Representation theory Lecture Notes: Chapter 4

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

Last update: 20 November 2013

Abstract.

Spaces

A topological space is a set $X$ with a specified collection of open subsets of $X$ which is closed under unions, finite intersections, complements and contains $\varnothing$ and $X\text{.}$ A continuous function $f:X\to Y$ is a map such that $f^{-1}\left(V\right)$ is open in $X$ for all open subsets $V\subseteq Y\text{.}$ The morphisms in the category of topological spaces are continuous functions.

(a)	A closed subset of $X$ is the complement of an open set of $X\text{.}$
(b)	The space $X$ is compact if every open cover has a finite subcover.
(c)	The space $X$ is locally compact if every point has a neighborhood with compact closure.
(d)	The space $X$ is totally disconnected if there is no connected subset with more than one element.
(e)	The space $X$ is Hausdorff if ${\Delta }_X=\left\{(x,x)\hspace{0.17em}\right|\hspace{0.17em}x\in X\}$ is a closed subspace of $X\times X,$ where $X\times X$ has the product topolgy.

The topological space $X$ is Hausdorff if and only if for any two points in $X$ there exist neighborhoods of each of them that do not intersect.

A metric space is a set $X$ with a metric $d:X\times X\to {ℝ}_{\ge 0}$ such that A Cauchy sequence is a sequence $(p_i\in V\hspace{0.17em}|\hspace{0.17em}i\in {\mathbb{Z}}_{>0})$ such that, for every positive real number $\epsilon$ there is a positive integer $N$ such that $d(p_n,p_m)<\epsilon$ for all $m,n>N\text{.}$ A sequence $(p_i\in V\hspace{0.17em}|\hspace{0.17em}i\in {\mathbb{Z}}_{\ge 0})$ converges if there is a $p\in V$ such that, for every $\epsilon \in {ℝ}_{>0},$ there is an $N\in {\mathbb{Z}}_{>0}$ such that $d(p_n,p)<\epsilon$ for all $n>N\text{.}$ A metric space is complete if all Cauchy sequences converge.

Sheaves

Let $X$ be a topological space. A sheaf on $X$ is a contravariant functor $$\begin{array}{cccc}{𝒪}_X:& \left\{\text{open sets of}\hspace{0.17em}X\right\}& ⟶& \left\{\text{rings}\right\}\\ & U& ⟼& {𝒪}_X\left(U\right)\end{array}$$ such that if $\left\{U_{\alpha }\right\}$ is an open cover of $U$ and $f_{\alpha }\in {𝒪}_X\left(U_{\alpha }\right)$ are such that $$f_{\alpha }{|}_{U_{\alpha }\cap U_{\beta }}=f_{\beta }{|}_{U_{\alpha }\cap U_{\beta }},\text{for all}\hspace{0.17em}\alpha ,\beta ,$$ then there is a unique $f\in {𝒪}_X\left(U\right)$ such that $f_{\alpha }=f{|}_{U_{\alpha }},$ for all $\alpha \text{.}$ A ringed space is a pair $(X,{𝒪}_X)$ where $X$ is a topological space and ${𝒪}_X$ is a sheaf on $X\text{.}$ The stalk of ${𝒪}_X$ at $x\in X$ is $${𝒪}_{X,x}=\text{ind}\hspace{0.17em}\underset{U}{lim}{𝒪}_X\left(U\right),$$ where the limit is over all neighborhoods $U$ of $x\text{.}$

Note: an alternate way of stating the condition in the definition of a sheaf is to say that the sequence $$𝒪\to {𝒪}_x\left(U\right)\stackrel{i}{⟶}\prod _{\alpha }{𝒪}_x\left(U_{\alpha }\right)\underset{\underset{k}{⟶}}{\overset{\stackrel{j}{⟶}}{}}\prod _{\alpha ,\beta }{𝒪}_x(U_{\alpha }\cap U_{\beta })$$ is exact where $i$ is the map induced by the inclusions $U_{\alpha }\hookrightarrow U,$
$j$ is the map induced by the inclusions $U_{\alpha }\cap U_{\beta }\hookrightarrow U_{\alpha },$
$k$ is the map induced by the inclusions $U_{\alpha }\cap U_{\beta }\hookrightarrow U_{\beta },$ and exactness of the sequence means $\text{im}\hspace{0.17em}i=\text{ker}(j-k)\text{.}$

Smooth manifolds

A manifold is a topological space $X$ which is locally homeomorphic to ${ℝ}^n\text{.}$ Locally homeomorphic to ${ℝ}^n$ means that for each $x\in X$ there is an open neighborhood $U$ of $x,$ an open set $V$ in ${ℝ}^n$ and a homeomorphism $\varphi :U\to V\text{.}$ The map $\varphi :U\to V$ is a chart. An atlas is an open covering $\left(U_{\alpha }\right)$ of $X,$ a set of open sets $\left(V_{\alpha }\right)$ of ${ℝ}^n$ and a collection of charts ${\varphi }_{\alpha }:U_{\alpha }\to V_{\alpha }\text{.}$ Examples of manifolds are $$\begin{array}{cc}PICTURE\; OF\; SPHERE& PICTURE\; OF\; TORUS\\ \text{sphere}& \text{torus}\end{array}$$ A smooth manifold is a manifold with an atlas $\left({\varphi }_{\alpha }\right)$ such that for each pair of charts ${\varphi }_{\alpha },{\varphi }_{\beta }$ the maps $${\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}:{\varphi }_{\alpha }(U_{\alpha }\cap U_{\beta })⟶{\varphi }_{\beta }(U_{\alpha }\cap U_{\beta })$$ are smooth (i.e. $C^{\infty }\text{).}$ Let $M$ be a smooth manifold and let $U$ be an open subset of $M\text{.}$ The ring of smooth functions on $U$ is the set of functions $f:U\to ℝ$ that are smooth at every point of $U,$ i.e. If $x\in U$ then there is a chart ${\varphi }_{\alpha }:U_{\alpha }\to V_{\alpha }$ such that $x\in U_{\alpha }$ and $$f\circ {\varphi }_{\alpha }^{-1}:V_{\alpha }\to ℝ,\text{is}\hspace{0.17em}{C}^{\infty }\text{.}$$

Let $V_{\alpha }$ be an open set of ${ℝ}^n\text{.}$ For each open set $V$ of $V_{\alpha }$ let $C^{\infty }\left(V\right)$ be the set of functions $f:V\to ℝ$ that are $C^{\infty }$ at every point of $V\text{.}$ If $V\hookrightarrow V\prime$ then we have a map $$\begin{array}{ccc}{C}^{\infty }\left(V\prime \right)& ⟶& C^{\infty }\left(V\right)\\ f& ⟼& f{|}_V\text{.}\end{array}$$ Thus $$\begin{array}{cccc}{C}^{\infty }:& \left\{\text{open sets of}\hspace{0.17em}{V}_{\alpha }\right\}& ⟶& \left\{\text{rings}\right\}\\ & V& ⟼& C^{\infty }\left(V\right)\end{array}$$ is a sheaf on $V_{\alpha }$ and $(V_{\alpha },C^{\infty })$ is a ringed space.

A smooth manifold is a Hausdorff topological space which is locally isomorphic to ${ℝ}^n,$ i.e. a Hausdorff ringed space $(M,C^{\infty })$ with an open cover $\left(U_{\alpha }\right)$ such that each $(U_{\alpha },C^{\infty })$ is isomorphic (as a ringed space) to an open set $(V_{\alpha },C^{\infty })$ of ${ℝ}^n\text{.}$

Varieties

A affine algebraic variety over $\overline{𝔽}$ is a set $$X=\left\{(x_1,\dots ,x_n)\hspace{0.17em}\right|\hspace{0.17em}{f}_{\alpha }(x_1,\dots ,x_n)=0\hspace{0.17em}\text{for all}\hspace{0.17em}{f}_{\alpha }\in S\}$$ where $S$ is a set of polynomials in $\overline{𝔽}[t_1,t_2,\dots ,t_n]\text{.}$ By definition, these are the closed sets in the Zariski topology on $\overline{𝔽}\text{.}$ Let $U$ be an open set of $X$ and define ${𝒪}_X\left(U\right)$ to be the set of functions $f:U\to \overline{𝔽}$ that are regular at every point of $x\in U,$ i.e. For each $x\in U$ there is a neigborhood $U_{\alpha }\subseteq U$ of $x$ and functions $g,h\in \overline{𝔽}[t_1,\dots ,t_n]$ such that $h\left(y\right)\ne 0$ and $f\left(y\right)=g\left(y\right)/h\left(y\right)$ for all $y\in U_{\alpha }\text{.}$ Then ${𝒪}_X$ is a sheaf on $X$ and $(X,{𝒪}_X)$ is a ringed space. The sheaf ${𝒪}_X$ is the structure sheaf of the affine algebraic variety $X\text{.}$

A variety is a ringed space $(X,𝒪)$ such that

(a)	$X$ has a finite open covering $\left\{U_{\alpha }\right\}$ such that each $(U_{\alpha },𝒪{|}_{U_{\alpha }})$ is isomorphic to an affine algebriac variety,
(b)	$(X,𝒪)$ satisfies the separation axiom, i.e. $${\Delta }_X=\left\{(x,x)\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\hspace{0.17em}\text{is closed in}\hspace{0.17em}X\times X,$$ where the topology on $X\times X$ is the Zariski topology. (Note that the Zariski topology on $X\times X$ is, in general, finer than the product topology on $X\times X\text{.)}$

A prevariety is a ringed space which satisfies (a).

Schemes

Let $A$ be a finitely generated commutative $\overline{𝔽}\text{-algebra}$ and let $$X={\text{Hom}}_{\overline{𝔽}\text{alg}}(A,\overline{𝔽})\text{.}$$ By definition, the closed sets of $X$ in the Zariski topology are the sets $$C_J=\{M\in X\hspace{0.17em}|\hspace{0.17em}J\subseteq M\}\text{for}\hspace{0.17em}J\subseteq A,$$ where we identify the points of $X$ with the maximal ideals in $A\text{.}$ Let $U$ be an open set of $X$ and let $${𝒪}_X\left(U\right)=\{g/h\hspace{0.17em}g,h\in A,\hspace{0.17em}x\left(h\right)\ne 0\hspace{0.17em}\text{for all}\hspace{0.17em}x\in U\}\text{.}$$ Then ${𝒪}_X$ is a sheaf on $X$ and $(X,{𝒪}_X)$ is a ringed space. The space $X$ is an affine $\overline{𝔽}\text{-scheme.}$

An $\overline{𝔽}\text{-variety}$ is a ringed space $(X,{𝒪}_X)$ such that

(a)	For each $x\in X$ the stalk ${𝒪}_{X,x}$ is a local ring,
(b)	$X$ has a finite open covering $\left\{U_{\alpha }\right\}$ such that each $(U_{\alpha },{𝒪}_X{|}_{U_{\alpha }})$ is isomorphic to an affine $\overline{𝔽}\text{-scheme,}$
(c)	$(X,{𝒪}_X)$ is reduced, i.e. for each $x\in X$ the local ring ${𝒪}_{X,x}$ has no nonzero nilpotent elements,
(d)	$(X,{𝒪}_X)$ satisfies the separation axiom, i.e. $${\Delta }_X=\left\{(x,x)\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\hspace{0.17em}\text{is closed in}\hspace{0.17em}X\times X\text{.}$$

A prevariety is a ringed space which satisfies (a), (b) and (c). An $\overline{𝔽}\text{-scheme}$ is a ringed space which satisfies (a) and (b). An $\overline{𝔽}\text{-space}$ is a ringed space which satisfies (a).

Groups

A group is a set $G$ with a multiplication such that

(a)	$\left(ab\right)c=a\left(bc\right),$ for all $a,b,c\in G,$
(b)	There is an identity $1\in G,$
(c)	Every element of $G$ is invertible. Let $$[x,y]=xy{x}^{-1}{y}^{-1},\text{for}\hspace{0.17em}x,y\in G\text{.}$$ The lower central series of $G$ is the sequence $$C^1\left(G\right)\supseteq C^2\left(G\right)\supseteq \cdots ,\text{where}\hspace{0.17em}{C}^1\left(G\right)=G\hspace{0.17em}\text{and}\hspace{0.17em}{C}^{i+1}\left(G\right)=[G,C^i\left(G\right)]\text{.}$$ The derived series of $G$ is the sequence $$D^0\left(G\right)\supseteq D^2\left(G\right)\supseteq \cdots ,\text{where}\hspace{0.17em}{D}^0\left(G\right)=G\hspace{0.17em}\text{and}\hspace{0.17em}{D}^{i+1}\left(G\right)=[D^i\left(G\right),D^i\left(g\right)]\text{.}$$

Let $G$ be a group.

(a)	$G$ is abelian if $[G,G]=\left\{1\right\}\text{.}$
(b)	$G$ is nilpotent if $C^n\left(G\right)=\left\{1\right\}$ for all sufficiently large $n\text{.}$
(c)	$G$ is solvable if $D^n\left(G\right)=\left\{1\right\}$ for all sufficiently large $n\text{.}$

The radical $R\left(G\right)$ of a Lie group $G$ is the largest connected solvable normal subgroup of $G\text{.}$

A topological group is a topological space $G$ which is also a group such that multiplication and inversion $$\begin{array}{ccc}G\times G& ⟶& G\\ (g,h)& ⟼& gh\end{array}\begin{array}{ccc}G& ⟶& G\\ g& ⟼& g^{-1}\end{array}$$ are morphisms of topological spaces, i.e. continuous maps.
A Lie group is a smooth manifold with a group structure such that multiplication and inversion are morphisms of smooth manifolds, i.e. smooth maps.
A complex Lie group is a complex analytic manifold which is also a group such that multiplication and inversion are morphisms of complex analytic manifolds, i.e. holomorphic maps.
A linear algebraic group is an affine algebraic variety which is also a group such that multiplication and inversion are morphisms of affine algebraic varieties.
A group scheme is a scheme which is also a group such that multiplication and inversion are morphisms of schemes.

Lie groups

The Lie group $S^1=ℝ/\mathbb{Z}=U_1\left(ℂ\right)\text{.}$ A torus is a Lie group $G$ is isomorphic to $S^1\times \cdots S^1$ $\text{(}k$ factors), for some $k\in {\mathbb{Z}}_{>0}\text{.}$

A connected Lie group is semisimple if $R\left(G\right)=\left\{1\right\}\text{.}$

Let $G$ be a Lie group and let $x\in G\text{.}$ A tangent vector at $x$ is a linear map ${\xi }_x:C^{\infty }\left(G\right)\to ℝ$ such that $${\xi }_x\left(f_1{f}_2\right)={\xi }_x\left(f_1\right)f_2\left(x\right)+f_1\left(x\right){\xi }_x\left(f_2\right),\text{for all}\hspace{0.17em}{f}_1,f_2\in C^{\infty }\left(G\right)\text{.}$$ A vector field is a linear map $\xi :C^{\infty }\left(G\right)\to C^{\infty }\left(G\right)$ such that $$\xi \left(f_1{f}_2\right)=\xi \left(f_1\right)f_2+f_1\xi \left(f_2\right),\text{for}\hspace{0.17em}{f}_1,f_2\in C^{\infty }\left(G\right)\text{.}$$ A left invariant vector field on $G$ is a vector field $\xi :C^{\infty }\left(G\right)\to C^{\infty }\left(G\right)$ such that $$L_g\xi =\xi L_g,\text{for all}\hspace{0.17em}g\in G\text{.}$$ A one parameter subgroup of $G$ is a smooth group homomorphism $\gamma :ℝ\to G\text{.}$ If $\gamma$ is a one parameter subgroup of $G$ define $$\frac{d}{dt}f\left(\gamma \left(t\right)\right)=\underset{h\to 0}{\text{lim}}\frac{f\left(\gamma (t+h)\right)-f\left(\gamma \left(t\right)\right)}{h}\text{.}$$ The following proposition says that we can identify the three vector spaces

(1)	{left invariant vector fields on $G\text{},}$
(2)	{one parameter subgroups of $G\text{},}$
(3)	{tangent vectors at $1\in G\text{}.}$

The maps $$\begin{array}{ccc}\left\{\text{left invariant vector fields}\right\}& ⟶& \left\{\text{tangent vectors at}\hspace{0.17em}1\right\}\\ \xi & ⟼& {\xi }_1\end{array}$$ and $$\begin{array}{ccc}\left\{\text{{one paramemeter subgr}\right\}& ⟶& \left\{\text{tangent vectors at}\hspace{0.17em}1\right\}\\ \gamma & ⟼& {\gamma }_1\end{array}$$ where $${\xi }_{1}f=\left(\xi f\right)\left(1\right),\text{and}{\gamma }_1=\left(\frac{d}{dt}f\left(\gamma \left(t\right)\right)\right){|}_{t=0},$$ are vector space isomorphisms.

