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
aram@unimelb.edu.au

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 $c_{\lambda },$ $$M\cong \underset{\lambda \in \hat{M}}{⨁}{c}_{\lambda }{V}^{\lambda }\text{.}$$ $M$ 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 $C\times 𝔘$ and $$M\cong \underset{\lambda \in \hat{M}}{⨁}{C}_{\lambda }\otimes V^{\lambda },$$ 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 $V^{\lambda },$ $\lambda \in \hat{M}\text{.}$ It is true that the set of $C_{\lambda },$ $\lambda \in \hat{M},$ 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 $$\xi \left(c_1{c}_2\right)=\xi \left(c_2{c}_1\right),$$ for all $c_1,c_2\in ℂ\text{.}$ Let $\hat{C}$ 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 ${\Lambda }_{\hat{M}}$ be the vector space which is the span of the ${\chi }^{\lambda },$ $\lambda \in \hat{M}\text{.}$ Define the characteristic map $\text{ch}$ to be the map given by $$\begin{array}{cccc}\text{ch}:& \hat{C}& ⟶& {\Lambda }_{\hat{M}}\\ & {\xi }_{\lambda }& ⟼& {\chi }^{\lambda }\end{array}$$

For each $\mu ={\sum }_i{\mu }_i{\omega }_i\in P$ define the $\mu \text{-weight}$ space of $M$ to be the vector space $$M_{\mu }=\{m\in M\hspace{0.17em}|\hspace{0.17em}\text{for each}\hspace{0.17em}1\le i\le n,\hspace{0.17em}{h}_{i}m={\mu }_{i}m\}\text{.}$$ 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 $𝔘\left(𝔥\right),$ $$\begin{array}{cc}M\cong \underset{\mu \in P}{⨁}{M}_{\mu }\text{.}& \text{(4.2)}\end{array}$$ 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 $$\begin{array}{cc}\text{bichar}M=\sum _{\mu \in P}{\eta }_{\mu }{e}^{\mu },& \text{(4.3)}\end{array}$$ 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 $$\begin{array}{cc}\text{bichar}M=\sum _{\lambda \in \hat{M}}{\xi }_{\lambda }{\chi }^{\lambda }\text{.}& \text{(4.4)}\end{array}$$

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 $\mu \in P$ $$x_i:M_{\mu }⟶M_{\mu +{\alpha }_i}\text{.}$$ Let ${\text{ker}}_{\mu }\left(x_i\right)$ denote the kernel of the map $x_i$ acting on $M_{\mu }\text{.}$ Let $${\bar{C}}_{\mu }=\bigcap _i{\text{ker}}_{\mu }\left(x_i\right)\text{.}$$

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

		Proof.
	Using (4.1) we have that $M_{\mu }={\oplus }_{\lambda \in \hat{M}}{C}_{\lambda }\otimes {\left(V^{\lambda }\right)}_{\mu }\text{.}$ 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 $$\begin{array}{ccc}{\bar{C}}_{\mu }& =& \bigcap _i{\text{ker}}_{\mu }\left(x_i\right)\\ & \cong & \underset{\lambda \in \hat{M}}{⨁}{C}_{\lambda }\otimes {\left(V^{\lambda }\right)}_{\mu }{\delta }_{\lambda \mu }\\ & \cong & C_{\mu },\end{array}$$ 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 $${\xi }_{\lambda }=\sum _{w\in W}\epsilon \left(w\right){\eta }_{\lambda +\rho -w\rho },$$ 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 $$\sum _{\lambda }{\xi }_{\lambda }\sum _{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )-\rho }=\sum _{\mu \in P}\sum _{w\in W}{\eta }_{\mu }\epsilon \left(w\right)e^{\mu +w\rho -\rho }\text{.}$$ Substitute $\gamma =\mu +w\rho -\rho$ to get $$\sum _{\lambda }{\xi }_{\lambda }\sum _{w\in W}\epsilon \left(w\right)e^{w(\lambda +\rho )-\rho }=\sum _{\mu \in P}\left(\sum _{w\in W}\epsilon \left(w\right){\eta }_{\gamma +\rho -w\rho }\right)e^{\gamma }\text{.}$$ 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 $w(\lambda +\rho )-\rho$ is not an element of $P^{+}$ for any $w\in W$ except the identity. Thus $${\xi }_{\lambda }=\sum _{w\in W}\epsilon \left(w\right){\eta }_{\lambda +\rho -w\rho }\text{.}$$ $\square$

The dimension $c_{\lambda }$ of the irreducible $C\text{-module}$ $C_{\lambda }$ is given by $$c_{\lambda }=\sum _{w\in W}\epsilon \left(w\right)d_{\lambda +\rho -w\rho },$$ 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.

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