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 9

Spaces

A Lie group is a group $G$ that is also a manifold such that the maps $$\begin{array}{ccc}G\times G& ⟶& G\\ (g_1,g_2)& ⟼& g_1{g}_2\end{array}\text{and}\begin{array}{ccc}G& ⟶& G\\ g& ⟼& g^{-1}\end{array}$$ are morphisms of manifolds.

An algebraic group is a group $G$ that is also a variety such that the maps $$\begin{array}{ccc}G\times G& ⟶& G\\ (g_1,g_2)& ⟼& g_1{g}_2\end{array}\text{and}\begin{array}{ccc}G& ⟶& G\\ g& ⟼& g^{-1}\end{array}$$ are morphisms of varieties.

A topological group is a group $G$ that is also a topological space such that $$\begin{array}{ccc}G\times G& ⟶& G\\ (g_1,g_2)& ⟼& g_1{g}_2\end{array}\text{and}\begin{array}{ccc}G& ⟶& G\\ g& ⟼& g^{-1}\end{array}$$ are morphisms of topological spaces.

A group scheme is a group $G$ that is also a scheme such that $$\begin{array}{ccc}G\times G& ⟶& G\\ (g_1,g_2)& ⟼& g_1{g}_2\end{array}\text{and}\begin{array}{ccc}G& ⟶& G\\ g& ⟼& g^{-1}\end{array}$$ are morphisms of schemes.

A complex Lie group is a group $G$ that is also a complex manifold such that $$\begin{array}{ccc}G\times G& ⟶& G\\ (g_1,g_2)& ⟼& g_1{g}_2\end{array}\text{and}\begin{array}{ccc}G& ⟶& G\\ g& ⟼& g^{-1}\end{array}$$ are morphisms of complex manifolds.

Remarks:

(a)	Morphisms of manifolds are called smooth functions. Lie groups have $ℝ$ in them in a crucial way.
(b)	Morphisms of varieties are called regular functions. Algebraic groups usually need to be based on an algebraically closed field. A variety is a topological space which is locally isomorphism to an affine variety.
(c)	Morphisms of topological spaces are called continuous functions.
(d)	Schemes are varieties over $\mathbb{Z}\text{.}$
(e)	Complex manifolds are not manifolds.
(f)	Morphisms of Lie groups, morphisms of algebraic groups, morphisms of topological groups, morphisms of schemes are all different things.

There are equivalences of categories

$$\begin{array}{cc}\left\{\text{connected reductive complex algebraic groups}\right\}& G\\ \updownarrow & ↧\\ \left\{\text{connected compact Lie groups}\right\}& K\\ \updownarrow & ↧\\ \left\{\text{complex semisimple Lie algebras}\right\}& 𝔤\\ \updownarrow & ↧\\ \left\{\mathbb{Z}\text{-reflection groups}\right\}& (W_0,{𝔥}_{\mathbb{Z}}^{*})\\ \updownarrow \\ \left\{\text{Dynkin diagrams}\right\}\end{array}$$

${GL}_n,{SL}_n,P{GL}_n\text{.}$ $${GL}_n\left(ℂ\right)=\{g\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g\hspace{0.17em}\text{is invertible}\}\text{.}$$ Let $V$ be a vector space over $𝔽\text{.}$ $$GL\left(V\right)=\{g\in \text{End}\left(V\right)\hspace{0.17em}|\hspace{0.17em}g\hspace{0.17em}\text{is invertible}\}\text{.}$$ ${GL}_n\left(ℂ\right)$ is a complex algebraic group.
${GL}_n\left(\overline{𝔽}\right)$ is an algebraic group.
${GL}_n$ is ???

The group homomorphism $$\text{det}:{GL}_n\left(𝔽\right)⟶{𝔽}^{\times }$$ is a $1\text{-dimensional}$ representation (character) of ${GL}_n\left(𝔽\right)\text{.}$ $$\begin{array}{rcl}{\displaystyle {SL}_n\left(𝔽\right)}& =& {\displaystyle \text{ker}\left(\text{det}\right)}\\ & =& {\displaystyle \{g\in {GL}_n\left(𝔽\right)\hspace{0.17em}|\hspace{0.17em}\text{det}\left(g\right)=1\}\text{.}}\end{array}$$ The center of ${GL}_n\left(𝔽\right)$ is $$𝒵\left({GL}_n\right)=\{c·\text{Id}\hspace{0.17em}|\hspace{0.17em}c\in {𝔽}^{\times }\}$$ $$P{GL}_n=\raisebox{1ex}{${GL}_n\left(𝔽\right)$}\!\left/ \!\raisebox{-1ex}{$𝒵\left({GL}_n\left(𝔽\right)\right)$}\right.$$ ${GL}_n\left(ℂ\right)$ is a complex reductive algebraic group.
${SL}_n\left(ℂ\right)$ is a complex semisimple algebraic group.
${PGL}_n\left(ℂ\right)$ is a complex semisimple algebraic group. In spite of $${SL}_n\left(ℂ\right)\subseteq {GL}_n\left(ℂ\right)\text{and}{GL}_n\left(ℂ\right)={SL}_n\left(ℂ\right)·{ℂ}^{\times }$$ and $$1⟶{SL}_n\left(ℂ\right)⟶{GL}_n\left(ℂ\right)\stackrel{\text{det}}{⟶}{ℂ}^{*}⟶1$$ being exact, $$P{GL}_n\left(ℂ\right)\not\simeq {SL}_n\left(ℂ\right)\text{.}$$ $$𝒵\left({SL}_n\left(ℂ\right)\right)=\left\{n^{\text{th}}\hspace{0.17em}\text{roots of}\hspace{0.17em}1\right\}={\mu }_n\left(ℂ\right)\text{.}$$

