Representation Theory

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updated: 2 October 2014

Lecture 11

A morphism $f:X\to Y$ of spaces provides $$df:T_x\left(X\right)⟶T_{f\left(x\right)}\left(y\right),$$ for $x\in X\text{.}$

Let $G$ be a Lie group or algebraic group. The conjugation action of $G$ on $G$ is given by $$\begin{array}{cccc}{\text{In}}_g:& G& ⟶& G\\ & h& ⟼& gh{g}^{-1}\end{array}$$ for $g\in G\text{.}$ The differential of these maps gives the Adjoint action of $G$ on $𝔤=T_1\left(G\right)$ $${\text{Ad}}_g:𝔤⟶𝔤,$$ for $g\in G\text{.}$

$G={GL}_n$ has Lie algebra ${𝔤𝔩}_n=M_n\left(ℂ\right)$ and the exponential map is $$\begin{array}{ccc}{𝔤𝔩}_n& ⟶& {GL}_n\\ x& ⟼& e^x\end{array}$$ where $$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots \text{.}$$ ${SO}_n,O_n,{\text{Sp}}_n$ etc. are subgroups of ${GL}_n$ and ${𝔰𝔬}_n,{𝔬}_n,{𝔰𝔭}_n$ etc. are Lie subalgebras of ${𝔤𝔩}_n\text{.}$

Since $$\begin{array}{cccc}{\text{In}}_g:& {GL}_n& ⟶& {GL}_n\\ & h& ⟼& gh{g}^{-1}\\ & e^{ty}& ⟼& g{e}^{ty}{g}^{-1}\end{array}$$ and $$g{e}^{tx}{g}^{-1}=g(1+tx+\frac{t^2{x}^2}{2!}+\frac{t^3{x}^3}{3!}+\cdots )g^{-1}=e^{t\left(gx{g}^{-1}\right)}$$ it follows that $$\begin{array}{cccc}{\text{Ad}}_g:& {𝔤𝔩}_n& ⟶& {𝔤𝔩}_n\\ & x& ⟼& gx{g}^{-1}\text{.}\end{array}$$ Let $M$ be a $G\text{-module,}$ $$\begin{array}{cccc}\rho :& G& ⟶& GL\left(M\right)\\ & g& ⟼& \rho \left(g\right)\\ & e^{tx}& ⟼& \rho \left(e^{tx}\right),\end{array}$$ the corresponding representation of $G\text{.}$ If $$\rho \left(x\right)=\frac{d\rho \left(e^{tx}\right)}{dt}{|}_{t=0}$$ then $$\rho \left(e^{tx}\right)=e^{t\rho \left(x\right)}$$ and we get a representation of $𝔤$ on $M$ $$\begin{array}{cccc}\rho :& 𝔤& ⟶& \text{End}\left(M\right)\\ & x& ⟼& \rho \left(x\right)\text{.}\end{array}$$ The group $G$ acts on $𝔤$ by the Adjoint action $$\begin{array}{cccc}{\text{Ad}}_g:& 𝔤& ⟶& 𝔤\\ & x& ⟼& gx{g}^{-1}\end{array}$$ and the Lie algebra $𝔤$ acts on $𝔤$ by the adjoint action $$\begin{array}{cccc}{\text{ad}}_y:& 𝔤& ⟶& 𝔤\\ & x& ⟼& [y,x],\end{array}$$ for $y\in 𝔤$ since $$\begin{array}{rcl}{\displaystyle {\text{Ad}}_{ty}\left(x\right)}& =& {\displaystyle e^{ty}x{e}^{-ty}}\\ & =& {\displaystyle (1+ty+\frac{t^2{y}^2}{2!}+\cdots )x(1-ty+\frac{t^2{y}^2}{2!}-\frac{t^3{y}^3}{3!}+\cdots )}\\ & =& {\displaystyle x+t(yx-xy)+\frac{t^2}{2!}(y^{2}x-2yxy+x{y}^2)+\cdots }\\ & =& {\displaystyle \left(e^{t{\text{ad}}_y}\right)\left(x\right)\text{.}}\end{array}$$ Note: $${\left({\text{ad}}_y\right)}^2\left(x\right)=[y,[y,x]]=[y,(yx-xy)]=y^{2}x-yxy-yxy+x{y}^2\text{.}$$ So we have three actions: $$\begin{array}{ccc}\text{conjugation action}& & \begin{array}{cccc}{\text{In}}_g:& G& ⟶& G\\ & h& ⟼& gh{g}^{-1}\end{array}\\ \text{Adjoint action}& & \begin{array}{cccc}{\text{Ad}}_g:& 𝔤& ⟶& 𝔤\\ & x& ⟼& gx{g}^{-1}\end{array}\\ \text{adjoint action}& & \begin{array}{cccc}{\text{ad}}_y:& 𝔤& ⟶& 𝔤\\ & x& [y,x]\end{array}\end{array}$$