The Lie algebra $𝔤=\text{Lie}\left(G\right)$ of the Lie group $G$ is the tangent space to $G$ at the identity with the bracket $[,]:𝔤\otimes 𝔤\to 𝔤$ given by $$[{\xi }_1,{\xi }_2]={\xi }_1{\xi }_2-{\xi }_2{\xi }_1,\text{for}{\xi }_1,{\xi }_2\in 𝔤\text{.}$$ Let $\varphi :G\to H$ be a Lie group homomorphism and let $𝔤=\text{Lie}\left(G\right)$ and $𝔥=\text{Lie}\left(G\right)\text{.}$ Then $$\begin{array}{ccc}{C}^{\infty }\left(H\right)& \stackrel{{\varphi }^{*}}{⟶}& C^{\infty }\left(G\right)\\ f& ⟼& f\circ \varphi \end{array}$$ and the differential of $\varphi$ is the Lie group homomorphism $𝔤\stackrel{d\varphi }{⟶}𝔥$ given by $$\begin{array}{cc}d\varphi \left({\xi }_1\right)={\xi }_1\circ {\varphi }^{*},& \text{if}\hspace{0.17em}{\xi }_1\hspace{0.17em}\text{is a tangent vectors at the identity,}\\ d\varphi \left(\xi \right)=\xi \circ {\varphi }^{*},& \text{if}\hspace{0.17em}\xi \hspace{0.17em}\text{is a left invariant vector field,}\\ d\varphi \left(\gamma \right)=\varphi \circ \gamma ,& \text{if}\hspace{0.17em}\gamma \hspace{0.17em}\text{is a one parameter subgroup.}\end{array}$$ (Note: It should be checked that (a) the map $d\varphi$ is well defined, (b) the three definitions of $d\varphi$ are the same, and (c) that $d\varphi$ is a Lie algebra homorphisms. These checks are not immediate, but are straightforward manipulations of the definitions.) The map $$\begin{array}{ccc}\text{the category of Lie groups}& ⟶& \text{the category of Lie algebras}\\ G& ⟼& \text{Lie}\left(G\right)\\ \varphi & ⟼& d\varphi \end{array}$$ is a functor. This functor is not one-to-one; for example, the Lie groups $O_n\left(ℝ\right)$ and ${SO}_n\left(ℝ\right)$ have the same Lie algebra. On the other hand, the Lie algebra contains the structure of the Lie groups in a neighborhood of the identity. The exponential map is $$\begin{array}{ccc}𝔤& ⟶& G\\ tX& ⟼& e^{tX},\end{array}\text{where}{e}^{t}X=\gamma \left(t\right)$$ is the one parameter subgroup corresponding to $X\in 𝔤\text{.}$ This map is a homeomorphism from a neighborhood of $0$ in $𝔤$ to a neighborhood of $1$ in $G\text{.}$

(Lie’s theorem) The functor $$\begin{array}{cccc}\text{Lie}:& \left\{\text{connected simply connected Lie groups}\right\}& ⟶& \left\{\text{Lie algebras}\right\}\\ & G& ⟼& 𝔤=\text{Lie}\left(G\right)=T_1\left(G\right)\end{array}$$ is an equivalence of categories.

If $𝔤$ is a Lie subalgebra of ${𝔤𝔩}_n$ then the matrices $$\left\{e^{tX}\hspace{0.17em}\right|\hspace{0.17em}t\in ℝ,\hspace{0.17em}X\in {𝔤𝔩}_n\},\text{where}{e}^{t}X=\sum _{k\ge 0}\frac{t^k{X}^k}{k!},$$ form a group with Lie algebra $𝔤\text{.}$ $$\begin{array}{c}{e}^{tX}{e}^{tY}=e^{t(X+Y)+(t^2/2)[X,Y]+\cdots },\\ e^{tX}{e}^{tY}{e}^{-tX}=e^{tY+t^2[X,Y]+\cdots },\\ e^{tX}{e}^{tY}{e}^{-tX}{e}^{-tY}=e^{t^2[X,Y]+\cdots },\end{array}$$

Let $G$ be a Lie group and let $𝔤=\text{Lie}\left(G\right)\text{.}$ Let $x\in \text{.}G$ Then the differential of the Lie group homomorphism $$\begin{array}{cccc}{\text{Int}}_x:& G& ⟶& G\\ & g& ⟼& xg{x}^{-1}\end{array}$$ is a Lie algebra homomorphism $$\begin{array}{cccc}{\text{Ad}}_x:& 𝔤& ⟶& 𝔤\text{.}\end{array}$$ Since there is a map ${\text{Ad}}_x$ for each $x\in G,$ there is a map $$\begin{array}{cccc}\text{Ad}:& G& ⟶& GL\left(𝔤\right)\\ & x& ⟼& {\text{Ad}}_x\end{array}\text{and}{\text{Ad}}_x{\text{Ad}}_y={\text{Ad}}_{xy},\text{for}\hspace{0.17em}x,y\in G,$$ since ${\text{Int}}_x{\text{Int}}_y={\text{Int}}_{xy}\text{.}$ The differential of Ad is $$\begin{array}{cccc}\text{ad}:& 𝔤& ⟶& \text{End}\left(𝔤\right)\\ & X& ⟼& {\text{ad}}_X\end{array},\text{where}\begin{array}{cccc}{\text{ad}}_X:& 𝔤& ⟶& 𝔤\\ & Y& ⟼& [X,Y]\end{array},$$ since $$\frac{d}{dt}\frac{d}{ds}{e}^{tX}{e}^{sY}{e}^{-tX}{|}_{s=0,t=0}=[X,Y],\text{for}\hspace{0.17em}X,Y\in 𝔤\text{.}$$ Define a (right) action of $G$ on $C^{\infty }\left(G\right)$ by $$\left(R_{x}f\right)\left(g\right)=f\left(gx\right),\text{for}\hspace{0.17em}x\in G,f\in C^{\infty }\left(G\right),g\in G\text{.}$$ Then $${\text{Ad}}_x\xi =R_x\xi R_{x^{-1}},\text{for all}\hspace{0.17em}x\in G,\xi \in 𝔤,$$ since, for $x\in G,$ ${\text{Int}}_x^{*}\left({\text{Ad}}_x\xi \right)=\xi \circ {\text{Int}}_x^{*}=\xi L_{x^{-1}}{R}_{x^{-1}}=L_{x^{-1}}\xi R_{x^{-1}}{L}_{x^{-1}}{R}_{x^{-1}}{R}_x\xi R_{x^{-1}}={\text{Int}}_x^{*}\left(R_x\xi R_{x^{-1}}\right)\text{.}$

Recall that the adjoint representation of $G$ is $$\begin{array}{cccc}\text{Ad}:& G& ⟶& GL\left(𝔤\right)\\ & x& ⟼& {\text{Ad}}_x\end{array}\text{where}\begin{array}{cccc}{\text{Ad}}_x:& 𝔤& ⟶& fg\\ & \xi & ⟼& R_x\xi R_{x^{-1}}\end{array}$$ is the differential of $$\begin{array}{cccc}{\text{Int}}_x:& G& ⟶& G\\ & g& ⟼& xg{x}^{-1}\end{array}\text{.}$$ The coadjoint representation of $G$ is the dual of the adjoint representation, i.e. the action of $G$ on ${𝔤}^{*}=\text{Hom}(𝔤,ℂ)$ given by $$\left(g\varphi \right)\left(X\right)=\varphi \left({\text{Ad}}_{g^{-1}}X\right),\text{for}\hspace{0.17em}g\in G,\hspace{0.17em}\varphi \in {𝔤}^{*},\hspace{0.17em}X\in 𝔤\text{.}$$ A coadjoint orbit is the set produced by the action of $G$ on an element $\varphi \in {𝔤}^{*},$ i.e. $G\varphi \subseteq {𝔤}^{*}$ is a coadjoint orbit.

Let $G$ be a Lie group and let $𝔤$ be the Lie algebra of $G\text{.}$ Then $G^0$ is nilpotent if and only if $\text{Lie}\left(G\right)$ is nilpotent, and $G^0$ is solvable if and only if $\text{Lie}\left(G\right)$ is solvable. A semisimple Lie group is a connected Lie group with semisimple Lie algebra.

The class of reductive Lie groups is the largest class of Lie groups which contains all the semisimple Lie groups and parabolic subgroups of them and for which the representation theory is still controllable. A real Lie group is reductive if there is a linear algebraic group $G$ over $ℝ$ whose identity component (in the Zariski topology) is reductive and a morphism $\nu :G\to G\left(ℝ\right)$ with finite kernel, whose image is an open subgroup of $G\left(ℝ\right)\text{.}$ For the definition of Harish-Chandra class see Knapp’s article.

(a)	$U\left(n\right)=\{x\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}x{\bar{x}}^t=\text{id}\}\text{.}$
(b)	$Sp(2n,ℂ)=\{A\in M_n\left(ℂ\right)\hspace{0.17em}|\hspace{0.17em}{A}^{t}JA=J\}\text{.}$
(c)	${Sp}_{2n}=Sp(2n,ℂ)\cap U\left(2n\right)\text{.}$

The simple compact Lie groups are

(a)	(Type $A\text{)}$ $S{U}_n\left(ℂ\right)$
(b)	(Type $B_n\text{)}$ ${SO}_{2n+1}\left(ℝ\right),$ $n\ge 1,$
(c)	(Type $C_n\text{)}$ ${Sp}_{2n}\left(ℂ\right)\cap U_n,$ $n\ge 1,$
(d)	(Type $D_n\text{)}$ ${SO}_{2n}\left(ℝ\right),$ $n\ge 4,$
(e)	???

If $G$ is a Lie group such that $G/G^0$ is finite then

(a)	$G$ has a maximal compact subgroup,
(b)	Any two maximal compact subgroups are conjugate,
(c)	$G$ is homeomorphic to $K\times {ℝ}^m$ under the map $$\begin{array}{ccc}K\times 𝔭& ⟶& G\\ (k,x)& ⟼& k{e}^x\end{array}$$ where $K$ is a maximal compact subgroup of $G$ and $𝔭=?????????????\text{.}$
(d)	If $G$ is a semisimple Lie group then $$K=\{g\in G\hspace{0.17em}|\hspace{0.17em}\Theta \left(g\right)=g\},$$ where $\Theta$ is the Cartan involution on $G,$ is a maximal compact subgroup of $G\text{.}$ For matrix groups $$\begin{array}{cccc}\Theta :& G& ⟶& G\\ & g& ⟼& \overline{{\left(g^{-1}\right)}^t}\end{array}$$ is the Cartan involution.

On the Lie algebra level $$\begin{array}{ccccc}\begin{array}{cccc}\theta :& 𝔤& ⟶& fg\\ & x& ⟼& -{\bar{x}}^t\end{array}& & 𝔩=\{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\theta x=x\},& & 𝔭=\{s\in 𝔤\hspace{0.17em}|\hspace{0.17em}\theta x=-x\},\\ 𝔤=𝔩\oplus 𝔭,& & 𝔲=𝔩\oplus i𝔭,& & {𝔤}_{ℂ}=𝔤\oplus i𝔤=𝔲\oplus i𝔲\text{.}\end{array}$$

(a)

There is an equivalence of categories $$\begin{array}{ccc}\left\{\text{compact connected Lie groups}\right\}& ⟷& \left\{\text{connected reductive algebraic groups over}\hspace{0.17em}ℂ\right\}\\ U& ⟷& G\end{array}$$ where $U$ is the maximal compact subgroup of $G$ and $G$ is the algebraic group with coordinate ring $C{\left(U\right)}^{\text{rep}}\text{.}$ The group $G$ is the complexification of $U\text{.}$

(b)

The functor $$\begin{array}{cccc}{\text{Res}}_K^G:& \left\{\text{holomorphic representations of}\hspace{0.17em}G\right\}& ⟶& \left\{\text{representations of}\hspace{0.17em}K\right\}\end{array}$$ is an equivalence of categories.

		Proof.
	(a) The point of (a) is that for compact groups the continuous functions separate the points of $G$ and for algebraic groups the polynomial functions separate the points of $G,$ and, for $ℂ$ and $ℝ$ the polynomial functions are dense in the continuous functions. $\square$

Examples: Under the equivalence of (???)

(a)	semisimple algebraic groups correspond exactly to the Lie groups with finite center,
(b)	algebraic tori correspond exactly to geometric tori.
(c)	irreducible finite dimensional representations of $G$ correspond exactly to irreducible finite dimensional representations of $U\text{.}$ $$\begin{array}{ccc}{U}_n& ⟷& {GL}_n\left(ℂ\right)\\ S{U}_n& ⟷& {SL}_n\left(ℂ\right)\\ {SO}_{2n+1}\left(ℝ\right)& ⟷& {SO}_{2n+1}\left(ℂ\right)\\ {Sp}_{2n}& ⟷& {Sp}_{2n}\left(ℂ\right)\\ {SO}_{2n}\left(ℝ\right)& ⟷& {SO}_{2n}\left(ℂ\right)\end{array}$$ Other examples are ${GL}_n\left(ℂ\right),$ ${SL}_n\left(ℂ\right),$ ${PGL}_n\left(ℂ\right),$ $O_n\left(ℂ\right),$ ${SO}_n\left(ℂ\right),$ ${\text{Pin}}_n,$ ${\text{Spin}}_n,$ ${Sp}_{2n},$ ${PSp}_{2n},$ $U_n\left(ℂ\right),$ ${SU}_n\left(ℂ\right),$ $U_n\left(ℂ\right)/Z\left(U_n\left(ℂ\right)\right),$ $O_n\left(ℝ\right),$ $S{O}_n\left(ℝ\right),$ $\dots \text{.}$

Equivalences: $$\begin{array}{ccc}\left\{\text{compact Lie groups}\right\}& ⟷& \left\{\text{complex semisimple Lie groups}\right\}\\ & ⟷& \left\{\text{semisimple algebraic groups}\right\}\\ & ⟶& \left\{\text{complex semisimple Lie algebras}\right\}\end{array}$$

A representation of $G$ is an action of $G$ on a vector space by linear transformations. The words representation and $G\text{-module}$ are used interchangably. A complex representation is a representation where $V$ is a vector space over $ℂ\text{.}$ In order to distinguish the group element $g$ from the linear transformation of $V$ given by the action of $g$ write $V\left(g\right)$ for the linear transformation. Then $$V:G⟶GL\left(V\right)$$ and the statement that the representation is a group action means $$V\left(xy\right)=V\left(x\right)V\left(y\right),\text{for all}\hspace{0.17em}x,y\in G\text{.}$$ Unless otherwise stated we shall assume that all representations of $G$ are Lie group homomorphisms. A holomorphic representation is a representation in the category of complex Lie groups.

A representation is irreducible, or simple, if it has no subrepresentations (except $0$ and itself). In the case when $V$ is a topological vector space then a subrepresentation is required to be a closed subspace of $V\text{.}$ The trivial $G\text{-module}$ is the representation $$\begin{array}{cccccc}1:& G& ⟶& C^{*}& =& {GL}_1\left(ℂ\right)\\ & g& ⟼& 1\end{array}$$ If $V$ and $W$ are $G\text{-modules}$ the tensor product is the action of $G$ on $V\otimes W$ given by $$g(v\otimes w)=gv\otimes gw,\text{for}\hspace{0.17em}v\in V,\hspace{0.17em}w\in W,\hspace{0.17em}g\in G\text{.}$$ If $V$ is a $G\text{-module}$ the dual $G\text{-module}$ to $V$ is the action of $G$ on $V^{*}=\text{Hom}(V,ℂ)$ (linear maps $\psi :V\to ℂ\text{)}$ given by $$\left(g\psi \right)\left(v\right)=\psi \left(g^{-1}v\right),\text{for}g\in G,\psi \in V^{*},v\in V\text{.}$$ The maps $$\begin{array}{ccc}1\otimes V& \stackrel{\sim }{⟶}& V\\ 1\otimes v& ⟼& v\end{array}\text{and}\begin{array}{ccc}V\otimes 1& \stackrel{\sim }{⟶}& V\\ v\otimes 1& ⟼& v\end{array}$$ are $G\text{-module}$ isomorphisms for any $V\text{.}$ The maps $$\begin{array}{ccc}{V}^{*}\otimes V& ⟶& 1\\ \varphi \otimes v& ⟼& \varphi \left(v\right)\end{array}\text{and}\begin{array}{ccc}1& ⟶& V\otimes V^{*}\\ 1& ⟼& {\sum }_i{b}_i\otimes {\beta }_i^{*}\end{array}$$ where $\left\{b_i\right\}$ is a basis of $V$ and $\left\{{\beta }_i^{*}\right\}$ is the dual basis in  $V^{*}$ are $G\text{-module}$ homomorphisms.

If $V:G\to GL\left(V\right)$ is a homomorphism of Lie groups then the differential of $V$ is a map $$dV:𝔤⟶\text{End}\left(V\right)$$ which satisfies $$dV\left([x,y]\right)=[dV\left(x\right),dV\left(y\right)]=dV\left(x\right)dV\left(y\right)-dV\left(y\right)dV\left(x\right),$$ for $x,y\in 𝔤\text{.}$ A representation of a Lie algebra $𝔤,$ or $𝔤\text{-module,}$ is an action of $𝔤$ on a vector space $V$ by linear transformations, i.e. a linear map $\varphi :𝔤\to \text{End}\left(V\right)$ such that $$V\left([x,y]\right)=[V\left(x\right),V\left(y\right)]=V\left(x\right)V\left(y\right)-V\left(y\right)V\left(x\right),\text{for all}\hspace{0.17em}x,y\in 𝔤,$$ where $V\left(x\right)$ is the linear transformation of $V$ determined by the action of $x\in 𝔤\text{.}$ The trivial representation of $𝔤$ is the map $$\begin{array}{cccc}1:& 𝔤& ⟶& ℂ\\ & x& ⟼& 0\end{array}$$ If $V$ is a $𝔤\text{-module,}$ the dual $𝔤\text{-module}$ is the $𝔤\text{-action}$ on $V^{*}=\text{Hom}(V,ℂ)$ given by $$\left(x\varphi \right)\left(v\right)=\varphi (-xv),\text{for}\hspace{0.17em}x\in 𝔤,\hspace{0.17em}\varphi \in V^{*},\hspace{0.17em}v\in V\text{.}$$ If $V$ and $W$ are $𝔤\text{-modules}$ the tensor product of $V$ and $W$ is the $𝔤\text{-action}$ on $V\otimes W$ given by $$x(v\otimes w)=xv\otimes w+v\otimes xw,x\in 𝔤,v\in V,w\in W\text{.}$$ The definitions of the trivial, dual and tensor product $𝔤\text{-modules}$ are accounted for by the following formulas: $$\begin{array}{ccc}\frac{d}{dt}1{|}_{t=0}& =& \frac{d}{dt}{e}^{t·0}{|}_{t=0}=0,\\ \frac{d}{dt}{\left(e^{tX}\right)}^{-1}{|}_{t=0}& =& \frac{d}{dt}{e}^{-tX}{|}_{t=0}=-X,\\ \frac{d}{dt}(e^{tX}\otimes e^{tX}){|}_{t=0}& =& \frac{d}{dt}(1+tX+\frac{t^2{X}^2}{2!}+\cdots )\otimes (1+tX+\frac{t^2{X}^2}{2!}+\cdots ){|}_{t=0}\\ & =& \frac{d}{dt}(1\otimes 1+t(X\otimes 1+1\otimes X)+\cdots ){|}_{t=0}\\ & =& X\otimes 1+1\otimes X\text{.}\end{array}$$

Lie algebras

A Lie algebra over a field $F$ is a vector space $𝔤$ over $F$ with a bracket $[,]:𝔤\times 𝔤\to 𝔤$ which is bilinear and satisfies

(1)	$[x,y]=-[y,x],$ for all $x,y\in 𝔤,$
(2)	(The Jacobi identity) $[x,[y,z]]+[z,[x,y]]+[y,[z,x]]=0,$ for all $x,y,z\in \text{.}𝔤$

The derived series of $𝔤$ is the sequence $$D^0𝔤\supseteq D^1𝔤\supseteq \cdots ,\text{where}\hspace{0.17em}{D}^0𝔤=𝔤\hspace{0.17em}\text{and}\hspace{0.17em}{D}^{i+1}𝔤=[D^i𝔤,D^i𝔤]\text{.}$$ The lower central series of $𝔤$ is the sequence $$C^1𝔤\supseteq C^2𝔤\supseteq \cdots ,\text{where}\hspace{0.17em}{C}^0𝔤=𝔤\hspace{0.17em}\text{and}\hspace{0.17em}{C}^{i+1}𝔤=[𝔤,C^i𝔤]\text{.}$$

Let $𝔤$ be a Lie algebra.