$U_n,O_n,$ {\text{Sp}}_n\text{.}$$

The unitary group $$U\left(n\right)=\{g\in {GL}_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{\bar{g}}^t=1\}$$ where $\bar{g}=\left({\bar{g}}_{ij}\right)$ if $g=\left(g_{ij}\right)\text{.}$

The orthogonal group $$O_n\left(ℂ\right)=\{g\in {GL}_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{g}^t=1\}\text{.}$$

The symplectic group $${\text{Sp}}_{2n}\left(ℂ\right)=\{g\in {GL}_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}gJ{g}^t=J\}$$ where $$J=\left(\begin{array}{cccccc}& & & 1& & 0\\ & 0& & & \ddots & \\ & & & 0& & 1\\ -1& & 0& & & \\ & \ddots & & & 0& \\ 0& & -1& & & \end{array}\right)\text{or}J=\left(\begin{array}{cccccc}& & & 0& & 1\\ & 0& & & ⋰& \\ & & & 1& & 0\\ 0& & -1& & & \\ & ⋰& & & 0& \\ -1& & 0& & & \end{array}\right)\text{.}$$

Let $V$ be a vector space over $𝔽\text{.}$ A symmetric bilinear form on $V$ is a map $$\begin{array}{cccc}⟨,⟩:& V\times V& ⟶& 𝔽\\ & (v_1,v_2)& ⟼& ⟨v_1,v_2⟩\end{array}$$ such that

(a)	$⟨,⟩$ is bilinear, i.e. $$\begin{array}{rcl}{\displaystyle ⟨v_1+v_2,v_3⟩}& =& {\displaystyle ⟨v_1,v_3⟩+⟨v_2,v_3⟩,}\\ {\displaystyle ⟨v_1,v_2+v_3⟩}& =& {\displaystyle ⟨v_1,v_2⟩+⟨v_1,v_3⟩,}\\ {\displaystyle ⟨c{v}_1,v_2⟩}& =& {\displaystyle c⟨v_1,v_2⟩,\text{and}}\\ {\displaystyle ⟨v_1,c{v}_2⟩}& =& {\displaystyle c⟨v_1,v_2⟩,}\end{array}$$ for $v_1,v_2,v_3\in V,$ $c\in 𝔽,$
(b)	$⟨,⟩$ is symmetric, i.e. $$⟨v_1,v_2⟩=⟨v_2,v_1⟩$$ for $v_1,v_2\in V\text{.}$

The orthogonal group is $$\begin{array}{rcl}{\displaystyle O_n\left(𝔽\right)}& =& {\displaystyle O(V,⟨,⟩)=O\left(⟨,⟩\right)}\\ & =& {\displaystyle \{g\in GL\left(V\right)\hspace{0.17em}|\hspace{0.17em}⟨g{v}_1,g{v}_2⟩=⟨v_1,v_2⟩\hspace{0.17em}\text{for}\hspace{0.17em}{v}_1,v_2\in V\},}\end{array}$$ the group of invertible linear transformations "preserving the metric".

A skew symmetric form on $V$ is a map $$⟨,⟩:V\times V⟶𝔽$$ such that

(a)	$⟨,⟩$ is bilinear,
(b)	$⟨v_2,v_1⟩=-⟨v_1,v_2⟩,$ for $v_1,v_2\in V\text{.}$

The symplectic group is $$\begin{array}{rcl}{\displaystyle {\text{Sp}}_n\left(𝔽\right)}& =& {\displaystyle \text{Sp}\left(V\right)}\\ & =& {\displaystyle \text{Sp}(V,⟨,⟩)}\\ & =& {\displaystyle \text{Sp}\left(⟨,⟩\right)}\\ & =& {\displaystyle \{g\in GL\left(V\right)\hspace{0.17em}|\hspace{0.17em}⟨g{v}_1,g{v}_2⟩=⟨v_1,v_2⟩\hspace{0.17em}\text{for}\hspace{0.17em}{v}_1,v_2\in V\}\text{.}}\end{array}$$ Let $$\begin{array}{cccc}\bar{\phantom{A}}:& 𝔽& ⟶& 𝔽\\ & z& ⟼& \bar{z}\end{array}$$ be an involution.