Let $M$ be a $G\text{-module.}$ The dual vector space $$M^{*}=\text{Hom}(M,ℂ)=\{\phi :M\to ℂ\hspace{0.17em}|\hspace{0.17em}\phi \hspace{0.17em}\text{is linear}\}$$ is a $G\text{-module}$ with action given by $$\left(g\phi \right)\left(m\right)=\phi \left(g^{-1}m\right),$$ for $g\in G,$ $m\in M\text{.}$ Since $$\left(e^{tx}\phi \right)\left(m\right)=\phi \left(e^{-tx}m\right),$$ if $M$ is a $𝔤\text{-module,}$ then $M^{*}$ is a $𝔤\text{-module}$ with action given by $$\left(x\phi \right)\left(m\right)=\phi (-xm),$$ for $x\in 𝔤,$ $m\in M\text{.}$ Thus we have $S\text{-actions:}$ $$\begin{array}{c}\text{conjugation:}\begin{array}{cccc}{\text{In}}_g:& G& ⟶& G\\ & h& ⟼& gh{g}^{-1}\end{array},\\ \text{Adjoint:}\begin{array}{cccc}{\text{Ad}}_g:& 𝔤& ⟶& 𝔤\\ & x& ⟼& gx{g}^{-1}\end{array},\text{co Adjoint:}\begin{array}{cccc}{\text{Ad}}_{𝔤}^{*}:& {𝔤}^{*}& ⟶& {𝔤}^{*}\end{array},\\ \text{adjoint:}\begin{array}{cccc}{\text{ad}}_y:& 𝔤& ⟶& 𝔤\\ & x& [y,x]\end{array},\text{coadjoint:}\begin{array}{cccc}{\text{ad}}_y^{*}:& {𝔤}^{*}& ⟶& {𝔤}^{*}\end{array}\text{.}\end{array}$$

Tori and Cartan subalgebras

Let $G$ be an algebraic group. A torus $H$ is a subgroup of $G$ such that $$H\simeq \underset{\underset{n}{⏟}}{{ℂ}^{\times }\times \cdots \times {ℂ}^{\times }},$$ for some $n\in {\mathbb{Z}}_{>0}\text{.}$

Let $K$ be a Lie group. A torus $T$ is a subgroup of $K$ such that $$K\simeq \underset{\underset{n}{⏟}}{S^1\times \cdots \times S^1},$$ for some $n\in {\mathbb{Z}}_{>0}$ where $$S^{\prime }=U\left(1\right)=\{z\in {ℂ}^{\times }\hspace{0.17em}|\hspace{0.17em}z\bar{z}=1\}\text{.}$$

Let $𝔤$ be a Lie algebra. An abelian Lie subalgebra is a Lie subalgebra $𝔥$ such that $$[h_1,h_2]=0,$$ for $h_1,h_2\in 𝔥\text{.}$