(a)	$𝔤$ is abelian if $[𝔤,𝔤]=0\text{.}$
(b)	$𝔤$ is nilpotent if $C^n\left(𝔤\right)=0$ for all sufficiently large $n\text{.}$
(c)	$𝔤$ is solvable if $D^n\left(𝔤\right)=0$ for all sufficiently large $n\text{.}$
(d)	The radical $\text{rad}\left(𝔤\right)$ is the largest solvable ideal of $𝔤\text{.}$
(e)	The nilradical $\text{nil}\left(𝔤\right)$ is the largest solvable ideal????????? of $𝔤\text{.}$
(f)	$𝔤$ is semisimple if $\text{rad}\left(𝔤\right)=0\text{.}$
(g)	$𝔤$ is reductive if $\text{nil}\left(𝔤\right)=0\text{.}$ $𝔤$ is reductive if all its representations are completely decomposable. $𝔤$ is reductive if $𝔤=Z\left(𝔤\right)\oplus [𝔤,𝔤]$ with $[𝔤,𝔤]$ semisimple.
(h)	A Cartan subalgebra is a maximal abelian subalgebra of semisimple elements.

Then $$0\subseteq \text{nil}\left(𝔤\right)\subseteq \text{rad}\left(𝔤\right)\subseteq 𝔤$$ where $\text{nil}\left(𝔤\right)$ is nilpotent, $\text{rad}\left(𝔤\right)$ is solvable, $𝔤/\text{rad}\left(𝔤\right)$ is semisimple, $\text{rad}\left(𝔤\right)/\text{nil}\left(𝔤\right)$ is abelian, and $\text{nil}\left(𝔤\right)$ is nilpotent.

Example. [Bou, Chap. I, §4, Prop. 5] The following are equivalent:

(a)	$𝔤$ is reductive,
(b)	The adjoint representation of $𝔤$ is semisimple,
(c)	$[𝔤,𝔤]$ is a semisimple Lie algebra,
(d)	$𝔤$ is the direct sum of a semisimple Lie algebra and a commutative Lie algebra.
(e)	$𝔤$ has a finite dimensional representation such that the associated bilinear form is nondegenerate.
(f)	$𝔤$ has a faithful finite dimensional representation.
(g)	$\text{rad}\left(𝔤\right)$ is the center of $𝔤\text{.}$

The finite dimensional simple Lie algebras over $ℂ$ are

(a)	(Type $A_{n-1}\text{)}$ ${𝔰𝔩}_n\left(ℂ\right),$ $n\ge 2,$
(b)	(Type $B_n\text{)}$ ${𝔰𝔬}_{2n+1}\left(ℂ\right),$ $n\ge 1,$
(c)	(Type $C_n\text{)}$ ${𝔰𝔭}_{2n}\left(ℂ\right),$ $n\ge 1,$
(d)	(Type $D_n\text{)}$ ${𝔰𝔬}_{2n}\left(ℂ\right),$ $n\ge 4,$ and
(e)	the five simple Lie algebras $E_6,$ $E_7,$ $E_8,$ $F_4,$ $G_2\text{.}$

The finite dimensional simple Lie algebras over $ℝ$ are ?????

Linear algebraic groups

A linear algebraic group is an afine algebraic variety $G$ which is also a group such that multiplication and inversion are morphisms of algebraic varieties.

The following fundamental theorem is reason for the terminology linear algebraic group.

If $G$ is a linear algebraic group then there is an injective morphism of algebraic groups $i:G\to {GL}_n\left(F\right)$ for some $n\in {\mathbb{Z}}_{>0}\text{.}$

The multiplicative group is the linear algebraic group ${𝔾}_m=F^{*}\text{.}$

A matrix $x\in M_n\left(F\right)$ is

(a)	semisimple if it is conjugate to a diagonal matrix,
(b)	nilpotent if all it eigenvalues are $0,$ or, equivalently, if $x^n=0$ for some $n\in {\mathbb{Z}}_{>0},$
(c)	unipotent if all its eigenvalues are $1,$ or equivalently, if $x-1$ is nilpotent.

Let $G$ be an linear algebraic group and let $i:G\to {GL}_n\left(F\right)$ be an injective homomorphism. An element $g\in G$ is

(a)	semisimple if $i\left(g\right)$ is semisimple in ${GL}_n\left(F\right),$
(b)	unipotent if $i\left(g\right)$ is unipotent in ${GL}_n\left(F\right)\text{.}$

The resulting notions of semisimple and unipotent elements in $G$ do not depend on the choice of the imbedding $i:G\to {GL}_n\left(ℂ\right)\text{.}$

(Jordan decomposition) Let $G$ be a linear algebraic group and let $g\in G\text{.}$ Then there exist unique $g_s,g_u\in G$ such that

(a)	$g_s$ is semisimple,
(b)	$g_u$ is unipotent,
(c)	$g=g_s{g}_u=g_u{g}_s\text{.}$

Let $G$ be a linear algebraic group.

(a)	The radical $R\left(G\right)$ is the unique maximal closed connected solvable normal subgroup of $G\text{.}$
(b)	The unipotent radical $R_u\left(G\right)$ is the unique maximal closed connected unipotent normal subgroup of $G\text{.}$
(c)	$G$ is semisimple if $R\left(G\right)=1\text{.}$
(d)	$G$ is reductive if $R_u\left(G\right)=1\text{.}$ $G$ is reductive if its Lie algebra is reductive.
(e)	$G$ is an (algebraic) torus if $G$ is isomorphic to ${𝔾}_m\times \cdots {𝔾}_m$ $\text{(}k$ factors) for some $k\in {\mathbb{Z}}_{>0}\text{.}$
(f)	A Borel subgroup of $G$ is a maximal connected closed solvable subgroup of $G^0\text{.}$

Let $G$ be a linear algebraic group and let $G^0$ be the connected component of the identity in $G\text{.}$ Then $$1\subseteq R_u\left(G\right)\subseteq R\left(G\right)\subseteq G^0\subseteq G$$ where $R_u\left(G\right)$ is unipotent, $R\left(G\right)$ is solvable, $G^0$ is connected, $G/G^0$ is finite, $G^0/R\left(G\right)$ is semisimple, $R\left(G\right)/R_u\left(G\right)$ is a torus, and $R_u\left(G\right)$ is unipotent.

A linear algebraic group is simple if it has no proper closed connected normal subgroups. This implies that proper normal subgroups are finite subgroups of the center.

Let $G$ be an algebraic group.

(a)	If $G$ is nilpotent then $G\cong TU$ where $T$ is a torus and $U$ is unipotent.
(b)	If $G$ is connected reductive then $G=[G,G]Z^{\circ },$ where $[G,G]$ is semisimple and $[G,G]\cap Z^{\circ }$ is finite.
(c)	If $[G,G]$ is semisimple then $G$ is an almost direct product of simple groups, i.e. there are closed normal subgroups $G_1,\dots ,G_k$ in $G$ such that $G=G_1·G_2\cdots G_k$ and $G_i\cap \left(G_1\cdots {\hat{G}}_i\cdots G_k\right)$ is finite.

Example. If $G={GL}_n\left(ℂ\right)$ then $$[G,G]={SL}_n\left(ℂ\right),Z^{\circ }=ℂ·\text{Id},\text{and}[G,G]\cap Z^{\circ }=\{\lambda ·\text{Id}\hspace{0.17em}|\hspace{0.17em}{\lambda }^n=1\}\cong \mathbb{Z}/n\mathbb{Z}\text{.}$$

Structure of a simple algebraic group

$$x_{\alpha }\left(t\right)=e^{t{X}_{\alpha }},w_{\alpha }\left(t\right)=x_{\alpha }\left(t\right)x_{-\alpha }\left(t^{-1}\right)x_{\alpha }\left(t\right),h_{\alpha }\left(t\right)=w_{\alpha }\left(t\right)w_{\alpha }{\left(1\right)}^{-1},$$ $$U=⟨x_{\alpha }\left(t\right)\hspace{0.17em}|\hspace{0.17em}\alpha >0⟩,T=⟨h_{\alpha }\left(t\right)⟩N=⟨w_{\alpha }\left(t\right)⟩B=TUW=N/T$$ The Langlands decomposition of a parabolic is $P=MAN$ where $$\begin{array}{ccc}M& =& \left(\begin{array}{c}{A}_1\\ & A_2& & 0\\ & & \ddots \\ & 0& & A_{\ell -1}\\ & & & & A_{\ell }\end{array}\right),\text{det}\left(A_i\right)=1,\\ A& =& \left(\begin{array}{c}{a}_1\text{Id}\\ & a_2\text{Id}& & 0\\ & & \ddots \\ & 0& & a_{\ell -1}\text{Id}\\ & & & & a_{\ell }\text{Id}\end{array}\right),a_i>0,\\ N& =& \left(\begin{array}{c}\text{Id}\\ & \text{Id}& & *\\ & & \ddots \\ & 0& & \text{Id}\\ & & & & \text{Id}\end{array}\right),\end{array}$$ and there is a corresponding decomposition $𝔭=𝔪\oplus 𝔞\oplus 𝔫$ at the Lie algebra level.

The Iwasawa decomposition of $G$ is $G=KAN$ where $$\begin{array}{ccc}K& =& \text{a maximal compact subgroup of}\hspace{0.17em}G,\\ A& =& \left(\begin{array}{c}{a}_1\\ & a_2& & 0\\ & & \ddots \\ & 0& & a_{\ell -1}\\ & & & & a_{\ell }\text{Id}\end{array}\right),\text{det}\left(A_i\right)=1,\\ N& =& \left(\begin{array}{c}1\\ & 1& & *\\ & & \ddots \\ & 0& & 1\\ & & & & 1\end{array}\right),\end{array}$$ and the corresponding Lie algebra decomposition is $$𝔤=𝔩\oplus 𝔭=𝔩\oplus 𝔞\oplus 𝔫,\text{where}\begin{array}{ccc}𝔩& =& \{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\theta x=x\},𝔭=\{x\in 𝔤\hspace{0.17em}|\hspace{0.17em}\theta x=-x\},\\ 𝔞& =& \text{a maximal abelian subspace of}\hspace{0.17em}𝔭,\\ 𝔫& =& \text{the set of positive roots with respect to}\hspace{0.17em}𝔞\text{.}\end{array}$$ The Cartan decomposition of $G$ is $G=KAK\text{.}$ The Bruhat decomposition of $G$ is $G=BWB\text{.}$

Let $𝔤$ be a semsimple complex Lie algebra.

(a)	There is an involutory semiautomorphism ${\sigma }_0$ of $𝔤$ (relative to complex conjugation) such that $${\sigma }_0\left(X_{\alpha }\right)=-X_{\alpha },{\sigma }_0\left(H_{\alpha }\right)=-H_{\alpha },\text{for all}\hspace{0.17em}\alpha \in R\text{.}$$

Let $G$ be a Chevalley group over $ℂ$ viewed as a (real) Lie group.

(b)	There is an (analytic) automorphism $\sigma$ of $G$ such that $$\sigma x_{\alpha }\left(t\right)=x_{-\alpha }(-\bar{t}),\sigma h_{\alpha }\left(t\right)=h_{\alpha }\left({\bar{t}}^{-1}\right),\text{for all}\hspace{0.17em}\alpha \in R,\hspace{0.17em}t\in ℂ\text{.}$$
(c)	A maximal compact subgroup of $G$ is $$K=\{g\in G\hspace{0.17em}|\hspace{0.17em}\sigma \left(g\right)=g\}\text{.}$$
(d)	$K$ is semisimple and connected.
(e)	The Iwasawa decomposition is $G=BK\text{.}$
(f)	The Cartan decomposition is $G=KAK$ where $$A=\{h\in H\hspace{0.17em}|\hspace{0.17em}\mu \left(h\right)>0\hspace{0.17em}\text{for all}\hspace{0.17em}\mu \in L\}\text{.}$$

Let $\Theta$ be a P.I.D., $k$ the quotient field, and ${\Theta }^{*}$ the group of units of $\Theta$ (examples: $\Theta =\mathbb{Z},$ $\Theta =F\left[t\right],$ $\Theta ={\mathbb{Z}}_p\text{).}$ If $G$ is a Chevalley group over $k$ let $G_{\Theta }$ be the subgroup of $G$ with coordinates relative to $M$ in $\Theta \text{.}$ Now let $G$ be a semisimple Chevalley group over $k\text{.}$

(a)	The Iwasawa decomposition is $G=BK$ where $$K=G_{\Theta }\text{.}$$
(b)	The Cartan decomposition is $K{A}^{+}K$ where $$A^{+}=\{h\in H\hspace{0.17em}|\hspace{0.17em}\alpha \left(h\right)\in \Theta \hspace{0.17em}\text{for all}\hspace{0.17em}\alpha \in R^{+}\}\text{.}$$
(c)	If $\Theta$ is not a field (in particular if $\Theta =\mathbb{Z}\text{)}$ then $K$ is maximal in its commensurability class.
(d)	If $\Theta ={\mathbb{Z}}_p$ and $k={\mathbb{Q}}_p$ the $K$ is a maximal compact subgroup in the $p\text{-adic}$ topology.
(e)	If $\Theta$ is a local PID and $p$ is its unique prime then (1) The Iwahori subgroup $I=U_p^{-}{H}_{\Theta }{U}_{\Theta }$ is a subgroup of $K\text{.}$ (2) $K=\bigcup _{w\in W}IwI\text{.}$ (3) $IwI=Iw{U}_{w,\Theta }$ with the last component determined uniquely mod $U_{w,p}\text{.}$

Classification Theorems

$$\begin{array}{ccc}\left\{\text{semisimple algebraic groups over}\hspace{0.17em}ℂ\right\}& \stackrel{1-1}{⟷}& \left\{\text{lattices and root systems}\right\}\\ \left\{\text{complex semisimple Lie groups}\right\}& \stackrel{1-1}{⟷}& \left\{\text{semisimple algebraic groups over}\hspace{0.17em}ℂ\right\}\\ \left\{\genfrac{}{}{0ex}{}{\text{connected reductive}}{\text{algebraic groups over}\hspace{0.17em}ℂ}\right\}& \stackrel{1-1}{⟷}& \left\{\text{compact connected Lie groups}\right\}\\ G& ⟼& U=\text{maximal compact subgroup of}\hspace{0.17em}G\\ \text{semisimple}& ⟼& \text{finite center}\\ \text{algebraic torus}& ⟼& geometric\; torus\\ \left\{\text{connected simply connected Lie groups}\right\}& \stackrel{1-1}{⟷}& \left\{\text{finite dimensional real Lie algebras}\right\}\\ \left\{\genfrac{}{}{0ex}{}{\text{finite dimensional}}{\text{complex simple Lie algebras}}\right\}& \stackrel{1-1}{⟷}& \left\{\genfrac{}{}{0ex}{}{\text{Root systems:}}{\text{4 infinite families and 5 exceptionals}}\right\}\\ \left\{\genfrac{}{}{0ex}{}{\text{finite dimensional}}{\text{real simple Lie algebras}}\right\}& \stackrel{1-1}{⟷}& \left\{\text{12 infinite families and 23 exceptionals}\right\}\end{array}$$

