## Weyl Group Symmetric Functions and the Representation Theory of Lie Algebras

Last update: 11 September 2013

## Centralizer algebras

Let $M$ be a finite dimensional module for a semisimple Lie algebra or its enveloping algebra $𝔘\text{.}$ By Weyl's complete reduciblity theorem we know that $M$ decomposes as a direct sum of irreducible $𝔘$ modules ${V}^{\lambda },$ with certain multiplicities ${c}_{\lambda },$ $M≅⨁λ∈M^ cλVλ.$ $M$ denotes the set of $\lambda \in {P}^{+}$ such that ${V}^{\lambda }$ appears in the decomposition of $M\text{.}$ Let $C$ be the centralizer of the action of $𝔘$ on $M,$ i.e. $C={\text{End}}_{𝔘}\left(M\right)\text{.}$ The following theorem is the basic result of the double centralizer theory.

$M$ is a bimodule for $C×𝔘$ and $M≅⨁λ∈M^ Cλ⊗Vλ,$ where ${C}_{\lambda }$ is an irreducible module for $C\text{.}$

This decomposition induces a pairing between the irreducible modules ${C}_{\lambda }$ of $C$ and irreducible $𝔘$ modules ${V}^{\lambda },$ $\lambda \in \stackrel{^}{M}\text{.}$ It is true that the set of ${C}_{\lambda },$ $\lambda \in \stackrel{^}{M},$ is a complete set of irreducible modules of $C\text{.}$ 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 $\xi :C\to ℂ$ such that $ξ(c1c2)= ξ(c2c1),$ for all ${c}_{1},{c}_{2}\in ℂ\text{.}$ Let $\stackrel{^}{C}$ be the vector space of characters of $C\text{.}$ Given a $C\text{-module}$ $M$ the character $\xi$ of $C$ corresponding to $M$ is given by defining $\xi \left(c\right),$ $c\in C,$ to be the trace of the action of $c$ on $M\text{.}$ Let ${\xi }_{\lambda }$ denote the irreducible character of $C$ corresponding to the irreducible $C\text{-module}$ ${C}_{\lambda }\text{.}$ For each $\lambda \in {P}^{+}$ let ${\chi }^{\lambda }$ denote the corresponding Weyl character for $𝔘\text{.}$ Let ${\Lambda }_{\stackrel{^}{M}}$ be the vector space which is the span of the ${\chi }^{\lambda },$ $\lambda \in \stackrel{^}{M}\text{.}$ Define the characteristic map $\text{ch}$ to be the map given by $ch: C^ ⟶ ΛM^ ξλ ⟼ χλ$

For each $\mu ={\sum }_{i}{\mu }_{i}{\omega }_{i}\in P$ define the $\mu \text{-weight}$ space of $M$ to be the vector space $Mμ= { m∈M | for each 1≤i≤n, him =μim } .$ Let $𝔘\left(𝔥\right)$ be the subalgebra of $𝔘$ generated by the ${h}_{i}\text{.}$ $M$ decomposes as a direct sum of weight spaces under the action of $𝔘\left(𝔥\right),$ $M≅⨁μ∈P Mμ. (4.2)$ Since $𝔘\left(𝔥\right)$ is a subalgebra of $𝔘$ each ${M}_{\mu }$ is a $C$ module. The bicharacter of $M$ is defined to be $bicharM=∑μ∈P ημeμ, (4.3)$ where ${\eta }_{\mu }$ denotes the character of ${M}_{\mu }$ as a $C\text{-module.}$ In view of the decomposition in (4.1) we also have $bicharM=∑λ∈M^ ξλχλ. (4.4)$

Construction of irreducible modules for centralizer algebras

Consider the action of the generators ${x}_{i}\in 𝔘$ on the weight spaces ${M}_{\mu }$ of $M\text{.}$ For each $\mu \in P$ $xi:Mμ⟶ Mμ+αi.$ Let ${\text{ker}}_{\mu }\left({x}_{i}\right)$ denote the kernel of the map ${x}_{i}$ acting on ${M}_{\mu }\text{.}$ Let $C‾μ= ⋂ikerμ (xi).$

 1) $\stackrel{‾}{C}\mu$ is either $0$ or an irreducible $C$ module. Furthermore all irreducible $C$ modules can be obtained in this fashion. 2) ${\stackrel{‾}{C}}_{\mu }\ne 0⟺\mu \in \stackrel{^}{M}\text{.}$

 Proof. Using (4.1) we have that ${M}_{\mu }={\oplus }_{\lambda \in \stackrel{^}{M}}{C}_{\lambda }\otimes {\left({V}^{\lambda }\right)}_{\mu }\text{.}$ But the only weight in ${V}^{\lambda }$ killed by all ${x}_{i}$ is $\lambda \text{.}$ So there are no elements of ${\left({V}^{\lambda }\right)}_{\mu }$ in the ${\cap }_{i}{\text{ker}}_{\mu }\left({x}_{i}\right)$ unless $\lambda =\mu \text{.}$ When $\lambda =\mu \text{.}$ we have ${\left({V}^{\mu }\right)}_{\mu }\subseteq {\text{ker}}_{\mu }\left({x}_{i}\right)\text{.}$ Thus $C‾μ = ⋂ikerμ(xi) ≅ ⨁λ∈M^ Cλ⊗ (Vλ)μ δλμ ≅ Cμ,$ as $C\text{-modules.}$ $\square$

A character formula for ${C}_{\lambda }$

The character of the irreducible $C\text{-module}$ ${C}_{\lambda }$ can be given by $ξλ=∑w∈W ε(w)ηλ+ρ-wρ,$ where ${\eta }_{\mu }$ is the character of ${M}_{\mu }$ as a $C\text{-module.}$

 Proof. Equating the expressions (4.3) and (4.4) and rewriting ${\chi }^{\lambda }$ by using (2.8) we have $∑λξλ ∑w∈Wε(w) ew(λ+ρ)-ρ =∑μ∈P ∑w∈Wημε (w)eμ+wρ-ρ.$ Substitute $\gamma =\mu +w\rho -\rho$ to get $∑λξλ ∑w∈Wε(w) ew(λ+ρ)-ρ= ∑μ∈P ( ∑w∈Wε(w) ηγ+ρ-wρ ) eγ.$ Compare coefficients of ${e}^{\gamma }$ for $\gamma \in {P}^{+}$ on each side of this equation. Since $\lambda \in {P}^{+}$ we know by (2.3) that $w\left(\lambda +\rho \right)-\rho$ is not an element of ${P}^{+}$ for any $w\in W$ except the identity. Thus $ξλ=∑w∈Wε (w)ηλ+ρ-wρ.$ $\square$

The dimension ${c}_{\lambda }$ of the irreducible $C\text{-module}$ ${C}_{\lambda }$ is given by $cλ=∑w∈Wε (w)dλ+ρ-wρ,$ where ${d}_{\mu }$ is the dimension of the weight space ${M}_{\mu }\text{.}$

 Proof. Evaluate the identity in Theorem (4.6) at the identity of $C\text{.}$ $\square$

## 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.