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

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

8. Let $𝔤$ be a Lie algebra. The enveloping algebra $U_{𝔤}$ of $𝔤$ of the associative algebra with 1 given by $$\text{Generators: }x\in 𝔤,\text{and}$$ $$\text{Relations: }xy-yx=[x,y],\text{ for all }x\in 𝔤.$$ The functor $$\begin{array}{rcll}U:& \left\{\text{Lie algebras}\right\}& \to & \left\{\text{associative algebras}\right\}\\ & 𝔤& \mapsto & U_{𝔤}\end{array}$$ is the left adjoint of the functor $$\begin{array}{rcll}L:& \left\{\text{associative algebras}\right\}& \to & \left\{\text{Lie algebras}\right\}\\ & \left(A,•\right)& \mapsto & \left(A,[,]\right)\end{array}$$ where $\left(A,[,]\right)$ is the Lie algebra generated by the vector space $A$ with the bracket $[,]:A\otimes A\to ℂ$ defined by $$[a_1,a_2]=a_1{a}_2-a_2{a}_1,\text{for all} a_1,a_2\in A.$$ This means that $${\mathrm{Hom}}_{\mathrm{Lie}}\left(𝔤,LA\right)\cong {\mathrm{Hom,}}_{\mathrm{alg}}\left(U_{𝔤},A\right),\text{for all associative algebras }A.$$ Let $i:𝔤\to U_{𝔤}$ be the map given by $i\left(x\right)=x.$ Then (???) is equivalent to the following universal property satisfied by $U_{𝔤}:$

If $\phi :𝔤\to A$ is a map from $𝔤$ to an associative algebra $A$ such that $$\phi \left([x,y]\right)=\phi \left(x\right)\phi \left(y\right)-\phi \left(y\right)\phi \left(x\right),\text{for all }x,y\in 𝔤$$ then there exists an algebra homomorphism $\tilde{\phi }:U_{𝔤}\to A$ such that $\tilde{\phi }\circ i=\phi .$

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

Let G be a Lie group and let $𝔤=ℂ{\otimes }_{ℝ}{𝔤}_{ℝ}$ be the complexification of the Lie algebra ${𝔤}_{ℝ}=\mathrm{Lie}\left(G\right)$ of G. Let $ℰ\left(G,\left\{1\right\}\right)$ be the algebra of distributions $\mu :C^{\infty }\left(G\right)\to ℂ$ on G such that $\mathrm{supp}\left(\mu \right)=1.$ Then $$\begin{array}{rcl}{U}_{𝔤}& \to & ℰ\left(G,\left\{1\right\}\right)\\ x& \mapsto & {\mu }_x\end{array}\text{where}{\mu }_x\left(f\right)=\frac{d}{dt}f\left(e^{tx}\right){|}_{t=0},\text{for }x\in 𝔤$$ 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), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)

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