Functions, measures and distributions

Let $G$ be a locally compact Hausdorff topological group and let $\mu$ be a Haar measure on $G\text{.}$ The support of a function $f$ is $$\text{supp}\hspace{0.17em}f=\{g\in G\hspace{0.17em}|\hspace{0.17em}f\left(g\right)\ne 0\}\text{.}$$ If it exists, the convolution of functions $f_1:G\to ℂ$ and $f_2:G\to ℂ$ is the function $(f_1*f_2):G\to ℂ$ given by $$\begin{array}{cc}(f_1*f_2)\left(g\right)={\int }_G{f}_1\left(h\right)f_2\left(h^{-1}g\right)d\mu \left(g\right)\text{.}& \text{(0.11)}\end{array}$$ Define an involution on functions $f:G\to ℂ$ by $$f^{*}\left(g\right)=f\left(g^{-1}\right),\text{for all}\hspace{0.17em}g\in G\text{.}$$ Useful norms on functions $f:G\to ℂ$ are defined by $$\begin{array}{ccc}{\Vert f\Vert }_1& =& {\int }_G\left|f\left(g\right)\right|d\mu \left(g\right),\\ {\Vert f\Vert }_2^2& =& {\int }_G{\left|f\left(g\right)\right|}^{2}d\mu \left(g\right),\\ {\Vert f\Vert }_{\infty }& =& \text{sup}\left\{\left|f\left(g\right)\right|\hspace{0.17em}\right|\hspace{0.17em}g\in G\},\end{array}$$ If it exists, the inner product of functions $f_1:G\to ℂ$ and $f_2:G\to ℂ$ is $$⟨f_1,f_2⟩={\int }_G{f}_1\left(g\right)\overline{f_2\left(g^{-1}\right)}d\mu \left(g\right)\text{.}$$ The left and right actions of $G$ on functions $f:G\to ℂ$ are defined by $$\left(L_{g}f\right)\left(x\right)=f\left(g^{-1}x\right),\text{and}\left(R_{g}f\right)\left(x\right)=f\left(xg\right),g,x\in G\text{.}$$ Some space of functions are $$\begin{array}{ccc}ℂG& =& \{\text{functions}\hspace{0.17em}f:G\to ℂ\hspace{0.17em}\text{with finite support}\}\text{.}\\ {\ell }^1\left(G\right)& =& \{\text{functions}\hspace{0.17em}f:G\to ℂ\hspace{0.17em}\text{with countable support and}\hspace{0.17em}\Vert f\Vert ={\sum }_{g\in G}\left|f\left(g\right)\right|<\infty \}\text{.}\\ L^1(G,\mu )& =& \{\text{functions}\hspace{0.17em}f:G\to ℂ\hspace{0.17em}\text{such that}\hspace{0.17em}\Vert f\Vert ={\int }_G\left|f\left(g\right)\right|d\mu \left(g\right)<\infty \}\text{.}\end{array}$$

Let $X$ be a topological space. A $\sigma \text{-algebra}$ is a collection of subsets of $X$ which is closed under countable unions and complements and contains the set $X\text{.}$ A Borel set is a set in the smallest $\sigma \text{-algebra}$ $ℬ$ containing all open sets of $X\text{.}$ A Borel measure is a function $\mu :ℬ\to [0,\infty ]$ which is countably additive, i.e. $$\mu \left(\underset{i=1}{\overset{\infty }{⨆}}{A}_i\right)=\sum _{i=1}^{\infty }\mu \left(A_i\right),$$ for every disjoint collection of $A_i$ from $ℬ\text{.}$ A regular Borel measure is a Borel measure which satisfies $$\mu \left(E\right)=\text{sup}\left\{\mu \left(K\right)\hspace{0.17em}\right|\hspace{0.17em}K\subseteq E\text{, for}\hspace{0.17em}K\hspace{0.17em}\text{compact}\}=\text{inf}\left\{\mu \left(U\right)\hspace{0.17em}\right|\hspace{0.17em}E\subseteq U\text{, for}\hspace{0.17em}U\hspace{0.17em}\text{open}\},$$ for all $E\in ℬ\text{.}$ A complex Borel measure is a function $\mu :ℬ\to ℂ$ which is countably additive. The total variation measure with respect to a complex Borel measure $\mu$ is the measure $\left|\mu \right|$ given by $$\left|\mu \right|\left(E\right)=\text{sup}\sum _i\left|\mu \left(E_i\right)\right|,\text{for}\hspace{0.17em}E\in ℰ,$$ where the sup is over all countable collections $\left\{E_i\right\}$ of disjoint sets of $ℬ$ such that ${\bigcup }_i{E}_i=E\text{.}$ A regular complex Borel measure is a Borel measure on $X$ such that the total variation measure $\left|\mu \right|$ is regular. A measure $\lambda$ is absolutely continuous with respect to a measure $\mu$ if $\mu \left(E\right)=0$ implies $\lambda \left(E\right)=0\text{.}$

Let $\mu$ be a Haar measure on a locally compact group $G\text{.}$ Under the map $$\begin{array}{ccc}\left\{\text{functions}\right\}& ⟶& \left\{\text{measures}\right\}\\ f& ⟼& f\left(g\right)d\mu \left(g\right)\end{array}$$ the group algebra $ℂG$ maps to measures $\nu$ with finite support, $\ell \left(G\right)$ maps to measures with countable support, and $L^1(G,\mu )$ maps to measures $\nu$ which are absolutely continuous with respect to $\mu \text{.}$

Let $X$ be a locally compact Hausdorff topological space. Define $$C_c\left(X\right)=\{\text{continuous functions}\hspace{0.17em}f:X\to ℂ\hspace{0.17em}\text{with compact support}\}\text{.}$$ Then $C_c\left(X\right)$ is a normed vector space (not always complete) under the norm $${\Vert f\Vert }_{\infty }=\text{sup}\left\{\left|f\left(x\right)\right|\hspace{0.17em}\right|\hspace{0.17em}x\in X\}\text{.}$$ The completion $C_0\left(X\right)$ of $C_c\left(X\right)$ with respect to ${\Vert ·\Vert }_{\infty }$ is a Banach space. A distribution is a bounded linear functional $\mu :C_c\left(X\right)\to ℂ\text{.}$ The Riesz representation theorem says that with the notation $$\mu \left(f\right)={\int }_{X}f\left(x\right)d\mu \left(x\right),\text{for}\hspace{0.17em}f\in C_c\left(X\right),$$ the regular complex Borel measures on $X$ are exactly the distributions on $X\text{.}$ The norm $\Vert \mu \Vert$ is the norm of $\mu$ as a linear functional $\mu :C_c\left(X\right)\to ℂ\text{.}$ Viewing $\mu$ as a measure, $\Vert \mu \Vert =\left|\mu \right|\left(X\right),$ where $\left|\mu \right|$ is the total variation measure of $\mu \text{.}$

The support supp $\mu$ of a distribution $\mu$ is the set of $x\in X$ such that for each neighborhood $U$ of $x$ there is $f\in C_c\left(X\right)$ such that $\text{supp}\left(f\right)\subseteq U$ and $\mu \left(f\right)\ne 0\text{.}$ Define $${ℰ}_c\left(X\right)=\left\{\text{distributions}\hspace{0.17em}\mu \hspace{0.17em}\text{on}\hspace{0.17em}X\hspace{0.17em}\text{with compact support}\right\}\text{.}$$ If $\varphi :X\to Y$ is a morphism of locally compact spaces then $${\varphi }_{*}:{ℰ}_c\left(X\right)\to {ℰ}_c\left(Y\right)\text{is given by}\left({\varphi }_{*}\mu \right)\left(f\right)=\mu (f\circ \varphi ),$$ for $f\in C_c\left(Y\right)\text{.}$

Let $G$ be a locally compact topological group. Define an involution on distributions by $${\mu }^{*}\left(f\right)=\mu \left(f^{*}\right),\text{for}\hspace{0.17em}f\in C_c\left(G\right)\text{.}$$ The convolution of distributions is defined by $${\int }_{G}f\left(g\right)d({\mu }_1*{\mu }_2)\left(g\right)={\int }_G{\int }_{G}f\left(g_1{g}_2\right)d{\mu }_1\left(g_1\right)d{\mu }_2\left(g_2\right)\text{.}$$ The left and right actions of $G$ on distributions are given by $$\left(L_g\mu \right)\left(f\right)=\mu \left(L_{g^{-1}}f\right),\text{and}\left(R_g\mu \right)\left(f\right)=\mu \left(R_{g^{-1}}f\right),\text{for all}\hspace{0.17em}f\in C_c\left(G\right)\text{.}$$

Let $X$ be a smooth manifold. The vector space $C^{\infty }\left(X\right)$ is a topological vector space under a suitable topology. A compactly supported distribution on $X$ is a continuous linear functional $\mu :C^{\infty }\left(X\right)\to ℂ\text{.}$ Let $${ℰ}^1\left(X\right)=\{\text{continuous linear functionals}\hspace{0.17em}\mu :C^{\infty }\left(X\right)\to ℂ\}$$ and, for a compact subset $K\subseteq X,$ $${ℰ}^1(X,K)=\{\mu \in {ℰ}^1\left(X\right)\hspace{0.17em}|\hspace{0.17em}\text{supp}\left(\mu \right)\subseteq K\}\text{.}$$ If $\varphi :X\to Y$ is a morphism of smooth manifolds then $${\varphi }_{*}:{ℰ}^1\left(X\right)\to {ℰ}^1\left(Y\right)\text{is given by}\left({\varphi }_{*}\mu \right)\left(f\right)=\mu (f\circ \varphi )\text{.}$$

Haar measures and the modular function

Let $G$ be a locally compact Hausdorff topological group. A Haar measure on $G$ is a linear functional $\mu :C_0\left(G\right)\to ℂ$ such that

(a)	(continuity) $\mu$ is continuous with respect to the topology on $C_0\left(G\right)$ given by $${\Vert f\Vert }_{\infty }=\text{sup}\left\{\left|f\left(g\right)\right|\hspace{0.17em}\right|\hspace{0.17em}g\in G\},$$
(b)	(positivity) If $f\left(g\right)\in R_{\ge 0}$ for all $g\in G$ then $\mu \left(f\right)\in R_{\ge 0},$
(c)	(left invariance) $\mu \left(L_{g}f\right)=\mu \left(f\right),$ for all $g\in G$ and $f\in C_0\left(G\right)\text{.}$

(Existence and uniqueness of Haar measure) If $G$ is a locally compact Hausdorff topological group then $G$ has a Haar measure and any two Haar measures are proportional.

Fix a (left) Haar measure $\mu$ on $G\text{.}$ A group is unimodular if $\mu$ is also a right Haar measure on $G\text{.}$ The modular function is the function $\Delta :G\to {ℝ}_{\ge 0}$ given by $$\mu \left(f\right)=\Delta \left(g\right)\mu \left(R_{g}f\right),\text{for all}\hspace{0.17em}f\in C_0\left(G\right)\text{.}$$ The fact that the image of $\Delta$ is in ${ℝ}_{\ge 0}$ is a consequence of the positivity condition in the definition of Haar measure. There are several equivalent ways of defining the modular function $$\mu \left(f^{*}\right)=\mu \left({\Delta }^{-1}f\right)\text{or}{\int }_{G}f\left(g\right)d\mu \left(gh\right)={\int }_{G}f\left(g\right)\Delta \left(h\right)d\mu \left(g\right),\text{or}\mu \left(f\right)={\mu }_R\left(\Delta f\right),$$ for all $f\in C_0\left(G\right),$ where ${\mu }_R$ is a right Haar measure on $G\text{.}$ The group $G$ is unimodular exactly when $\Delta =1\text{.}$

Finite groups, abelian groups, compact groups, semisimple Lie groups, reductive Lie groups, and nilpotent groups are all unimodular.

(a)	On a Lie group the Haar measure is given by $$\mu \left(f\right)={\int }_{G}f\omega ,\text{for all}\hspace{0.17em}f\in C_0\left(G\right),$$ where $\omega$ is the unique positive left invariant $n$ form on $G\text{.}$
(b)	For a Lie group $G$ the modular function is given by $$\Delta \left(g\right)=\left|\text{det}\hspace{0.17em}A{d}_g\right|,\text{for all}\hspace{0.17em}g\in G\text{.}$$

Examples

(1)	$ℝ,$ under addition. Haar measure is the usual Lebesgue measure $dx$ on $ℝ\text{.}$
(2)	${ℝ}_{\ge 0},$ under multiplication. Haar measure is given by $(1/x)dx\text{.}$
(3)	${GL}_n\left(ℝ\right)$ has Haar measure $\frac{1}{{\left|\text{det}\left(x_{ij}\right)\right|}^n}\prod _{i,j=1}^{n}d{x}_{ij}\text{.}$
(4)	The group $B_n$ of upper triangular matrices in ${GL}_n\left(ℝ\right)$ has Haar measure $\frac{1}{{\prod }_{i=1}^n{\left|x_{ii}\right|}^i}\prod _{1\le i<j\le n}d{x}_{ij}\text{.}$ This group is not unimodular unless $n=1\text{.}$
(5)	A finite group has Haar measure $\mu \left(f\right)=\frac{1}{\left|G\right|}\sum _{g\in G}f\left(g\right)\text{.}$

Vector spaces and linear transformations

A vector space is a set $V$ with an addition $+:V\times V\to V$ and a scalar multiplication $ℂ\times V\to V$ such that addition makes $V$ into an abelian group and $$\begin{array}{ccc}c(v_1+v_2)& =& c{v}_1+c{v}_2,\\ c_1\left(c_{2}v\right)& =& \left(c_1{c}_2\right)v,\end{array}\text{and}\begin{array}{ccc}(c_1+c_2)v& =& c_{1}v+c_{2}v,\\ 1v& =& v\end{array}$$ for all $c,c_1,c_2\in ℂ$ and $v,v_1,v_2\in V\text{.}$ A linear transformation from a vector space $X$ to a vector space $Y$ is a map $T:X\to Y$ such that $T(c_1{v}_1+c_2{v}_2)=c_{1}T\left(v_1\right)+c_{2}T\left(v_2\right),$ for all $c_1,c_2\in ℂ$ and $v_1,v_2\in V\text{.}$ The morphisms in the category of vector spaces are linear transformations.

A topological vector space is a vector space $V$ with a topology such that addition and scalar multiplication are continuous maps. The morphisms in the category of topological vector spaces are continuous linear transformations. A set $C\subseteq V$ is convex if $tx+(1-t)y\in C,$ for all $x,y\in C,$ $t\in [0,1]\text{.}$ A topological vector space $V$ is locally convex if it has a basis of neighbourhoods of $0$ consisting convex sets.

A normed linear space is a vector space $V$ with a norm $\Vert ·\Vert :V\to {ℝ}_{\ge 0}$ such that

(a)	$\Vert x+y\Vert \le \Vert x\Vert +\Vert y\Vert ,$ for $x,y\in V,$
(b)	$\Vert \alpha x\Vert =\left|\alpha \right|\Vert x\Vert ,$ for $\alpha \in ℂ,$ $x\in V,$
(c)	$\Vert x\Vert =0$ implies $x=0\text{.}$

A linear transformation $T:X\to Y$ between normed vector spaces $X$ and $Y$ is an isometry if $\Vert Tx\Vert =\Vert x\Vert$ for all $x\in X\text{.}$ The norm of a linear transformation $T:X\to Y$ is $$\begin{array}{cc}\Vert T\Vert =\text{sup}\left\{\Vert Tx\Vert \hspace{0.17em}\right|\hspace{0.17em}x\in X,\Vert x\Vert \le 1\}\text{.}& \text{(0.15)}\end{array}$$ A linear transformation $T$ is bounded if $\Vert T\Vert <\infty \text{.}$ If $X$ and $Y$ are normed linear spaces such that points are closed then linear transformation $T:X\to Y$ is continuous if and only if it is bounded (reference??)

A Banach space is a normed linear space which is complete with respect to the metric defined by $d(x,y)=\Vert x-y\Vert \text{.}$ A Hilbert space is a vector space $V$ with an inner product $⟨,⟩:V\times V\to ℂ$ such that for all $c,c_1,c_2\in ℂ$ and $v,v_1,v_2,v_3\in V,$

(a)	$⟨v_1,v_2⟩=\overline{⟨v_2,v_1⟩},$
(b)	$⟨c_1{v}_1+c_2{v}_2,v_3⟩=c_1⟨v_1,v_3⟩+c_2⟨v_2,v_3⟩,$
(c)	$⟨v,v⟩=0$ only if $v=0,$
(d)	$V$ is a Banach space with respect to the norm given by ${\Vert v\Vert }^2=⟨v,v⟩\text{.}$

If $H$ is a Hilbert space the adjoint $T^{*}$ of a linear transformation $T:H\to H$ is the linear transformation defined by $$\begin{array}{cc}⟨T{h}_1,h_2⟩=⟨h_1,T^{*}{h}_2⟩,\text{for all}\hspace{0.17em}{h}_1,h_2\in H,& \text{(0.16)}\end{array}$$ and $T$ is unitary if $⟨T{x}_1,T{x}_2⟩=⟨x_1,x_2⟩$ for all $x_1,x_2\in H\text{.}$

Algebras

An algebra is a vector space $A$ with an associative multiplication $A\times A$ which satisfies the distributive laws, i.e. such that $A$ is a ring. A Banach algebra is a Banach space $A$ with a multiplication such that $A$ is an algebra and $$\Vert a_1{a}_2\Vert \le \Vert a_1\Vert \Vert a_2\Vert ,\text{for all}\hspace{0.17em}{a}_1,a_2\in A\text{.}$$ A $*\text{-algebra}$ is a Banach algebra with an involution $*:A\to A$ such that $$$$ An element $a$ in a $*\text{-algebra}$ is hermitian, or self adjoint, if $a^{*}=a\text{.}$ A $C^{*}\text{-algebra}$ is a $*\text{-algebra}$ $A$ such that $$\Vert a^{*}a\Vert ={\Vert a\Vert }^2,\text{for all}\hspace{0.17em}a\in A\text{.}$$ An idempotented algebra is an algebra $A$ with a set of idempotents $ℰ$ such that

(1)	For each pair $e_1,e_2\in ℰ$ there is an $e_0\in ℰ$ such that $e_0{e}_1=e_1{e}_0=e_1$ and $e_0{e}_2=e_2{e}_0=e_2,$ and
(2)	For each $a\in A$ there is an $e\in ℰ$ such that $ae=ea=a\text{.}$ A von-Neumann algebra is an algebra $A$ of operators on a Hilbert space $H$ such that (a) $A$ is closed under taking adjoints, (b) $A$ coincides with its bicommutant.

Examples

1.	The algebra $B\left(H\right)$ of bounded linear operators on a Hilbert space $H$ with the operator norm (???) and involution given by adjoint (???) is a Banach algebra.
2.	Let $G$ be a locally compact Hausdorff topological group $G$ and let $\mu$ be a Haar measure on $G\text{.}$ The vector space $$L^2(G,\mu )=\{f:G\to ℂ\hspace{0.17em}|\hspace{0.17em}{\Vert f\Vert }_2<\infty \}$$ is a Hilbert space under the operations defined in (???).
3.	Let $V$ be a vector space. Then $\text{End}\left(V\right)$ is an algebra.

