Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 11 September 2013

Centralizer algebras

Let M be a finite dimensional module for a semisimple Lie algebra or its enveloping algebra 𝔘. By Weyl's complete reduciblity theorem we know that M decomposes as a direct sum of irreducible 𝔘 modules Vλ, with certain multiplicities cλ, MλM^ cλVλ. M denotes the set of λP+ such that Vλ appears in the decomposition of M. Let C be the centralizer of the action of 𝔘 on M, i.e. C=End𝔘(M). The following theorem is the basic result of the double centralizer theory.

M is a bimodule for C×𝔘 and MλM^ CλVλ, where Cλ is an irreducible module for C.

This decomposition induces a pairing between the irreducible modules Cλ of C and irreducible 𝔘 modules Vλ, λM^. It is true that the set of Cλ, λM^, is a complete set of irreducible modules of C. The pairing between irreducible C modules and the irreducible 𝔘 modules appearing as factors in M can be given in terms of characters. By definition, a character of C is a linear functional ξ:C such that ξ(c1c2)= ξ(c2c1), for all c1,c2. Let C^ be the vector space of characters of C. Given a C-module M the character ξ of C corresponding to M is given by defining ξ(c), cC, to be the trace of the action of c on M. Let ξλ denote the irreducible character of C corresponding to the irreducible C-module Cλ. For each λP+ let χλ denote the corresponding Weyl character for 𝔘. Let ΛM^ be the vector space which is the span of the χλ, λM^. Define the characteristic map ch to be the map given by ch: C^ ΛM^ ξλ χλ

For each μ=iμiωiP define the μ-weight space of M to be the vector space Mμ= { mM|for each 1in,him =μim } . Let 𝔘(𝔥) be the subalgebra of 𝔘 generated by the hi. M decomposes as a direct sum of weight spaces under the action of 𝔘(𝔥), MμP Mμ. (4.2) Since 𝔘(𝔥) is a subalgebra of 𝔘 each Mμ is a C module. The bicharacter of M is defined to be bicharM=μP ημeμ, (4.3) where ημ denotes the character of Mμ as a C-module. In view of the decomposition in (4.1) we also have bicharM=λM^ ξλχλ. (4.4)

Construction of irreducible modules for centralizer algebras

Consider the action of the generators xi𝔘 on the weight spaces Mμ of M. For each μP xi:Mμ Mμ+αi. Let kerμ(xi) denote the kernel of the map xi acting on Mμ. Let Cμ= ikerμ (xi).

1) Cμ is either 0 or an irreducible C module. Furthermore all irreducible C modules can be obtained in this fashion.
2) Cμ0μM^.


Using (4.1) we have that Mμ=λM^Cλ(Vλ)μ. But the only weight in Vλ killed by all xi is λ. So there are no elements of (Vλ)μ in the ikerμ(xi) unless λ=μ. When λ=μ. we have (Vμ)μkerμ(xi). Thus Cμ = ikerμ(xi) λM^ Cλ (Vλ)μ δλμ Cμ, as C-modules.

A character formula for Cλ

The character of the irreducible C-module Cλ can be given by ξλ=wW ε(w)ηλ+ρ-wρ, where ημ is the character of Mμ as a C-module.


Equating the expressions (4.3) and (4.4) and rewriting χλ by using (2.8) we have λξλ wWε(w) ew(λ+ρ)-ρ =μP wWημε (w)eμ+wρ-ρ. Substitute γ=μ+wρ-ρ to get λξλ wWε(w) ew(λ+ρ)-ρ= μP ( wWε(w) ηγ+ρ-wρ ) eγ. Compare coefficients of eγ for γP+ on each side of this equation. Since λP+ we know by (2.3) that w(λ+ρ)-ρ is not an element of P+ for any wW except the identity. Thus ξλ=wWε (w)ηλ+ρ-wρ.

The dimension cλ of the irreducible C-module Cλ is given by cλ=wWε (w)dλ+ρ-wρ, where dμ is the dimension of the weight space Mμ.


Evaluate the identity in Theorem (4.6) at the identity of C.

Notes and references

This is an excerpt of a paper entitled Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras, written by Arun Ram.

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