Last update: 11 September 2013
Let be a finite dimensional module for a semisimple Lie algebra or its enveloping algebra By Weyl's complete reduciblity theorem we know that decomposes as a direct sum of irreducible modules with certain multiplicities denotes the set of such that appears in the decomposition of Let be the centralizer of the action of on i.e. The following theorem is the basic result of the double centralizer theory.
is a bimodule for and where is an irreducible module for
This decomposition induces a pairing between the irreducible modules of and irreducible modules It is true that the set of is a complete set of irreducible modules of The pairing between irreducible modules and the irreducible modules appearing as factors in can be given in terms of characters. By definition, a character of is a linear functional such that for all Let be the vector space of characters of Given a the character of corresponding to is given by defining to be the trace of the action of on Let denote the irreducible character of corresponding to the irreducible For each let denote the corresponding Weyl character for Let be the vector space which is the span of the Define the characteristic map to be the map given by
For each define the space of to be the vector space Let be the subalgebra of generated by the decomposes as a direct sum of weight spaces under the action of Since is a subalgebra of each is a module. The bicharacter of is defined to be where denotes the character of as a In view of the decomposition in (4.1) we also have
Construction of irreducible modules for centralizer algebras
Consider the action of the generators on the weight spaces of For each Let denote the kernel of the map acting on Let
|1)||is either or an irreducible module. Furthermore all irreducible modules can be obtained in this fashion.|
Using (4.1) we have that But the only weight in killed by all is So there are no elements of in the unless When we have Thus as
A character formula for
The character of the irreducible can be given by where is the character of as a
Equating the expressions (4.3) and (4.4) and rewriting by using (2.8) we have Substitute to get Compare coefficients of for on each side of this equation. Since we know by (2.3) that is not an element of for any except the identity. Thus
The dimension of the irreducible is given by where is the dimension of the weight space
Evaluate the identity in Theorem (4.6) at the identity of
This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.