Representations

A representation of a group $G,$ or $G\text{-module,}$ is an action of $G$ on a vector space $V$ by automorphisms (invertible linear transformations). A representation of an algebra $A,$ or $A\text{-module,}$ is an action of $A$ on a vector space $V$ by endomorphisms (linear transformations). A morphism $T:V_1\to V_2$ of $A\text{-modules}$ is a linear transformation such that $T\left(av\right)=aT\left(v\right),$ for all $a\in A$ and $v\in V\text{.}$ An $A\text{-module}$ $M$ is simple, or irreducible, if it has no submodules except $0$ and itself.

A representation of a topological group $G,$ or $G\text{-module,}$ is an action of $G$ on a topological vector space $V$ by automorphisms (continuous invertible linear transformations) such that the map $$\begin{array}{ccc}G\times V& ⟶& V\\ (g,v)& ⟼& gv\end{array}$$ is continuous. When dealing with representations of topological groups all submodules are assumed to be closed subspaces.

A $*\text{-representation}$ of a $*\text{-algebra}$ $A$ is an action of $A$ on a Hilbert space $H$ by bounded operators such that $$⟨a{v}_1,v_2⟩=⟨v_1,a^{*}{v}_2⟩,\text{for all}\hspace{0.17em}{v}_1,v_2\in V,\hspace{0.17em}a\in A\text{.}$$ A $*\text{-representation}$ of $A$ on $H$ is nondegenerate if $AV=\left\{av\hspace{0.17em}\right|\hspace{0.17em}a\in A,v\in V\}$ is dense in $V\text{.}$

A unitary representation of a topological group $G,$ or $G\text{-module,}$ is an action of $G$ on a Hilbert space $V$ by automorphisms (unitary continuous invertible linear transformations) such that the action $G\times V\to V$ is a continuous map.

An admissible representation of an idempotented algebra $(A,ℰ)$ is an action of $A$ on a vector space $V$ by linear transformations such that

(a)	$V=\bigcup _{e\in ℰ}eV,$
(b)	each $eV$ is finite dimensional.

A representation of an idempotented algebra is smooth if it satisfies (a).

Group algebras

(1)	Let $G$ be a group. Then $ℂG$ is the algebra with basis $G$ and multiplication forced by the multiplication in $G$ and the distributive law. A representation of $G$ on a vector space $V$ extends uniquely to a representation of $ℂG$ on $V$ and this induces an equivalence of categories between the representations of $G$ and the representations of $ℂG\text{.}$
(2)	Let $G$ be a locally compact topological group and fix a Haar measure $\mu$ on $G\text{.}$ Let $$L^1(G,\mu )=\{f:G\to ℂ\hspace{0.17em}|\hspace{0.17em}\Vert f\Vert ={\int }_G\left|f\left(g\right)\right|d\mu \left(g\right)<\infty \}\text{.}$$ Then $L^1(G,\mu )$ is a $*\text{-algebra}$ under the operations defined in (???). Any unitary representation of $G$ on a Hilbert space $H$ extends uniquely to a representation of $L^1(G,\mu )$ on $H$ by the formula $$fv={\int }_{G}f\left(g\right)gvd\mu \left(g\right),f\in L^1(G,\mu ),g\in G,$$ and this induces an equivalence of categories between the unitary representations of $G$ and the nondegenerate $*\text{-representations}$ of $L^1(G,\mu )\text{.}$
(3)	Let $G$ be a locally compact topological group. and fix a Haar measure $\mu$ on $G\text{.}$ Let $${ℰ}_c=\left\{\text{distributions on}\hspace{0.17em}G\hspace{0.17em}\text{with compact support}\right\}$$ Then ${ℰ}_c$ is a ???-algebra under the operations defined in (???). Any representation of the topological group $G$ on a complete locally convex vector space $V$ extends uniquely to a representation of ${ℰ}_c$ on $V$ by the formula $$\mu v={\int }_{G}gvd\mu \left(g\right),f\in {ℰ}_c,g\in G,$$ and this induces an equivalence of categories between the representations of $G$ on a complete locally convex vector space $V$ and the representations of ${ℰ}_c\left(g\right)$ on a complete locally convex vector space $V\text{.}$
(4)	Let $G$ be a totally disconnected locally compact unimodular group and fix a Haar measure $\mu$ on $G\text{.}$ Let $$C_c\left(G\right)=\{\text{locally constant compactly supported functions}\hspace{0.17em}f:G\to ℂ\}\text{.}$$ Then $C_c\left(G\right)$ is a idempotented algebra with with the operations in (???) and with idempotents given by $$e_K=\frac{1}{\mu \left(K\right)}{\chi }_K,\text{for open compact subgroups}\hspace{0.17em}K\subseteq G,$$ where ${\chi }_K$ denotes the characteristic function of the subgroup $K\text{.}$ Any smooth representation of $G$ extends uniquely to a smooth representation of $C_c\left(G\right)$ on $V$ by the formula in (???) and this induces an equivalence of categories between the smooth representations of $G$ and the smooth representations of $C_c\left(G\right)$ (see Bump Prop. 3.4.3 and Prop. 3.4.4). This correspondence takes admissible representations for $G$ (see Bump p. 425) to admissible representations for $C_c\left(G\right)\text{.}$
(5)	Let $G$ be a Lie group. Let $$C_c^{\infty }\left(G\right)=\left\{\text{compactly supported smooth functions on}\hspace{0.17em}G\right\}\text{.}$$ Then $C_c^{\infty }\left(G\right)$ is a ???-algebra under the operations defined in (???). Any representation of the topological group $G$ on a complete locally convex vector space $V$ extends uniquely to a representation of $C_c^{\infty }\left(G\right)$ on $V$ by the formula in (???) and this induces an equivalence of categories between the representations of $G$ on a complete locally convex vector space V and the representations of $C_c^{\infty }\left(G\right)$ on a complete locally convex vector space $V\text{.}$
(6)	Let $G$ be a reductive Lie group and let $K$ be a maximal compact subgroup of $G\text{.}$ Let $$ℰ{(G,K)}^{\text{fin}}=\{\mu \in {ℰ}_c\left(G\right)\hspace{0.17em}|\hspace{0.17em}\text{supp}\left(\mu \right)\subseteq K\hspace{0.17em}\text{and}\hspace{0.17em}\mu \hspace{0.17em}\text{is left and right}\hspace{0.17em}K\hspace{0.17em}\text{finite}\}\text{.}$$ Then $ℰ{(G,K)}^{\text{fin}}$ is a idempotented algebra with with the operations in (???) and with idempotents given by $$e_K=\frac{1}{\mu \left(K\right)}{\chi }_K,\text{for open compact subgroups}\hspace{0.17em}K\subseteq G,$$ where ${\chi }_K$ denotes the characteristic function of the subgroup $K\text{.}$ Any $(𝔤,K)\text{-module}$ extends uniquely to a smooth representation of $ℰ{(G,K)}^{\text{fin}}$ on $V$ by the formula in (???) and this induces an equivalence of categories between the $(𝔤,K)\text{-modules}$ and the smooth representations of $ℰ{(G,K)}^{\text{fin}}$ (see Bump Prop. 3.4.8). This correspondence takes admissible modules for $G$ (see Bump p. 280 and p. 193) to admissible modules for $ℰ{(G,K)}^{\text{fin}}\text{.}$ By Knapp and Vogan Cor. 1.7.1 $$ℰ{(G,K)}^{\text{fin}}=C{\left(K\right)}^{\text{fin}}{\otimes }_{U\left({𝔩}_{ℂ}\right)}U\left({𝔤}_{ℂ}\right)\text{.}$$
(7)	Let $G$ be a compact Lie group. Let $$C{\left(G\right)}^{\text{fin}}=\{f\in C^{\infty }\left(G\right)\hspace{0.17em}|\hspace{0.17em}f\hspace{0.17em}\text{is}\hspace{0.17em}G\hspace{0.17em}\text{finite}\}\text{.}$$ Then $C{\left(G\right)}^{\text{fin}}$ is an idempotented algebra with idempotents corresponding to the identity on a finite sum of blocks ${⨁}_{\lambda }{G}^{\lambda }\otimes {\bar{G}}^{\lambda }\text{.}$

The category of representations of $G$ in a Hilbert space $V$ and the category of smooth representations of ${C\left(G\right)}^{\text{fin}}$ are equivalent.

(8)	Let $𝔤$ be a Lie algebra. The enveloping algebra $U𝔤$ of $𝔤$ is the associative algebra with $1$ given by Generators: $x\in 𝔤,$ and Relations: $xy-yx=[x,y],$ for all $x\in 𝔤\text{.}$

The functor $$\begin{array}{cccc}U:& \left\{\text{Lie algebras}\right\}& ⟶& \left\{\text{associative algebras}\right\}\\ & 𝔤& ⟼& U𝔤\end{array}$$ is the left adjoint of the functor $$\begin{array}{cccc}L:& \left\{\text{associative algebras}\right\}& ⟶& \left\{\text{Lie algebras}\right\}\\ & (A,·)& ⟼& (A,[,])\end{array}$$ where $(A,[,])$ is the Lie algebra given by the vector space $A$ with the bracket $[,]:A\otimes A\to ℂ$ defined by $$[a_1,a_2]=a_1{a}_2-a_2{a}_1,\text{for all}\hspace{0.17em}{a}_1,a_2\in A\text{.}$$ This means that $$\begin{array}{cc}{\text{Hom}}_{\text{Lie}}(𝔤,LA)\cong {\text{Hom}}_{\text{alg}}(U𝔤,A),\text{for all associative algebras}\hspace{0.17em}A\text{.}& \text{(0.18)}\end{array}$$ Let $\iota :𝔤\to U𝔤$ be the map given by $\iota \left(x\right)=x\text{.}$ Then (???) is equivalent to the following universal property satisfied by $U𝔤\text{:}$ If $\varphi :𝔤\to A$ is a map from $𝔤$ to an associative algebra $A$ such that $$\varphi \left([x,y]\right)=\varphi \left(x\right)\varphi \left(y\right)-\varphi \left(y\right)\varphi \left(x\right),\text{for all}\hspace{0.17em}x,y\in 𝔤,$$ then there exists an algebra homomorphism $\stackrel{\sim }{\varphi }:U𝔤\to A$ such that $\stackrel{\sim }{\varphi }\circ \iota =\varphi \text{.}$

A representation of $𝔤$ on a vector space $V$ extends uniquely to a representation of $U𝔤$ on $V$ and this induces an equivalence of categories between the representations of $𝔤$ and the representations of $U𝔤\text{.}$

Let $G$ be a Lie group and let $𝔤=ℂ{\otimes }_{ℝ}{𝔤}_{ℝ}$ be the complexification of the Lie algebra ${𝔤}_{ℝ}=\text{Lie}\left(G\right)$ of $G\text{.}$ Let $ℰ(G,\left\{1\right\})$ be the algebra of distributions $\mu :C^{\infty }\left(G\right)\to ℂ$ on $G$ such that $\text{supp}\left(\mu \right)=\left\{1\right\}\text{.}$ Then $$\begin{array}{ccc}U𝔤& ⟶& ℰ(G,\left\{1\right\})\\ x& ⟼& {\mu }_x\end{array}\text{where}{\mu }_x\left(f\right)=\frac{d}{dt}f\left(e^{tx}\right){|}_{t=0},\text{for}\hspace{0.17em}x\in 𝔤,$$ is an isomorphism of algebras.

Compact groups

Let $G$ be a compact Lie group and let $\mu$ be a Haar measure on $G\text{.}$ Assume that $\mu$ is normalized so that $\mu \left(G\right)=1\text{.}$ The algebra $C_c\left(G\right)$ (under convolution) of continuous complex valued functions on $G$ with compact support is the same as the algebra $C\left(G\right)$ of continuous functions on $G\text{.}$ The vector space $C\left(G\right)$ is a $G\text{-module}$ with $G\text{-action}$ given by $$\left(xf\right)\left(g\right)=f\left(x^{-1}g\right),\text{for}\hspace{0.17em}x\in G,\hspace{0.17em}f\in C\left(G\right)\text{.}$$ The group $G$ acts on $C\left(G\right)$ in two ways, $$\left(L_{g}f\right)\left(x\right)=f\left(g^{-1}x\right),\text{and}\left(R_{g}f\right)\left(x\right)=f\left(xg\right),$$ and these two actions commute with each other.

Suppose that $V$ is a representation of $G$ in a complete locally convex vector space. Let $(,):V\otimes V\to ℂ$ be an inner product on $V$ and define a new innner product $⟨,⟩:V\otimes V\to ℂ$ by $$⟨v_1,v_2⟩={\int }_G(g{v}_1,g{v}_2)d\mu \left(g\right),v_1,v_2\in V\text{.}$$ Under the inner product $⟨,⟩$ the representation $V$ is unitary. If $V$ is a finite dimensional representation of $G,$ $$\begin{array}{cccc}V:& G& ⟶& M_n\left(ℂ\right)\\ & g& ⟼& V\left(g\right),\end{array}\text{then}\begin{array}{cccc}\bar{V}:& G& ⟶& M_n\left(ℂ\right)\\ & g& ⟼& \overline{V\left(g\right)}=V{\left(g^{-1}\right)}^t,\end{array}$$ is another finite dimensional representation of $G\text{.}$

Every finite dimensional representation of a compact group is unitary and completely decomposable.

The representation $C\left(G\right)$ is an example of an infinite dimensional representation of $G$ which is not unitary.

If $V$ is a representation of $G$ in a complete locally convex normed vector space $V$ then the representation $V$ can be extended to be a representation of the algebra (under convolution) of continuous functions $C\left(G\right)$ on $G$ by $$\begin{array}{cc}fv={\int }_{G}f\left(g\right)gvd\mu \left(g\right),f\in C\left(G\right),\hspace{0.17em}v\in V\text{.}& \text{(0.21)}\end{array}$$ The complete locally convex assumption on $V$ is necessary to define the integral in (???).

If $V$ is a representation of $G$ define $$V^{\text{fin}}=\{v\in V\hspace{0.17em}|\hspace{0.17em}\text{the}\hspace{0.17em}G\text{-module generated by}\hspace{0.17em}v\hspace{0.17em}\text{is finite dimensional}\}\text{.}$$

The vector space ${C\left(G\right)}^{\text{rep}}$ of representative functions consists of all functions $f:G\to ℂ$ given by $$f\left(g\right)=⟨v,gw⟩,$$ for some vectors $v,$ $w$ in a finite dimensional representation of $G\text{.}$

Let G be a compact group. Then ${C\left(G\right)}^{\text{fin}}={C\left(G\right)}^{\text{rep}}\text{.}$

		Proof.
	Let $f\in {C\left(G\right)}^{\text{rep}}\text{.}$ Let $v,$ $w$ be vectors in a finite dimensional representation $V$ such that $f\left(g\right)=⟨v,gw⟩$ for all $g\in G\text{.}$ Let $\{v_1,\dots ,v_k\}$ be an orthonormal basis of $V$ and let $W$ be the vector space of linear combinations of the functions $f_j=⟨v_j,gw⟩,$ $1\le j\le k\text{.}$ Since $v$ can be written as a linear combination of the $v_j,$ the function $f$ can be written as a linear combination of the $f_j$ and so $f\in W\text{.}$ For each $1\le i\le k$ $$\left(x{f}_i\right)\left(g\right)={\stackrel{\sim }{f}}_i\left(x^{-1}g\right)=⟨v_i,x^{-1}gw⟩=⟨x{v}_i,gw⟩=⟨\sum _{j=1}^k{c}_j{v}_j,gw⟩=\sum _{j=1}^k{c}_j{f}_j\left(g\right)$$ for some constants $c_j\in ℂ\text{.}$ So the $G\text{-module}$ generated by $f$ is contained in the finite dimensional representation $W\text{.}$ So $f\in {C\left(G\right)}^{\text{fin}}\text{.}$ So ${C\left(G\right)}^{\text{rep}}\subseteq {C\left(G\right)}^{\text{fin}}\text{.}$ Let $f\in {C\left(G\right)}^{\text{fin}}$ and let $f_1=f,f_2,\dots ,f_k$ be an orthonormal basis of the finite dimensional representation $W$ generated by $f\text{.}$ Then $$f\left(g\right)=\left(g^{-1}{f}_1\right)\left(1\right)=\sum _{j=1}^k⟨f_j,g^{-1}{f}_1⟩f_j\left(1\right),\text{where}\hspace{0.17em}{c}_j=⟨f_j,g^{-1}{f}_1⟩\text{.}$$ Define a new finite dimensional representation $\bar{W}$ of $G$ which has orthonormal basis $\{{\bar{w}}_1,\dots ,{\bar{w}}_k\}$ and $G$ action given by $$g{\bar{w}}_i=\sum _{j=1}^k\overline{⟨f_j,g^{-1}{f}_i⟩}{\bar{w}}_j,1\le i\le k\text{.}$$ It is straightforward to check that $g_1\left(g_2\bar{w}\right)=\left(g_1{g}_2\right)\bar{w},$ for all $g_1,g_2\in G\text{.}$ Since $⟨{\bar{w}}_j,g{\bar{w}}_i⟩=⟨f_j,g^{-1}{f}_i⟩,$ $$f\left(g\right)=⟨\sum _{j=1}^k{c}_j{\bar{w}}_j,g{\bar{w}}_1⟩\text{where}\hspace{0.17em}{c}_j=f_j\left(1\right)$$ and so $f\in {C\left(G\right)}^{\text{rep}}\text{.}$ So ${C\left(G\right)}^{\text{fin}}\subseteq {C\left(G\right)}^{\text{rep}}\text{.}$ $\square$

(Peter-Weyl) Let $G$ be a compact Lie group. Then

(a)	${C\left(G\right)}^{\text{rep}}$ is dense in $C\left(G\right),$ under the topology defined by the sup norm.
(b)	$V^{\text{fin}}$ is dense in $V$ for all representations $V$ of $G\text{.}$
(c)	$G$ is linear, i.e. there is an injective map $i:G\to {GL}_n\left(ℂ\right)$ for some $n\text{.}$
(d)	Let $\hat{G}$ be an index set for the finite dimensional representations of $G\text{.}$ For each finite dimensional irreducible representation $G^{\lambda },$ $\lambda \in \hat{G},$ fix an orthonormal basis $\left\{v_i^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}1\le i\le d_{\lambda }\}$ of $G^{\lambda }\text{.}$ Define $M_{ij}^{\lambda }\in {C\left(G\right)}^{\text{rep}}$ by $$M_{ij}^{\lambda }\left(g\right)=⟨v_i^{\lambda },g{v}_j^{\lambda }⟩,g\in G\text{.}$$ Then $$\begin{array}{ccc}{⨁}_{\lambda \in \hat{G}}{G}^{\lambda }\otimes G^{\lambda }& ⟶& {C\left(G\right)}^{\text{rep}}\\ v_i^{\lambda }\otimes v_j^{\lambda }& ⟼& M_{ij}^{\lambda }\end{array}$$ is an isomorphism of $G\times G\text{-modules.}$
(e)	The map $$\begin{array}{ccc}{⨁}_{\lambda \in \hat{G}}{M}_{d_{\lambda }}\left(ℂ\right)& ⟶& {C\left(G\right)}^{\text{rep}}\\ E_{ij}^{\lambda }& ⟼& M_{ij}^{\lambda }\end{array}$$ is an isomorphism of algebras.

and (a), (b), (c), (d) and (e) are all equivalent.