A Cartan subalgebra is a maximal abelian Lie subalgebra of $𝔤\text{.}$

A maximal torus of ${GL}_n$ is $$H=\left\{\left(\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{x}_1,\dots ,x_n\in {ℂ}^{\times }\}\text{.}$$ A Cartan subalgebra of ${𝔤𝔩}_n$ is $$𝔥=\left\{\left(\begin{array}{ccc}{h}_1& & 0\\ & \ddots & \\ 0& & h_n\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}{h}_1,\dots ,h_n\in ℂ\}\text{.}$$ Note that $𝔥=\text{Lie}\left(H\right)=T_1\left(H\right)\text{.}$ Since $𝔥\subseteq 𝔤$ and $H\subseteq G,$ $H$ acts on $G$ by conjugation,
$H$ acts on $𝔤$ by the Adjoint action,
$𝔥$ acts on $𝔤$ by the adjoint action. The irreducible (rational) representations of $H$ are $$\begin{array}{rcl}{\displaystyle X^{\mu }}& =& {\displaystyle X^{{\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n}}\\ & =& {\displaystyle X^{{\mu }_1{\epsilon }_1}\cdots X^{{\mu }_n{\epsilon }_n}}\\ & =& {\displaystyle {\left(X^{{\epsilon }_1}\right)}^{{\mu }_1}\cdots {\left(X^{{\epsilon }_n}\right)}^{{\mu }_n},}\end{array}$$ with ${\mu }_1,\dots ,{\mu }_n\in \mathbb{Z}$ where $$\begin{array}{cccc}{X}^{{\epsilon }_i}:& H& ⟶& {ℂ}^{\times }\\ & \left(\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right)& ⟼& x_i\text{.}\end{array}$$ The irreducible representations of $𝔥$ are $$\mu :𝔥⟶ℂ,$$ so that $\mu \in {𝔥}^{*},$ and $$\mu ={\mu }_1{\epsilon }_1+\cdots +{\mu }_n{\epsilon }_n,$$ with ${\mu }_1,\dots ,{\mu }_n\in ℂ$ and $$\begin{array}{cccc}{\epsilon }_i:& 𝔥& ⟶& ℂ\\ & \left(\begin{array}{ccc}{h}_1& & 0\\ & \ddots & \\ 0& & h_n\end{array}\right)& ⟼& h_i\text{.}\end{array}$$ Hence ${𝔥}^{*}$ indexes irreducible representations of $𝔥,$ and $\left\{X^{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in {𝔥}_{\mathbb{Z}}^{*}\}$ are the irreducible representations of $H\text{.}$

Weights and roots

Let $M$ be a $G\text{-module}$ and $X^{\mu }:H\to {ℂ}^{\times }$ an irreducible representation of $H\text{.}$ The $\mu \text{-weight}$ space of $M$ is $$\begin{array}{rcl}{\displaystyle M_{\mu }}& =& {\displaystyle \{m\in M\hspace{0.17em}|\hspace{0.17em}\text{for each}\hspace{0.17em}t\in H,tm=X^{\mu }\left(t\right)m\}}\\ & =& {\displaystyle \{m\in M\hspace{0.17em}|\hspace{0.17em}\text{for each}\hspace{0.17em}h\in 𝔥,hm=\mu \left(h\right)m\}\text{.}}\end{array}$$ The generalized $\mu \text{-weight}$ space of $M$ is $$\begin{array}{rcl}{\displaystyle M_{\mu }^{\text{gen}}}& =& {\displaystyle \{m\in M\hspace{0.17em}|\hspace{0.17em}\text{for each}\hspace{0.17em}t\in H,{(t-X^{\mu }\left(t\right))}^{\ell }m=0\text{, for some}\hspace{0.17em}\ell \in {\mathbb{Z}}_{>0}\}}\\ & =& {\displaystyle \{m\in M\hspace{0.17em}|\hspace{0.17em}\text{for each}\hspace{0.17em}h\in 𝔥,{(h-\mu \left(h\right))}^{\ell }m=0\text{, for some}\hspace{0.17em}\ell \in {\mathbb{Z}}_{>0}\}\text{.}}\end{array}$$ Note that $M_{\mu }\subseteq M_{\mu }^{\text{gen}}$ and $M_{\mu }^{\text{gen}}\ne 0$ implies $M_{\mu }\ne 0\text{.}$ $$M=\underset{\mu \in {𝔥}^{*}}{⨁}{M}_{\mu }^{\text{gen}}$$