A sesquilinear form, or Hermitian form, is a map $$⟨,⟩:V\times V⟶𝔽$$ such that

(a)	$⟨,⟩$ is not bilinear, instead $$\begin{array}{rcl}{\displaystyle ⟨v_1+v_2,v_3⟩}& =& {\displaystyle ⟨v_1,v_3⟩+⟨v_2,v_3⟩,}\\ {\displaystyle ⟨v_1,v_2+v_3⟩}& =& {\displaystyle ⟨v_1,v_2⟩+⟨v_1,v_3⟩,}\\ {\displaystyle ⟨c{v}_1,v_2⟩}& =& {\displaystyle c⟨v_1,v_2⟩,\text{and}}\\ {\displaystyle ⟨v_1,c{v}_2⟩}& =& {\displaystyle \bar{c}⟨v_1,v_2⟩,}\end{array}$$ for $v_1,v_2,v_3\in V$ and $c\in 𝔽\text{.}$
(b)	$⟨v_2,v_1⟩=\overline{⟨v_1,v_2⟩},$ for $v_1,v_2\in V\text{.}$

The unitary group $$U_n=\{g\in GL\left(V\right)\hspace{0.17em}|\hspace{0.17em}⟨v_1,v_2⟩=⟨g{v}_1,g{v}_2⟩\hspace{0.17em}\text{for all}\hspace{0.17em}{v}_1,v_2\in V\}\text{.}$$

Maximal compacts and maximal tori

$$\begin{array}{ccc}\left\{\text{connected reductive linear algebraic groups over}\hspace{0.17em}ℂ\right\}& ⟷& \left\{\text{compact connected Lie groups}\right\}\\ G& ⟼& K\end{array}$$ where $K$ is the maximal compact subgroup of $G\text{.}$

A torus in a compact Lie group is a subgroup isomorphic to $S^1\times \cdots \times S^1\text{.}$

A torus in an algebraic group is a subgroup isomorphic to ${𝔽}^{\times }\times \cdots \times {𝔽}^{\times }\text{.}$

${GL}_1\left(𝔽\right)={𝔽}^{\times }$ and ${GL}_1\left(ℂ\right)={ℂ}^{\times }$ has maximal compact subgroup $$U\left(1\right)=\{z\in {ℂ}^{\times }\hspace{0.17em}|\hspace{0.17em}z\bar{z}=1\}=S^1\text{.}$$ So the maximal compact subgroup of ${ℂ}^{\times }\times \cdots \times {ℂ}^{\times }$ is $S^1\times \cdots \times S^1\text{.}$ $$\begin{array}{ccccc}& & & \stackrel{\genfrac{}{}{0ex}{}{\text{maximal}}{\text{compact}}}{⟶}& \\ & & G& ⟼& K\\ \downarrow \genfrac{}{}{0ex}{}{\text{maximal}}{\text{torus}}& & ↧& & ↧\\ & & T& ⟼& T_k\end{array}$$

$SU\left(2\right)$

$$SU\left(2\right)=\{g\in {SL}_2\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}g{\bar{g}}^t=1\}\text{.}$$ If $g=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in SU\left(2\right)$ then $$g^{-1}=\left(\begin{array}{cc}d& -b\\ -c& a\end{array}\right)={\bar{g}}^t=\left(\begin{array}{cc}\bar{a}& \bar{c}\\ \bar{b}& \bar{d}\end{array}\right)$$ so that $$g=\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)$$ with ${\mid a\mid }^2+{\mid b\mid }^2=1\text{.}$ So $$SU\left(2\right)=\left\{\left(\begin{array}{cc}a& b\\ -\bar{b}& \bar{a}\end{array}\right)\hspace{0.17em}\right|\hspace{0.17em}a,b\in ℂ,{\mid a\mid }^2+{\mid b\mid }^2=1\}\text{.}$$ Define an involution $$\begin{array}{cccc}\sigma :& {SL}_2\left(ℂ\right)& ⟶& {SL}_2\left(ℂ\right)\\ & g& ⟼& {\left({\bar{g}}^t\right)}^{-1}\end{array}$$ Then $$SU\left(2\right)={SL}_2{\left(ℂ\right)}^{\sigma }=\{g\in {SL}_2\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}\sigma \left(g\right)=g\}\text{.}$$ Let $G$ be a complex reductive algebraic group, $$\begin{array}{cccc}\sigma :& G& ⟶& G\\ & x_{\alpha }\left(t\right)& ⟼& x_{-\alpha }(-\bar{t})\\ & h_{\alpha }\left(t\right)& ⟼& h_{\alpha }\left({\bar{t}}^{-1}\right)\end{array}$$ an involution. Then $$K=G^{\sigma }=\{g\in G\hspace{0.17em}|\hspace{0.17em}\sigma \left(g\right)=g\}$$ is a maximal compact subgroup.

Notes and References

These are a typed copy of Lecture 9 from a series of handwritten lecture notes for the class Representation Theory given on October 7, 2008.

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