		Proof.
	(b) $⟹$ (a) is immediate. (a) $⟹$ (b): Note that ${C\left(G\right)}^{\text{fin}}V\subseteq V^{\text{fin}}\text{.}$ Since ${C\left(G\right)}^{\text{fin}}$ is dense in $C\left(G\right),$ the closure of ${C\left(G\right)}^{\text{fin}}\left(V\right)$ contains $C\left(G\right)V\text{.}$ Let $f_1,\dots ,f_2$ be a sequence of functions in $C\left(G\right)$ such that $\mu \left(f_i\right)=1$ and the sequence approaches the $\delta$ function at $1,$ i.e. the function ${\delta }_1$ which has $\text{supp}\left({\delta }_1\right)=\left\{1\right\}\text{.}$ If $v\in V$ then the sequence $f_{1}v,f_{2}v,\dots$ approaches $1v=v$ and so $v$ is in the closure of $C\left(G\right)V\text{.}$ So the closure of $C\left(G\right)V$ is $V\text{.}$ So $V^{\text{fin}}$ is dense in $V\text{.}$ The following method of making this precise is taken more or less from Bröcker and tom Dieck. An operator $K:C\left(G\right)\to C\left(G\right)$ is compact if, for every bounded $B\subseteq C\left(G\right),$ every sequence $\left(f_n\right)\subseteq K\left(B\right)$ converges in $K\left(B\right)\text{.}$ An operator $K:C\left(G\right)\to C\left(G\right)$ is symmetric if $⟨K{f}_1,f_2⟩=⟨f_1,K{f}_2⟩$ for all $f_1,f_2\in C\left(G\right)\text{.}$ See Bröcker-tom Dieck Theorem (2.6) If $K:C\left(G\right)\to C\left(G\right)$ is a compact symmetric operator then (a) $\Vert K\Vert =\text{sup}\left\{\Vert Kf\Vert \hspace{0.17em}\right|\hspace{0.17em}\Vert f\Vert \le 1\}$ or $-\Vert K\Vert$ is an eigenvalue of $K,$ (b) All eigenspaces of $K$ are finite dimensional, (c) ${⨁}_{\lambda }{C\left(G\right)}_{\lambda }$ is dense in $C\left(G\right)\text{.}$ Proof. (b) The reason eigenspaces are finite dimensional: Let $x_1,x_2,\dots$ be an orthonormal basis. Then $K{x}_i=\lambda x_i\text{.}$ So $${\Vert K{x}_i-K{x}_j\Vert }^2=\left|{\lambda }^2\right|{\Vert x_1-x_j\Vert }^2=2{\Vert \lambda \Vert }^2$$ and this never goes to zero. (c) If not then $U^{\perp }={\left(\overline{{⨁}_{\lambda }C{\left(G\right)}_{\lambda }}\right)}^{\perp }$ is nonzero. Then $K:U^{\perp }\to U^{\perp }$ is a compact symmetric operator. So this operator has a finite dimensional eigenspace. This is a contradiction. So $U^{\perp }=0\text{.}$ So ${⨁}_{\lambda }{C\left(G\right)}_{\lambda }$ is dense in $C\left(G\right)\text{.}$ $\square$ Take $K$ to be the operator given by convolution by an approximation $\varphi$ to the $\delta$ function. Then $Kf$ is close to $f,$ $$\begin{array}{ccc}{\Vert Kf-f\Vert }_{\infty }& =& \mid {\int }_G(\delta \left(g\right)f\left(xg\right)-f\left(g\right))d\mu \left(g\right)\mid \le {\int }_G\epsilon \delta \left(g\right)d\mu \left(g\right)=\epsilon \\ & =& {\Vert \delta \left(1\right)-1\Vert }_{\infty }\le \epsilon ,\end{array}$$ and $Kf$ can be approximated by the action of $\varphi$ on finite dimensional subspaces. The symmetric condition on $K$ translates to $$\varphi \left(g\right)=\varphi \left(g^{-1}\right)$$ and the compactness condition translates to $${\int }_G\varphi \left(g\right)d\mu \left(g\right)=1\text{.}$$ Note that $${\Vert f\Vert }_2^2=\int f\left(g\right)\overline{f\left(g\right)}d\mu \left(g\right)\le \int \left|f\left(g\right)\overline{f\left(g\right)}\right|d\mu \left(g\right)\le {\Vert f\Vert }_{\infty }^2\text{.}$$ So the $L^2$ and sup norms compare. For norms of operators ${\Vert \delta *f\Vert }_{\infty }\le {\Vert \delta \Vert }_{\infty }{\Vert f\Vert }_{\infty }\text{.}$ (c) $⟹$ (a): If $\iota :G\to {GL}_n\left(ℂ\right)$ is an injection then the algebra ${C\left(G\right)}^{\text{alg}}$ generated (under pointwise multiplication) by the functions ${\iota }_{ij}$ and ${\bar{\iota }}_{ij},$ where $${\iota }_{ij}\left(g\right)=\iota {\left(g\right)}_{ij},\text{and}{\iota }_{ij}\left(g\right)=\overline{{\iota }_{ij}\left(g\right)},\text{for}\hspace{0.17em}g\in G,$$ is contained in ${C\left(G\right)}^{\text{fin}}\text{.}$ This subalgebra separates points of $G$ and is closed under pointwise multiplication, and conjugation and so, by the Stone-Weierstrass theorem, is dense in $C\left(G\right)\text{.}$ So ${C\left(G\right)}^{\text{fin}}$ is dense in $C\left(G\right)\text{.}$ (a) $⟹$ (c): The elements of $C\left(G\right)$ distinguish the points of $G$ and so the functions in ${C\left(G\right)}^{\text{rep}}$ distinguish the points of $G\text{.}$ For each $g\in G$ fix a function $f_g$ such that $\left(g{f}_g\right)\left(1\right)=f_g\left(g^{-1}\right)\ne f_g\left(1\right)$ and let $V_g$ be the finite dimensional representation of $G$ generated by $f_g\text{.}$ By choosing $g_i\notin K_{i-1}$ we can find a sequence $g_1,g_2,\dots$ of elements of $G$ such that $$K_1\supseteq K_2\supseteq \dots ,\text{where}\hspace{0.17em}{K}_j=\text{ker}(V_{g_1}\oplus \cdots \oplus V_{g_j}),$$ and $K_i\ne K_{i+1}\text{.}$ Since each $K_i$ is a closed subgroup of $G,$ and $G$ is compact there is a finite $n$ such that $K_n=\left\{1\right\}\text{.}$ Then $W=V_{g_1}\oplus \cdots V_{g_n}$ is a finite dimensional representation of $G$ with trivial kernel. So there is an injective map from $G$ into $GL\left(W\right)\text{.}$ (d) By construction this an algebra isomorphism. After all the algebra multiplication is designed to extend the $G\times G$ module structure, and this is a $G\times G$ module homomorphism since $$\begin{array}{ccc}\left((x\otimes y)(v_i^{\lambda }\otimes v_j^{\lambda })\right)\left(g\right)& =& \left(\Phi (x{v}_i^{\lambda }\otimes y{v}_j^{\lambda })\right)\left(g\right)\\ & =& x{v}_i^{\lambda }\otimes gy{v}_j^{\lambda }\\ & =& v_i^{\lambda }\otimes x^{-1}gy{v}_j^{\lambda }\\ & =& M_{ij}^{\lambda }\left(x^{-1}gy\right)\\ & =& \left(L_x{R}_y{M}_{ij}^{\lambda }\right)\left(g\right)\text{.}\end{array}$$ $\square$

Note that $$\text{Tr}\left(E_{ij}^{\lambda }\right)=⟨v_i^{\lambda },v_j^{\lambda }⟩={\delta }_{ij}\text{.}$$ Consider the $L^2$ norm on ${C\left(G\right)}^{\text{rep}}\text{.}$ $$\begin{array}{ccc}{\Vert f\Vert }_2^2& =& {\int }_{G}f\left(g\right)\overline{f\left(g\right)}d\mu \left(g\right)\\ & =& {\int }_{G}f\left(g\right)f^{*}\left(g^{-1}\right)d\mu \left(g\right)\text{where}\hspace{0.17em}{f}^{*}\left(g\right)=\overline{f\left(g^{-1}\right)}\\ & =& (f*f^{*})\left(1\right)\text{.}\end{array}$$ More generally, ${⟨f_1,f_2⟩}_2=(f_1*f_2)\left(1\right)\text{.}$ Now $$\begin{array}{cccc}\tau :& {C\left(G\right)}^{\text{rep}}& ⟶& ℂ\\ & f& ⟼& f\left(1\right)\end{array}$$ is a trace on ${C\left(G\right)}^{\text{rep}},$ i.e. $\tau (f_1*f_2)=\tau (f_2*f_1)$ for all $f_1,f_2\in {C\left(G\right)}^{\text{rep}}\text{.}$ In fact this is trace of the action of ${C\left(G\right)}^{\text{rep}}$ on itself: $$\begin{array}{ccc}\tau \left(f\right)& =& {\int }_{G}f\left(g\right)gh{|}_{h}d\mu \left(g\right)\\ & =& {\int }_{G}f\left(g\right){\delta }_{g_1}d\mu \left(g\right)\\ & =& {\int }_{G}f\left(1\right)d\mu \left(g\right)=f\left(1\right)\mu \left(G\right)=f\left(1\right)\text{.}\end{array}$$ Now consider the action of ${⨁}_{\lambda }{M}_{d_{\lambda }}\left(ℂ\right)$ on itself. Then, if $f=\left({\hat{f}}^{\lambda }\right)$ then $$\tau \left(f\right)=\sum _{\lambda \in \hat{G}}{d}_{\lambda }\text{Tr}\left(f^{\lambda }\right)\text{.}$$ So $${\Vert f\Vert }_2^2=(f*f^{*})\left(1\right)=\tau (f*f^{*})=\tau \left({\hat{f}}^{\lambda }{\left({\hat{f}}^{\lambda }\right)}^t\right)=\sum _{\lambda \in \hat{G}}{d}_{\lambda }\text{Tr}\left({\hat{f}}^{\lambda }{\left({\hat{f}}^{\lambda }\right)}^t\right)\text{.}$$ Note that $\text{Tr}\left({\text{Id}}_{\lambda }\right)=d_{\lambda }$ and $\tau \left({\text{Id}}_{\lambda }\right)=\text{???.}$

Fourier analysis for compact groups

A function $f:G\to ℂ$ is

(a)	representative if there is a finite dimensional representation $V$ of $G$ and vectors $v,w\in V$ such that $f\left(g\right)=⟨v,gw⟩$ for all $g\in G\text{.}$
(b)	square integrable if $${\Vert f\Vert }_2^2={\int }_{G}f\left(g\right)\overline{f\left(g\right)}d\mu \left(g\right)<\infty \text{.}$$
(c)	smooth if all derivatives of $f$ exist.
(d)	real analytic if $f$ has a power series expansion at every point. $$\begin{array}{ccc}{C\left(G\right)}^{\text{rep}}& =& \{\text{representative functions}\hspace{0.17em}f:G\to ℂ\},\\ L^2\left(G\right)& =& \{\text{square integrable functions}\hspace{0.17em}f:G\to ℂ\},\\ C^{\infty }\left(G\right)& =& \{\text{smooth functions}\hspace{0.17em}f:G\to ℂ\},\\ C^{\omega }\left(G\right)& =& \{\text{real analytic functions}\hspace{0.17em}f:G\to ℂ\},\end{array}$$

We have a map $$\prod _{\lambda \in \hat{G}}{M}_{d_{\lambda }}\left(ℂ\right)⟶\text{functions}\hspace{0.17em}f:G\to ℂ\text{.}$$ The set $\hat{G}$ has a norm $\Vert ·\Vert :\hat{G}\to {ℝ}_{\ge 0}\text{.}$ For $\left({\hat{f}}^{\lambda }\right)\in {\prod }_{\lambda \in \hat{G}}{M}_{d_{\lambda }}\left(ℂ\right)$ define

(a)	$\left({\hat{f}}^{\lambda }\right)$ is finite if all but a finite number of the blocks ${\hat{f}}^{\lambda }$ in $\left({\hat{f}}^{\lambda }\right)$ are $0,$
(b)	$\left({\hat{f}}^{\lambda }\right)$ is square summable if $$\sum _{\lambda \in \hat{G}}\frac{1}{d_{\lambda }}{\Vert f^{\lambda }\Vert }^2<\infty \text{.}$$
(c)	$\left({\hat{f}}^{\lambda }\right)$ is rapidly decreasing if, for all $k\in {\mathbb{Z}}_{>0},$ $\left\{{\Vert \lambda \Vert }^k\Vert {\hat{f}}^{\lambda }\Vert \hspace{0.17em}\right|\hspace{0.17em}\lambda \in \hat{G}\}$ is bounded,
(d)	$\left({\hat{f}}^{\lambda }\right)$ is exponentially decreasing if, for some $K\in {ℝ}_{>1},$ $\left\{K^{\Vert \lambda \Vert }\Vert {\hat{f}}^{\lambda }\Vert \hspace{0.17em}\right|\hspace{0.17em}\lambda \in \hat{G}\}$ is bounded.

Under the map $$\begin{array}{ccc}\{\text{functions}\hspace{0.17em}f:G\to ℂ\}& ⟶& {\prod }_{\lambda \in \hat{G}}{M}_{d_{\lambda }}\left(ℂ\right),\\ {C\left(G\right)}^{\text{rep}}& ⟼& \left\{\text{finite}\hspace{0.17em}\left({\hat{f}}^{\lambda }\right)\right\}\\ L^2(G,\mu )& ⟼& \left\{\text{square summable}\hspace{0.17em}\left({\hat{f}}^{\lambda }\right)\right\}\\ C^{\infty }\left(G\right)& ⟼& \left\{\text{rapidly decreasing}\hspace{0.17em}\left({\hat{f}}^{\lambda }\right)\right\}\\ C^{\omega }\left(G\right)& ⟼& \left\{\text{exponentially decreasing}\hspace{0.17em}\left({\hat{f}}^{\lambda }\right)\right\}\end{array}$$ The space ${C\left(G\right)}^{\text{rep}}$ is dense in $C\left(G\right)$ and $C\left(G\right)\subseteq L^2\left(G\right)\text{.}$ In fact the sup norm on $C\left(G\right)$ is related to the $L^2$ norm on $L^2\left(G\right)$ and $C\left(G\right)$ is dense in $L^2\left(G\right)\text{.}$

Abelian Lie groups

(a)	If $G$ is a connected abelian Lie group then $$G\cong {\left(S^1\right)}^k\times {ℝ}^{n-k},$$ for some $n\in {\mathbb{Z}}_{>0},$ $0\le k\le n\text{.}$
(b)	If $G$ is a compact abelian Lie group then $$G\cong {\left(S^1\right)}^k\times \mathbb{Z}/m_1\mathbb{Z}\times \mathbb{Z}/m_2\mathbb{Z}\times \cdots \times \mathbb{Z}/m_{\ell }\mathbb{Z},$$ for some $k\in {\mathbb{Z}}_{\ge 0},$ $m_1,\dots ,m_{\ell }\in {\mathbb{Z}}_{>0}\text{.}$

		Proof.
	(Sketch) (a) $$0⟶K⟶𝔤\stackrel{\text{exp}}{⟶}G⟶0,\text{where}K=\text{ker}\left(\text{exp}\right)\text{.}$$ The map exp is surjective since the image contains a set of generators of $G\text{.}$ The group $K$ is discrete since exp is a local bijection. So $K\cong {\mathbb{Z}}^k$ since it is a discrete subgroup of a vector space. So $$G\cong 𝔤/K\cong {ℝ}^n/{\mathbb{Z}}^k\cong ({ℝ}^k/{\mathbb{Z}}^k)\times {ℝ}^{n-k}\text{.}$$ Let $T=G^0\text{.}$ Then $0\to T\to G\to G/T\to 0$ and $G/T$ is discrete and compact since $T$ is open in $G\text{.}$ Thus, by part (a), $T\cong {\left(S^1\right)}^k,$ and $G/T$ is finite. So $$G\cong {\left(S^1\right)}^k\times (\mathbb{Z}/m_1\mathbb{Z})\times (\mathbb{Z}/m_2\mathbb{Z})\times \cdots \times (\mathbb{Z}/m_{\ell }\mathbb{Z})\text{.}$$ $\square$

(a)	The finite dimensional irreducible representations of $\mathbb{Z}/r\mathbb{Z}$ are $$\begin{array}{cccc}{X}^{\lambda }:& \mathbb{Z}/r\mathbb{Z}& ⟶& {ℂ}^{*}\\ & e^{2\pi ik/r}& ⟼& e^{2\pi ik\lambda /r}\end{array},0\le \lambda \le r-1\text{.}$$
(b)	The finite dimensional irreducible representations of $S^1$ are $$\begin{array}{cccc}{X}^{\lambda }:& \mathbb{Z}/r\mathbb{Z}& ⟶& {ℂ}^{*}\\ & e^{2\pi i\beta }& ⟼& e^{2\pi i\beta \lambda }\end{array},\lambda \in \mathbb{Z}\text{.}$$
(c)	The finite dimensional irreducible representations of $\mathbb{Z}$ are $$\begin{array}{cccc}z:& \mathbb{Z}& ⟶& {ℂ}^{*}\\ & r& ⟼& z^r=e^{2\pi i\lambda r}\end{array},z\in {ℂ}^{*},\lambda \in ℂ\text{.}$$
(d)	The finite dimensional irreducible representations of $ℝ$ are $$\begin{array}{cccc}z:& ℝ& ⟶& {ℂ}^{*}\\ & r& ⟼& z^r=e^{2\pi i\lambda r}\end{array},z\in {ℂ}^{*},\lambda \in ℂ\text{.}$$

Weights and roots

Let $G$ be a compact connected group. A maximal torus of $G$ is a maximal connected subgroup of $G$ isomorphic to ${\left(S^1\right)}^k$ for some positive integer $k\text{.}$

Fix a maximal torus $T$ in $G\text{.}$ The group $T$ is a maximal connected abelian subgroup of $G\text{.}$ The Weyl group is $$W=N_G\left(T\right)/T,\text{where}\hspace{0.17em}{N}_G\left(T\right)=\{g\in G\hspace{0.17em}|\hspace{0.17em}gT{g}^{-1}=T\}\text{.}$$ The Weyl group $W$ acts on $T$ by conjugation. The map $$\begin{array}{ccc}G/T\times T& \stackrel{\varphi }{⟶}& G\\ (gT,t)& ⟼& \left(gt{g}^{-1}\right)\end{array}$$ is surjective and $\text{Card}\left({\varphi }^{-1}\left(g\right)\right)=\left|W\right|$ for any $g\in G\text{.}$ It follows from this that

(a)	Every element $g\in G$ is in some maximal torus.
(b)	Any two maximal tori in $G$ are conjugate.

