8)
I have to explain the meaning of these notations.
by definition consists of polynomials
of the form 
(finite sum) where 
, with the identification
; this is the “universal
enveloping algebra”
means that we consider the tensor product
with identifications:
For example, if
then
; and
if
then
where
is the action
of
on
as
belongs to a representation
of
.
Notice that 
is naturally a
representation of 
. The action of 
on
is just a multiplication on the
left: 
For example, if