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 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 μ on G. Let L1Gμ=f:G→ℂ|f=∫Gfgdμg<∞. Then L1Gμ is a *-algebra under the operations defined in (???). Any unitary representation of G on a Hilbert space H extends uniquely to a representation of L1Gμ on H by the formula fv=∫Gfggvdμg,f∈L1Gμ,g∈G, and this induces an equivalence of categories between the unitary representations of G and the nondegenerate * -representations of L1Gμ.
3. Let G be a locally compact topological group and fix a Haar measure μ on G. Let ℰc=distributions on  G  with compact support. Then ℰc 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 ℰc on V by the formula μv=∫Ggvdμg,f∈ℰc,g∈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 ℰcG on a complete locally convex vector space V.
4. Let G be a totally disconnected locally compact unimodular group and fix a Haar measure μ on G. Let CcG=locally constant compactly supported functions  f:G→ℂ. Then CcG is an idempotented algebra with the operations in (???) and with idempotents given by eK=1μKχK, for open compact subgroups K⊆G, where χK denotes the characteristic function of the subgroup K. Any smooth representation of G extends uniquely to a smooth representation of CcG on V by the formula in (???) and this induces an equivalence of categories between the smooth representations of CcG (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 CcG.
5. Let G be a Lie group. Let Cc∞G=compactly supported smooth functions on G. Then Cc∞G 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 Cc∞G 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 Cc∞G 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 ℰGKfin=μ∈ℰcG|suppμ⊆K  and μ   is left and right  K  finite . Then ℰGKfin is an idempotented algebra with the operations in (???) and with the idempotents given by eK=1μKχK,for open compact subgroups K⊆G, where χK denotes the characteristic function of the subgroup K. Any 𝔤K -module extends uniquely to a smooth representation of ℰGKfin on V by the formula in (???) and this induces an equivalence of categories between the 𝔤K -modules and the smooth representations of ℰGKfin (see Bump Prop 3.4.8). This correspondence takes admissible modules for G (see Bump p.280 and p.193) to admissible modules for ℰGKfin. By Knapp and Vogan Cor 1.7.1 ℰGKfin=CKfin⊗𝔱ℂU𝔤ℂ.
7. Let G be a compact Lie group. Let CGfin=f∈C∞G|f is finite . Then CGfin is an idempotented algebra with idempotents corresponding to the identity on a finite sum of blocks ⨁λGλ ⊗Gλ.

The category of representations of G in a Hilbert space V and the category of smooth representations of CGfin are equivalent.

8. Let 𝔤 be a Lie algebra. The enveloping algebra U𝔤 of 𝔤 of the associative algebra with 1 given by Generators:  ∈𝔤,and Relations: xy-yx=[x,y], for all x∈𝔤. The functor U:Lie algebras→associative algebras𝔤↦U𝔤 is the left adjoint of the functor L:associative algebras→Lie algebrasA•↦A[,] where A[,] is the Lie algebra generated by the vector space A with the bracket [,]:A⊗A→ℂ defined by [a1,a2]=a1a2-a2a1,a1,a2∈A. This means that HomLie𝔤LA≅Hom,algU𝔤A,for all associative algebras A. Let i:𝔤→U𝔤 be the map given by ix=x. Then (???) is equivalent to the following universal property satisfied by U𝔤:

If φ:𝔤→A is a map from 𝔤 to an associative algebra A such that φ[x,y]=φxφy-φyφx,for all x,y∈𝔤 then there exists an algebra homomorphism φ~:U𝔤→A such that φ~∘i=φ.

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.

Let G be a Lie group and let 𝔤=ℂ⊗ℝ𝔤ℝ be the complexification of the Lie algebra 𝔤ℝ=LieG of G. Let ℰG1 be the algebra of distributions μ:C∞G→ℂ on G such that suppμ=1. Then U𝔤→ℰG1x ↦μxwhereμxf=ddtfetx|t=0,for x∈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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