Thus, maximal tori exist, are unique up to conjugacy, and cover the group $G\text{.}$

Let $P$ be an index set for the irreducible representations of $T\text{.}$ Since the irreducible representations of $S^1$ are indexed by $\mathbb{Z},$ $P\cong {\mathbb{Z}}^k\text{.}$ The set $P$ is called the weight lattice of $G\text{.}$ $$\text{If}\hspace{0.17em}\lambda \in P\text{then}{X}^{\lambda }:T\to {ℂ}^{*},$$ denotes the corresponding irreducible representation of $T\text{.}$ The $W\text{-action}$ on $T$ induces a $W\text{-action}$ on $P$ via $$X^{w\lambda }\left(t\right)=X^{\lambda }\left(w^{-1}t\right),\text{for all}\hspace{0.17em}t\in T\text{.}$$ A representation $V$ of $G$ is a representation of $T,$ by restriction, and, as a $T\text{-module,}$ $$V=\underset{\lambda \in P}{⨁}{V}_{\lambda },\text{where}{V}_{\lambda }=\{v\in V\hspace{0.17em}|\hspace{0.17em}tv=X^{\lambda }\left(t\right)v\hspace{0.17em}\text{for all}\hspace{0.17em}t\in T\}\text{.}$$ The vector space $V_{\lambda }$ is the $X^{\lambda }$ isotopic component of the $T\text{-module}$ $V\text{.}$ The $W\text{-action}$ on $T$ gives $$\text{dim}\left(V_{\lambda }\right)=\text{dim}\left(V_{w\lambda }\right),\text{for all}\hspace{0.17em}w\in W\hspace{0.17em}\text{and}\hspace{0.17em}\lambda \in P\text{.}$$ The vector space $V_{\lambda }$ is the $\lambda \text{-weight}$ space of $V\text{.}$ A weight vector of weight $\lambda$ in $V$ is a vector $v$ in $V_{\lambda }\text{.}$

Let $G$ be a compact connected Lie group and let $𝔲=\text{Lie}\left(G\right)\text{.}$ The group $G$ acts on $𝔲$ by the adjoint representation. Extend the adjoint representation to be a representation of $G$ on the complex vector space $${𝔤}_{ℂ}=𝔲\oplus i𝔲=ℂ\otimes ℝ𝔲\text{.}$$ By ???, this representation extends to a representation of the complex algebraic group $G_{ℂ}$ which is the complexification of $G\text{.}$ Since $G$ is compact, the adjoint representation of $G_{ℂ}$ on ${𝔤}_{ℂ},$ and thus the adjoint representation of ${𝔤}_{ℂ}$ on itself, is completely decomposable. This shows that ${𝔤}_{ℂ}$ is a complex semisimple Lie algebra.

The adjoint representation ${𝔤}_{ℂ}$ of $G$ has a weight decomposition $${𝔤}_{ℂ}=\underset{\alpha \in P}{⨁}{𝔤}_{\alpha },$$ and the root system of $G$ is the set $$R=\{\alpha \in P\hspace{0.17em}|\hspace{0.17em}\alpha \ne 0,{𝔤}_{\alpha }\ne 0\}$$ of nonzero weights of the adjoint representation. The roots are the elements of $R\text{.}$ Set $𝔥={𝔤}_0\text{.}$ Then $${𝔤}_{ℂ}=𝔥⨁\left(\underset{\alpha \in R}{⨁}{𝔤}_{\alpha }\right)$$ is the decomposition of ${𝔤}_{ℂ}$ into the Cartan subalgebra $𝔥$ and the root spaces ${𝔤}_{\alpha }\text{.}$ (Note that the usual notation is ${𝔥}_{ℝ}=i𝔥,$ ${𝔥}_{ℂ}=𝔥\oplus i𝔥,$ where $𝔥$ is a Cartan subalgebra of $𝔤,$ i.e. a maximal abelian subspace of $𝔤\text{.}$ Also ${𝔤}_0={𝔥}_{ℂ}$ since $𝔥$ is maximal abelian in $𝔤\text{.}$ Also $𝔥=𝔱\oplus i𝔱$ where $𝔱$ is the Lie algebra of the maximal torus $T$ of $G,$ and the maximal abelian subalgebra in $𝔤\text{.}$ Don’t forget to think of $$\begin{array}{cccc}{X}^{\lambda }:& T& ⟶& {ℂ}^{*}\\ & t& ⟼& X^{\lambda }\left(t\right)\\ & e^h& ⟼& e^{\lambda \left(h\right)}\end{array},\begin{array}{cccc}\lambda :& 𝔥& ⟶& ℂ\\ & h& ⟼& \lambda \left(h\right)\end{array}$$

(?)	The Weyl group $W$ is generated by $s_{\alpha },$ $\alpha \in R\text{.}$ The action of $W$ on ${𝔥}^{\alpha }$ is generated by the transformations $$\begin{array}{cccc}{s}_{\alpha }:& {𝔥}^{*}& ⟶& {𝔥}^{*}\\ & \lambda & ⟼& \lambda -⟨\lambda ,{\alpha }^{\vee }\alpha ⟩\end{array}\text{where}{\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩},$$ and $⟨,⟩:{𝔥}^{*}\times {𝔥}^{*}\to ℝ$ is a nondegenerate symmtric bilinear form.
(1)	If $\alpha$ is a root then $-\alpha$ is a root and $\pm \alpha$ are the only multiples of $\alpha$ which are root. (The thing that makes this work is that the root spaces are pure imaginary.)
(2)	If $\alpha$ is a root then $\text{dim}\left({𝔤}_{\alpha }\right)=1\text{.}$
(3)	The only connected compact Lie groups with $\text{dim}\left(T\right)=1$ are ${SO}_3\left(ℝ\right)$ and the two fold simply connected cover of ${SO}_3\left(ℝ\right)\text{.}$

		Proof.
	(1) Suppose that $\alpha$ is a root and that $x\in {𝔤}_{\alpha }\text{.}$ $$\begin{array}{cccc}{X}_{\alpha }:& T& ⟶& {ℂ}^{*}\\ & e^h& ⟼& e^{\alpha \left(h\right)}\end{array}\text{and}\begin{array}{cccc}{X}^{-\alpha }:& T& ⟶& {ℂ}^{*}\\ & e^h& ⟼& \overline{e^{\alpha \left(h\right)}}=e^{-\alpha \left(h\right)}\end{array}$$ since $\alpha \left(h\right)\in iℝ$ for $h\in t\text{.}$ Then, for all $h\in 𝔱,$ $$\left[h\bar{x}\right]=[\bar{h},\bar{x}]=\overline{[h,x]}=\overline{\alpha \left(h\right)}\bar{x}=-\alpha \left(h\right)\bar{x},$$ and so $\bar{x}\in {𝔤}_{-\alpha }\text{.}$ Thus ${𝔤}_{-\alpha }\ne 0$ and $-\alpha$ is a root. Note that $[x,\bar{x}]\in 𝔥$ since it has weight $0\text{.}$ (2) Consider $X_{\alpha }:T\to {ℂ}^{*}\text{.}$ Then $T_{\alpha }=\text{ker}\hspace{0.17em}{X}^{\alpha }$ is closed in $T$ and is of codimension $1\text{.}$ Let $T_{\alpha }^{\circ }$ be the connected component of the identity in $T_{\alpha }$ and let $Z_{\alpha }=Z_G\left(T_{\alpha }^{\circ }\right)$ be the centralizer of $T_{\alpha }^{\circ }$ in $U$ (this is connected). Then $$ℂ{\otimes }_{ℝ}\text{Lie}\left(Z_{\alpha }\right)=𝔱\oplus i𝔱\oplus \left({\oplus }_{\underset{\beta \left(h\right)=1}{h\in T_{\alpha }}}{𝔤}_{\beta }\right)=𝔥\oplus \underset{k\in \mathbb{Z}}{⨁}{𝔤}_{k\alpha }\text{.}$$ Now $$\begin{array}{ccc}{Z}_{\alpha }& ⟶& Z_{\alpha }/T_{\alpha }^{\circ }\\ \cup |& & \cup |\\ T& ⟶& T/T_{\alpha }^{\circ }\end{array}$$ So $T/T_{\alpha }^{\circ }$ is a maximal torus of $Z_{\alpha }/T_{\alpha }^{\circ }$ and $\text{dim}\hspace{0.17em}T/T_{\alpha }^{\circ }=1\text{.}$ Then $$ℂ{\otimes }_{ℝ}\text{Lie}\left(Z_{\alpha }\right)={𝔥}_{\alpha }\oplus ℂH_{\alpha }\oplus \left(\underset{k\in \mathbb{Z}}{⨁}{𝔤}_{k\alpha }\right)\text{.}$$ If $X_{\alpha }={𝔤}_{\alpha }$ then $[X_{\alpha },X_{-\alpha }]=\lambda H_{\alpha }$ and $\lambda \ne 0$ since $ℂH$ is maximal abelian in $$\text{Lie}(Z_{\alpha }/T_{\alpha }^{\circ })=ℂH\oplus \left(\underset{k\in \mathbb{Z}}{⨁}{𝔤}_{k\alpha }\right)\text{.}$$ Now consider the action of $H_{\alpha }$ on $$ℂH\oplus \left(\underset{k\in {\mathbb{Z}}_{>0}}{⨁}{𝔤}_{k\alpha }\right)\oplus ℂX_{\alpha }\text{.}$$ Then $$\text{Tr}\left(H\right)=\frac{1}{\lambda }\text{Tr}\left([X_{\alpha },X_{-\alpha }]\right)=\frac{1}{\lambda }\text{Tr}({\text{ad}}_{X_{\alpha }}{\text{ad}}_{X_{-\alpha }}-{\text{ad}}_{X_{-\alpha }}{\text{ad}}_{X_{\alpha }})=0\text{.}$$ But this implies $$0=0+\sum _{k\in {\mathbb{Z}}_{>0}}\text{dim}\left({𝔤}_{k\alpha }\right)k\alpha \left(H_{\alpha }\right)-\alpha \left(H_{\alpha }\right)\text{.}$$ So ${𝔤}_{k\alpha }=0$ for $k>1$ and ${𝔤}_{\alpha }=ℂX_{\alpha }\text{.}$ So $\text{span}\{X_{\alpha },X_{-\alpha },H_{\alpha }\}$ is a $3$ dimensional subalgebra of $𝔤\text{.}$ If $U$ is a compact connected Lie group such that $\text{dim}\hspace{0.17em}T=1$ then $U$ has Lie algebra $$𝔤=\text{span}\{X_{\alpha },X_{-\alpha },H_{\alpha }\}=𝔲\oplus i𝔲\text{.}$$ Then the Weyl group of $U$ is $\{1,s_{\alpha }\}\cong S_2$ where $s_{\alpha }$ comes from conjugation by an element of $Z_{\alpha }$ and so $s_{\alpha }$ leaves $T_{\alpha }$ fixed. So the Weyl group of $G$ contains all the $s_{\alpha },$ $\alpha \in R\text{.}$ $\square$

Example. There are only two compact connected groups of dimension $3,$ $$SO\left(3\right)\text{and}\text{Spin}\left(3\right)\text{.}$$

		Proof.
	$G$ acts on $𝔤$ and this gives an imbedding $\text{Ad}:G\to SO\left(𝔤\right)$ (with respect to an Ad invariant form on $𝔤\text{).}$ This is an immersion since everything is connected. So $G$ is a cover of $SO\left(3\right)\text{.}$ $\square$

Weyl’s integral formula

Let $G$ be a compact connected Lie group. Let $T$ be a maximal torus of $G$ and let $W$ be the Weyl group. Let $R$ be the set of roots. Then $$\left|W\right|{\int }_{G}f\left(x\right)dx={\int }_T\prod _{\alpha \in R}(X^{\alpha }\left(t\right)-1){\int }_{G}f\left(gt{g}^{-1}\right)dgdt\text{.}$$

		Proof.
	First note that the map $G/T\times T\to G$ given by $(gT,t)\mapsto gt,$ can be used to define a (left) $G$ invariant measure on $G/T$ so that $${\int }_{G}f\left(g\right)dg={\int }_{G/T\times T}f\left(gt\right)dtd\left(gT\right),$$ and thus, for $y\in T,$ $$\begin{array}{cc}\begin{array}{ccc}{\int }_{G}f\left(gy{g}^{-1}\right)dg& =& {\int }_{G/T\times T}f\left(gty{t}^{-1}{g}^{-1}\right)dtd\left(gT\right)\\ & =& {\int }_{G/T\times T}f\left(gy{g}^{-1}\right)dtd\left(gT\right)={\int }_{G/T}f\left(gy{g}^{-1}\right)d\left(gT\right)\text{.}\end{array}& \left(a\right)\end{array}$$ Then the map $\varphi :G/T\times T\to G$ given by $(gT,t)\mapsto gt{g}^{-1}$ yields $$\begin{array}{cc}\left|W\right|{\int }_{G}f\left(g\right)dg={\int }_{G/T\times T}f\left(gt{g}^{-1}\right)J_{(gT,t)}dtd\left(gT\right),& \left(b\right)\end{array}$$ where $J_{(gT,t)}$ is the determinant of the differential at $(gT,t)$ of the map $\varphi \text{.}$ By translation, $J_{(gT,t)}$ is the same as the determinant of the differential at the identity, $(T,e),$ of the map $L_{g{t}^{-1}{g}^{-1}}\circ \varphi \circ L_{g,t},$ $$\begin{array}{ccccccc}G/T\times T& ⟶& G/T\times T& ⟶& G& ⟶& G\\ (xT,y)& ⟼& (gxT,ty)& ⟼& \left(gx\right)ty{\left(gx\right)}^{-1}& ⟼& \left(g{t}^{-1}{g}^{-1}\right)\left(gx\right)ty{\left(gx\right)}^{-1}\text{.}\end{array}$$ Since $\left(g{t}^{-1}{g}^{-1}\right)\left(gx\right)ty{\left(gx\right)}^{-1}=g{t}^{-1}xty{x}^{-1}{g}^{-1}$ this differential is $$\begin{array}{ccc}𝔤/𝔥\oplus 𝔥& ⟼& 𝔤\\ (X,Y)& ⟼& {\text{Ad}}_{𝔤}({\text{Ad}}_{t^{-1}}\left(X\right)+Y-X)\text{.}\end{array}$$ So $J_{(gT,t)}$ is the determinant of the linear transformation of $𝔤$ given by $${\text{Ad}}_{𝔤}\left(g\right)\left(\begin{array}{cc}{\text{Ad}}_{𝔤/𝔥}\left(t^{-1}\right)-{\text{id}}_{𝔤/h}& 0\\ 0& {\text{id}}_{𝔥}\end{array}\right),$$ where the second factor is a block $2\times 2$ matrix with respect to the decomposition $𝔤/𝔥\oplus 𝔥$ and ${\text{Ad}}_{𝔤/𝔥}$ is the adjoint action of $T$ restricted to the subspace $𝔤/𝔥$ in $𝔤\text{.}$ The element $t^{-1}$ acts on the root space ${𝔤}_{\alpha }$ by the value $X^{\alpha }\left(t^{-1}\right)$ where $X^{\alpha }:T\to {ℂ}^{*}$ is the character of $T$ associated to the root $\alpha \text{.}$ Since $G$ is unimodular $\text{det}\left({\text{Ad}}_g\right)=1,$ and since $𝔤/𝔥={⨁}_{\alpha \in R}{𝔤}_{\alpha },$ $$\begin{array}{cc}{J}_{(gT,t)}=\prod _{\alpha \in R}(X^{\alpha }\left(t^{-1}\right)-1)=\prod _{\alpha \in R}(X^{\alpha }\left(t\right)-1),& \left(c\right)\end{array}$$ where the last equality follows from the fact that if $\alpha$ is a root then $-\alpha$ is also a root. The theorem follows by combining (a), (b) and (c). $\square$