The weights of $M$ are the $\mu$ such that $M_{\mu }\ne 0\text{.}$

The adjoint representation $𝔤$ $\text{(}G$ acts on $𝔤$ or $𝔤$ acts on $𝔤\text{)}$ is a $G\text{-module.}$

The roots of $G$ (or $𝔤\text{)}$ are the nonzero weights of $𝔤\text{.}$

Note that ${𝔤}_0=𝔥,$ so the "interesting" weights of $𝔤$ are the nonzero ones.

$𝔤={𝔤𝔩}_n$ has basis $\left\{E_{ij}\hspace{0.17em}\right|\hspace{0.17em}1\le i,j\le n\}\text{.}$ If $$t=\left(\begin{array}{ccc}{x}_1& & 0\\ & \ddots & \\ 0& & x_n\end{array}\right)\in H\text{and}h=\left(\begin{array}{ccc}{h}_1& & 0\\ & \ddots & \\ 0& & h_n\end{array}\right)\in 𝔥$$ then $$t{E}_{ij}{t}^{-1}=x_i{x}_j^{-1}{E}_{ij}=X^{{\epsilon }_i-{\epsilon }_j}\left(t\right)E_{ij}$$ and $$[h,E_{ij}]=(h_i-h_j)E_{ij}=({\epsilon }_i-{\epsilon }_j)\left(h\right)E_{ij}$$ and hence $${𝔤}_{{\epsilon }_i-{\epsilon }_j}=ℂE_{ij},$$ for $1\le i,j\le n$ (and ${𝔤}_0=𝔥\text{).}$ Note that $𝔤$ contains lots of ${𝔰𝔩}_2\text{-subalgebras}$ $$\begin{array}{rcl}{\displaystyle E_{ij}}& =& {\displaystyle \left(\begin{array}{c}0\\ & \ddots & 1\\ & & & \\ & & & 0\end{array}\right),\text{where}\hspace{0.17em}1\hspace{0.17em}\text{is entry}\hspace{0.17em}(i,j),}\\ {\displaystyle E_{ji}}& =& {\displaystyle \left(\begin{array}{c}0\\ & \ddots \\ & 1& & \\ & & & 0\end{array}\right),\text{where}\hspace{0.17em}1\hspace{0.17em}\text{is entry}\hspace{0.17em}(j,i),}\\ {\displaystyle h_{ij}}& =& {\displaystyle \left(\begin{array}{c}0\\ & \ddots \\ & & 0\\ & & & 1\\ & & & & 0\\ & & & & & \ddots \\ & & & & & & 0\\ & & & & & & & -1\\ & & & & & & & & 0\\ & & & & & & & & & \ddots \\ & & & & & & & & & & 0\end{array}\right)\text{.}}\end{array}$$

A one parameter subgroup is an "embedding" of ${ℂ}^{\times }$ in $G\text{.}$

An $\text{"}{SL}_2$ imbedding" is a homomorphism $${𝔤}_{\alpha }:{SL}_2\left(ℂ\right)⟶G\text{.}$$

The Weyl group is $$W_0=\frac{N\left(H\right)}{H}$$ where $N\left(H\right)$ is the normalizer of $H$ in $G,$ $$N\left(H\right)=\{n\in G\hspace{0.17em}|\hspace{0.17em}nH{n}^{-1}=H\}\text{.}$$ The Weyl group acts on $H,$ by conjugation, and $W_0$ acts on $𝔥$ and, hence $$W_0\hspace{0.17em}\text{acts on}\hspace{0.17em}{𝔥}^{*},$$ and $$w:M_{\mu }⟶M_{w\mu },$$ for $w\in W\text{.}$

Notes and References

These are a typed copy of Lecture 11 from a series of handwritten lecture notes for the class Representation Theory given on October 19, 2008.

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