Group algebras
Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au
and
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA
ram@math.wisc.edu
Last updates: 10 April 2010
Group algebras

Let $G$
be a group. Then $\u2102G$
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 $V$
and this induces an equivalence of categories between the representations of $G$
and the representations of $\u2102G.$

Let $G$ be a locally compact topological group and fix a Haar measure $\mu $
on $G.$ Let $${L}^{1}\left(G,\mu \right)=\left\{f:G\to \u2102\phantom{\rule{.5em}{0ex}}\phantom{\rule{.5em}{0ex}}\u2225f\u2225={\int}_{G}\leftf\left(g\right)\rightd\mu \left(g\right)<\infty \right\}.$$ Then ${L}^{1}\left(G,\mu \right)$ is a $*$algebra under the operations defined in (???). Any unitary representation of $G$
on a Hilbert space $H$
extends uniquely to a representation of ${L}^{1}\left(G,\mu \right)$ on $H$
by the formula $$fv={\int}_{G}f\left(g\right)gvd\mu \left(g\right),\phantom{\rule{2em}{0ex}}f\in {L}^{1}\left(G,\mu \right),g\in G,$$
and this induces an equivalence of categories between the unitary representations of $G$
and the nondegenerate $*$
representations of ${L}^{1}\left(G,\mu \right).$

Let $G$ be a locally compact topological group and fix a Haar measure $\mu $
on $G.$
Let $${\mathcal{E}}_{c}=\left\{\text{distributions on}G\text{with compact support}\right\}.$$
Then ${\mathcal{E}}_{c}$
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 ${\mathcal{E}}_{c}$ on $V$
by the formula $$\mu v={\int}_{G}gvd\mu \left(g\right),\phantom{\rule{2em}{0ex}}f\in {\mathcal{E}}_{c},g\in G,$$
amd this induces am equivalence of categories between the representations of $G$
on a complete locally convex vector space $V$
and the representations on
${\mathcal{E}}_{c}\left(G\right)$
on a complete locally convex vector space $V.$

Let $G$
be a totally disconnected locally compact unimodular group and fix a Haar measure $\mu $
on $G.$
Let $${C}_{c}\left(G\right)=\left\{\text{locally constant compactly supported functions}f:G\to \u2102\right\}.$$
Then ${C}_{c}\left(G\right)$
is an idempotented algebra with the operations in (???) and with idempotents given by $${e}_{K}=\frac{1}{\mu \left(K\right)}{\chi}_{K},\phantom{\rule{2em}{0ex}}\text{for open compact subgroups}K\subseteq G,$$
where ${\chi}_{K}$
denotes the characteristic function of the subgroup $K.$
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 ${C}_{c}\left(G\right)$
(see Bump Prop 3.4.3 and Prop 3.4.4). This correspondence takes admissible representations for representations for $G$
(see Bump p.425) to admissible representations for ${C}_{c}\left(G\right).$

Let $G$
be a Lie group. Let $${C}_{c}^{\infty}\left(G\right)=\left\{\text{compactly supported smooth functions on}G\right\}.$$
Then ${C}_{c}^{\infty}\left(G\right)$ is a ???algebra under the operations defined in (???). Any representation of a topological group $G$
on a complete locally connected 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 $V.$

Let $G$
be a reductive Lie group and let $K$
be a maximal compact subgroup of $G.$
Let $$\mathcal{E}{\left(G,K\right)}^{\mathrm{fin}}=\left\{\mu \in {\mathcal{E}}_{c}\left(G\right)\phantom{\rule{.5em}{0ex}}\phantom{\rule{.5em}{0ex}}\mathrm{supp}\left(\mu \right)\subseteq K\text{and}\mu \text{is left and right}K\text{finite}\right\}.$$
Then $\mathcal{E}{\left(G,K\right)}^{\mathrm{fin}}$
is an idempotented algebra with the operations in (???) and with the idempotents given by $${e}_{K}=\frac{1}{\mu \left(K\right)}{\chi}_{K},\phantom{\rule{2em}{0ex}}\text{for open compact subgroups}K\subseteq G,$$
where ${\chi}_{K}$ denotes the characteristic function of the subgroup $K.$
Any $\left(\U0001d524,K\right)$
module extends uniquely to a smooth representation of $\mathcal{E}{\left(G,K\right)}^{\mathrm{fin}}$
on $V$
by the formula in (???) and this induces an equivalence of categories between the $\left(\U0001d524,K\right)$
modules and the smooth representations of $\mathcal{E}{\left(G,K\right)}^{\mathrm{fin}}$ (see Bump Prop 3.4.8). This correspondence takes admissible modules for $G$
(see Bump p.280 and p.193) to admissible modules for $\mathcal{E}{\left(G,K\right)}^{\mathrm{fin}}.$
By Knapp and Vogan Cor 1.7.1 $$\mathcal{E}{\left(G,K\right)}^{\mathrm{fin}}=C{\left(K\right)}^{\mathrm{fin}}{\otimes}_{{\U0001d531}_{\u2102}}U\left({\U0001d524}_{\u2102}\right).$$