It follows from this theorem that, if $\chi$ and $\eta$ are class functions on $G$ then $$\begin{array}{ccc}{\int }_G\chi \left(g\right)\overline{\eta \left(g\right)}dg& =& \frac{1}{\left|W\right|}{\int }_T\prod _{\alpha \in R}(X^{\alpha }\left(t\right)-1){\int }_G\chi \left(gt{g}^{-1}\right)\overline{\eta \left(gt{g}^{-1}\right)}dgdt\\ & =& \frac{1}{\left|W\right|}{\int }_T\prod _{\alpha >0}(X^{\alpha }\left(t\right)-1)(X^{-\alpha }\left(t\right)-1)\chi \left(t\right)\overline{\eta \left(t\right)}dt\\ & =& \frac{1}{\left|W\right|}{\int }_T\prod _{\alpha >0}(X^{\alpha /2}\left(t\right)-X^{-\alpha /2}\left(t\right))(X^{-\alpha /2}\left(t\right)-X^{\alpha /2}\left(t\right))\chi \left(t\right)\overline{\eta \left(t\right)}dt\\ & =& \frac{1}{\left|W\right|}{\int }_T\prod _{\alpha >0}\left(a_{\rho }\chi \right)\left(t\right)\overline{\left(a_{\rho }\eta \right)\left(t\right)}dt\text{.}\end{array}$$

Weyl’s character formula

The adjoint representation $𝔤$ is a unitary representation of $G\text{.}$ So the Weyl group $W$ acts on $𝔥$ by unitary operators. So $W$ acts on $𝔱$ by orthogonal matrices. Identify $𝔱$ and ${𝔱}^{*}=\text{Hom}(𝔱,ℝ)=\{\alpha :𝔱\to ℝ\}$ with the inner product, $$\begin{array}{ccc}𝔱& \stackrel{\sim }{⟶}& {𝔱}^{*}\\ \alpha & ⟼& ⟨\alpha ,·⟩\text{.}\end{array}$$ For a root $\alpha$ define $${\alpha }^{\vee }=\frac{2\alpha }{⟨\alpha ,\alpha ⟩}\text{and}{H}_{\alpha }=\{x\in 𝔱\hspace{0.17em}|\hspace{0.17em}\alpha \left(x\right)=0\}\text{.}$$ Then, the reflection $s_{\alpha }$ in the hyperplane $H_{\alpha },$ which comes from $Z_{\alpha }=Z_G\left(T_{\alpha }^{\circ }\right)/T_{\alpha }^{\circ },$ is $$\begin{array}{c}\begin{array}{cccc}{s}_{\alpha }:& 𝔱& ⟶& 𝔱\\ & \lambda & ⟼& \lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha \text{.}\end{array}\\ \text{PICTURE OF HYPERPLANE AND REFLECTION.}\end{array}$$ So

(a)	$W$ acts on $𝔱,$ and
(b)	$𝔱-\bigcup _{\alpha \in R}{H}_{\alpha }={ℝ}^n\backslash \left(\bigcup _{\alpha \in R}{H}_{\alpha }\right)$ is a union of chambers (these are the connected components). $$\text{PICTURE OF CHAMBERS AND WEIGHT LATTICE}$$

The Weyl group $W$ permutes these chambers and if we fix a choice of a chamber C$$then we can identify the chambers are $wC,$ $w\in C\text{.}$ (See Bröcker-tom Dieck V (2.3iv) and the Claim at the bottom of p. 193. $$\text{PICTURE OF CHAMBERS LABELED BY}\hspace{0.17em}wC$$

Let $$\begin{array}{ccc}R\left(T\right)& =& \text{representation ring of}\hspace{0.17em}T\\ & =& \text{Grothendieck ring of representations of}\hspace{0.17em}G\text{, and}\\ R\left(G\right)& =& \text{representation ring of}\hspace{0.17em}G\text{.}\end{array}$$ This means that $R\left(G\right)=\text{span-}\left\{\left[G^{\lambda }\right]\hspace{0.17em}\right|\hspace{0.17em}\lambda \in \hat{G}\}$ with

(a)	addition given by $\left[G^{\lambda }\right]+\left[G^{\mu }\right]=[G^{\lambda }\oplus G^{\mu }],$ and
(b)	multiplication given by $\left[G^{\lambda }\right]\left[G^{\mu }\right]=[G^{\lambda }\otimes G^{\mu }]\text{.}$

Thus, in $R\left(G\right)$ it makes sense to write $$\sum _{\lambda \in \hat{G}}{m}_{\lambda }\left[G^{\lambda }\right]\text{instead of}\underset{\lambda \in \hat{G}}{⨁}{\left(G^{\lambda }\right)}^{\oplus m_{\lambda }}\text{.}$$ Define $$ℂP=\text{span-}\left\{e^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P\}\text{with multiplication}{e}^{\lambda }{e}^{\mu }=e^{\lambda +\mu },$$ for $\lambda ,\mu \in P\text{.}$ Then $$ℂP\cong R\left(T\right),\text{since}R\left(T\right)=\text{span-}\left\{\left[X^{\lambda }\right]\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P\}\text{.}$$ The action of $W$ on $R\left(T\right)$ (see (???)) induces an action of $W$ on $ℂP$ given by $$w{e}^{\lambda }=e^{w\lambda },\text{for}\hspace{0.17em}w\in W,\hspace{0.17em}\lambda \in P\text{.}$$ Note that $$\epsilon \left(w\right)=\underset{𝔥}{\text{det}}\left(w\right)=\pm 1$$ since the action of $w$ on $𝔥$ is by an orthogonal matrix. The vector spaces of symmetric and alternating functions are $$\begin{array}{ccc}ℂ{\left[P\right]}^W& =& \{f\in ℂP\hspace{0.17em}|\hspace{0.17em}wf=f\hspace{0.17em}\text{for all}\hspace{0.17em}w\in W\},\text{and}\\ 𝒜& =& \{f\in ℂP\hspace{0.17em}|\hspace{0.17em}wf==\epsilon \left(w\right)f\hspace{0.17em}\text{for all}\hspace{0.17em}w\in W\},\end{array}$$ respectively. Note that ${ℂ\left[P\right]}^W$ is a ring but $𝒜$ is only a vector space.

Define $$P^{+}=P\cap \bar{C}\text{and}{P}^{++}=P\cap C\text{.}$$ The set $P^{+}$ is the set of dominant weights. Every $W\text{-orbit}$ on $P$ contains a unique element of $P^{+}$ and so the set of monomial symmetric functions $$m_{\lambda }=\sum _{\gamma \in W\lambda }{e}^{\gamma },\lambda \in P^{+},$$ forms a basis of ${ℂ\left[P\right]}^W\text{.}$ Define $$a_{\mu }=\sum _{w\in W}\epsilon \left(w\right)e^{w\mu },$$ for $\mu \in P\text{.}$ Then

(a)	$w{a}_{\mu }=\epsilon \left(w\right)a_{\mu },$ for all $w\in W$ and all $\mu \in P,$
(b)	$a_{\mu }=0,$ if $\mu \in H_{\alpha }$ for some $\alpha ,$ and
(c)	$\left\{a_{\mu }\hspace{0.17em}\right|\hspace{0.17em}\mu \in P^{++}\}$ is a basis of $𝒜\text{.}$

The fundamental weights ${\omega }_1,\dots ,{\omega }_n$ in $𝔱$ are defined by $$⟨{\omega }_i,{\alpha }_j^{\vee }⟩={\delta }_{ij},$$ where $H_{{\alpha }_j}$ are the walls of $C\text{.}$ Write $$\alpha >0\text{if}\hspace{0.17em}⟨\lambda ,\alpha ⟩>0\hspace{0.17em}\text{for all}\hspace{0.17em}\lambda \in C\text{.}$$ Then $$\begin{array}{ccc}\rho & =& \sum _{i=1}^n{\omega }_i\\ & =& \frac{1}{2}\sum _{\alpha >0}\alpha ,\end{array}$$ is the element of $𝔱$ defined by $$⟨\rho ,{\alpha }_i^{\vee }⟩=1,\text{for all}\hspace{0.17em}{\alpha }_1,\dots ,{\alpha }_n\text{.}$$

The map $$\begin{array}{ccc}{P}^{+}& ⟶& P^{++}\\ \lambda & ⟼& \lambda +\rho \end{array}$$ is a bijection, and $$\begin{array}{ccc}{ℂ\left[P\right]}^W& ⟶& 𝒜\\ f& ⟼& a_{\rho }f\end{array}$$ is a vector space isomorphism.

		Proof.
	Since $$w\left(a_{\rho }f\right)=\left(w{a}_{\rho }\right)\left(wf\right)=\epsilon \left(w\right)a_{\rho }f,$$ the second map is well defined. Let $$g=\sum _{\lambda \in P}{g}_{\lambda }{e}^{\lambda }\in 𝒜\text{.}$$ Then, for a positive root $\alpha ,$ $$-g=s_{\alpha }g=\sum _{\lambda \in P}{g}_{\lambda }{e}^{s_{\alpha }\lambda },$$ and so $$g=\sum _{\underset{⟨\lambda ,\alpha ⟩>0}{\lambda }}{g}_{\lambda }(e^{\lambda }-s^{s_{\alpha }\lambda })\text{.}$$ Since $$e^{\lambda }-e^{s_{\alpha }\lambda }=(e^{\lambda -\alpha }+\cdots +e^{\lambda -⟨\lambda ,{\alpha }^{\vee }⟩\alpha })(e^{\alpha }-1),$$ the element $g$ is divsible by $e^{\alpha }-1\text{.}$ Thus, since all the factors in the product are coprime in $ℂP,$ $g$ is divisible by $$\prod _{\alpha >0}(e^{\alpha }-1)=e^{\rho }\prod _{\alpha >0}(e^{\alpha /2}-e^{-\alpha /2})=e^{\rho }{a}_{\rho },$$ where the last equality follows from the fact that $a_{\rho }$ is divisible by the product ${\prod }_{\alpha >0}(e^{\alpha /2}-e^{-\alpha /2})$ and these two expressions have the same top monomial, $e^{\rho }\text{.}$ Since $g\in 𝒜$ is divisible by $a_{\rho }$ the map $ℂP\to 𝒜$ is invertible. $\square$

Define $${\chi }^{\lambda }=\frac{a_{\lambda +\rho }}{a_{\rho }},\text{for}\hspace{0.17em}\lambda \in P^{+},$$ so that the $\left\{{\chi }^{\lambda }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ are the basis of ${ℂ\left[P\right]}^W$ obtained by taking the inverse image of the basis $\left\{a_{\lambda +\rho }\hspace{0.17em}\right|\hspace{0.17em}\lambda \in P^{+}\}$ of $A\text{.}$ Extend these functions to all of $U$ by setting $${\chi }^{\lambda }\left(gt{g}^{-1}\right)={\chi }^{\lambda }\left(t\right),\text{for all}\hspace{0.17em}g\in U\text{.}$$ Since ${\int }_T{X}^{\lambda }\left(t\right)X^{\mu }\left(t\right)dt={\delta }_{\lambda \mu },$ for $\lambda ,\mu \in P,$ $${\int }_T{a}_{\lambda +\rho }\left(t\right)\overline{a_{\mu +\rho }\left(t\right)}dt={\delta }_{\lambda \mu }\left|W\right|,$$ and thus, by (???), $${\delta }_{\lambda \mu }={\int }_G{\chi }^{\lambda }\left(g\right)\overline{{\chi }^{\mu }\left(g\right)}dg,\text{for all}\hspace{0.17em}\lambda ,\mu \in P^{+}\text{.}$$ Thus the ${\chi }^{\lambda },$ $\lambda \in P^{+}$ are an orthonormal basis of the set of class functions in ${C\left(G\right)}^{\text{rep}}\text{.}$ If $U^{\lambda }$ is an irreducible representation of $U$ then $${\text{Tr}}_{U^{\lambda }}\left(G\right)=\sum _{i=1}^d{M}_{ii}^{\lambda }\left(g\right),\text{where}{M}_{ij}^{\lambda }=⟨v_i^{\lambda },g{v}_j^{\lambda }⟩,$$ for an orthonormal basis $v_1^{\lambda },\dots ,v_n^{\lambda }$ of $U^{\lambda }\text{.}$ Then $${\int }_G{\text{Tr}}_{U^{\lambda }}\left(g\right)\overline{{\text{Tr}}_{U^{\mu }}\left(g\right)}dg={\delta }_{\lambda \mu },$$ and so the functions ${\text{Tr}}_{U^{\lambda }}$ are another orthonormal basis of the set of class functions in ${C\left(G\right)}^{\text{rep}}\text{.}$ It follows that ${\chi }^{\lambda }=\pm {\text{Tr}}_{U^{\lambda }}\text{.}$

It only remains to check that the sign is positive to show that the ${\chi }^{\lambda }$ are the irreducible characters of $U\text{.}$ This follows from the following computation. $$\begin{array}{ccc}{\chi }^{\lambda }\left(1\right)& =& \underset{t\to 0}{\text{lim}}{\chi }^{\lambda }\left(e^{t\rho }\right)\\ & =& \underset{t\to 0}{\text{lim}}\frac{{\sum }_{w\in W}\epsilon \left(w\right)X^{w(\lambda +\rho )}\left(e^{t\rho }\right)}{{\sum }_{w\in W}\epsilon \left(w\right)X^{w\rho }\left(e^{t\rho }\right)}\\ & =& \underset{t\to 0}{\text{lim}}\frac{{\sum }_{w\in W}\epsilon \left(w\right)$ e^{⟨w(\lambda +\rho ),t\rho ⟩}$}{{\sum }_{w\in W}\epsilon \left(w\right)$ e^{⟨w\rho ,t\rho ⟩}$}\\ & =& \underset{t\to 0}{\text{lim}}\frac{{\sum }_{w\in W}\epsilon \left(w\right)$ e^{t⟨\lambda +\rho ,w^{-1}\rho ⟩}$}{{\sum }_{w\in W}\epsilon \left(w\right)$ e^{t⟨\rho ,w^{-1}\rho ⟩}$}\\ & =& \underset{t\to 0}{\text{lim}}\frac{a_{\rho }\left(e^{t(\lambda +\rho )}\right)}{a_{\rho }\left(e^{t\rho }\right)}\\ & =& \underset{t\to 0}{\text{lim}}\frac{{\prod }_{\alpha >0}(X^{\alpha /2}-X^{-\alpha /2})\left(e^{t(\lambda +\rho )}\right)}{{\prod }_{\alpha >0}(X^{\alpha /2}-X^{-\alpha /2})\left(e^{t\rho }\right)}\\ & =& \underset{t\to 0}{\text{lim}}\frac{{\prod }_{\alpha >0}(e^{t⟨\lambda +\rho ,\alpha /2⟩}-e^{-t⟨\lambda +\rho ,\alpha /2⟩})}{{\prod }_{\alpha >0}(e^{t⟨\rho ,\alpha /2⟩}-e^{-t⟨\rho ,\alpha /2⟩})}\\ & =& \underset{t\to 0}{\text{lim}}\prod _{\alpha >0}\frac{\text{sinh}\left(t⟨\lambda +\rho ,\alpha /2⟩\right)}{\text{sinh}\left(t⟨\rho ,\alpha /2⟩\right)}\\ & =& \prod _{\alpha >0}\frac{⟨\lambda +\rho ,\alpha /2⟩}{⟨\rho ,\alpha /2⟩}\\ & =& \prod _{\alpha >0}\frac{⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{⟨\rho ,{\alpha }^{\vee }⟩}\text{.}\end{array}$$

Let $U$ be a compact connected Lie group and let $T$ be a maximal torus and $L$ the corresponding lattice.

(a)	The irreducible representations of $U$ are indexed by dominant integral weights $\lambda \in L^{+}$ under the correspondence $$\begin{array}{ccc}\text{irreducible representations}& \stackrel{1-1}{⟶}& P^{+}\\ V^{\lambda }& ⟼& \text{highest weight of}\hspace{0.17em}{V}^{\lambda }\end{array}$$
(b)	The character of $V^{\lambda }$ is $${\chi }^{\lambda }=\frac{{\sum }_{w\in W}\epsilon \left(w\right)w^{w(\lambda +\rho )}}{{\sum }_{w\in W}\epsilon \left(w\right)w^{w\rho }},$$ where $\rho \in P^{+}$ is defined by $⟨\rho ,{\alpha }_i^{\vee }⟩=1$ for $1\le i\le n$ and $\epsilon \left(w\right)=\text{det}\left(w\right)\text{.}$
(c)	The dimension of $V^{\lambda }$ is $$d_{\lambda }=\frac{{\prod }_{\alpha >0}⟨\lambda +\rho ,{\alpha }^{\vee }⟩}{{\prod }_{\alpha >0}⟨\rho ,{\alpha }^{\vee }⟩}\text{.}$$
(d)	$${\chi }^{\lambda }=\sum _{p\in {𝒫}_{\lambda }}{e}^{p\left(1\right)},$$ where ${𝒫}_{\lambda }$ is the set of all paths obtained by acting on $p_{\lambda }$ by root operators.

Remark. By part (d) $$\text{dim}\left({\left(V^{\lambda }\right)}_{\mu }\right)=\text{\# paths in}\hspace{0.17em}{𝒫}_{\lambda }\hspace{0.17em}\text{which end at}\hspace{0.17em}\mu \text{.}$$ (For the path model some copying can be done from the Barcelona abstract.)

Remark. Point out that $R\left(T\right)=\mathbb{Z}L,$ where $L$ is the lattice corresponding to $T\text{.}$ Also point out that $R\left(U\right)={R\left(T\right)}^W\cong {\left(\mathbb{Z}L\right)}^W\text{.}$

Notes and references

This is a typed exert of Representation theory Lecture notes: Chapter 4 by Arun Ram. Research supported in part by National Science Foundation grant DMS-9622985.

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