Let $G$
be a compact Lie group. Let $$C{\left(G\right)}^{\mathrm{fin}}=\left\{f\in {C}^{\infty}\left(G\right)\phantom{\rule{.5em}{0ex}}\phantom{\rule{.5em}{0ex}}f\text{is finite}\right\}.$$
Then $C{\left(G\right)}^{\mathrm{fin}}$
is an idempotented algebra with idempotents corresponding to the identity on a finite sum of blocks ${\u2a01}_{\lambda}{G}^{\lambda}\otimes {}^{}$G
λ
.
The category of representations of $G$ in a Hilbert space $V$ and the category of smooth representations of $C{\left(G\right)}^{\mathrm{fin}}$
are equivalent.

Let $\U0001d524$
be a Lie algebra. The enveloping algebra ${U}_{\U0001d524}$
of $\U0001d524$
of the associative algebra with 1 given by $$\text{Generators:}x\in \U0001d524,\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}$$
$$\text{Relations:}xyyx=[x,y],\text{for all}x\in \U0001d524.$$
The functor $$\begin{array}{rcll}U:& \left\{\text{Lie algebras}\right\}& \to & \left\{\text{associative algebras}\right\}\\ & \U0001d524& \mapsto & {U}_{\U0001d524}\end{array}$$ is the left adjoint of the functor $$\begin{array}{rcll}L:& \left\{\text{associative algebras}\right\}& \to & \left\{\text{Lie algebras}\right\}\\ & \left(A,\u2022\right)& \mapsto & \left(A,[,]\right)\end{array}$$
where $\left(A,[,]\right)$
is the Lie algebra generated by the vector space $A$
with the bracket $[,]:A\otimes A\to \u2102$ defined by $$[{a}_{1},{a}_{2}]={a}_{1}{a}_{2}{a}_{2}{a}_{1},\phantom{\rule{2em}{0ex}}\text{for all}{a}_{1},{a}_{2}\in A.$$
This means that $${\mathrm{Hom}}_{\mathrm{Lie}}\left(\U0001d524,LA\right)\cong {\mathrm{Hom,}}_{\mathrm{alg}}\left({U}_{\U0001d524},A\right),\phantom{\rule{2em}{0ex}}\text{for all associative algebras}A.$$
Let $i:\U0001d524\to {U}_{\U0001d524}$
be the map given by $i\left(x\right)=x.$
Then (???) is equivalent to the following universal property
satisfied by ${U}_{\U0001d524}:$
If $\phi :\U0001d524\to A$
is a map from $\U0001d524$
to an associative algebra $A$
such that $$\phi \left([x,y]\right)=\phi \left(x\right)\phi \left(y\right)\phi \left(y\right)\phi \left(x\right),\phantom{\rule{2em}{0ex}}\text{for all}x,y\in \U0001d524$$
then there exists an algebra homomorphism $\stackrel{~}{\phi}:{U}_{\U0001d524}\to A$
such that $\stackrel{~}{\phi}\circ i=\phi .$
A representation of $\U0001d524$
on a vector space $V$
extends uniquely to a representation of ${U}_{\U0001d524}$
on $V$
and this induces an equivalence of categories between the representations of $\U0001d524$
and the representations of ${U}_{\U0001d524}.$
Let G be a Lie group and let $\U0001d524=\u2102{\otimes}_{\mathbb{R}}{\U0001d524}_{\mathbb{R}}$
be the complexification of the Lie algebra ${\U0001d524}_{\mathbb{R}}=\mathrm{Lie}\left(G\right)$
of G. Let $\mathcal{E}\left(G,\left\{1\right\}\right)$
be the algebra of distributions $\mu :{C}^{\infty}\left(G\right)\to \u2102$
on G such that $\mathrm{supp}\left(\mu \right)=1.$
Then $$\begin{array}{rcl}{U}_{\U0001d524}& \to & \mathcal{E}\left(G,\left\{1\right\}\right)\\ x& \mapsto & {\mu}_{x}\end{array}\phantom{\rule{2em}{0ex}}\text{where}\phantom{\rule{2em}{0ex}}{\mu}_{x}\left(f\right)=\frac{d}{dt}f\left({e}^{tx}\right){}_{t=0},\phantom{\rule{2em}{0ex}}\text{for}x\in \U0001d524$$
is an isomorphism of algebras.
References [PLACEHOLDER]
[BG]
A. Braverman and
D. Gaitsgory,
Crystals via the affine Grassmanian,
Duke Math. J.
107 no. 3, (2001), 561575;
arXiv:math/9909077v2,
MR1828302 (2002e:20083)
page history