Representation Theory

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

Last updated: 2 October 2014

Lecture 9

Spaces

A Lie group is a group G that is also a manifold such that the maps G×G G (g1,g2) g1g2 and G G g g-1 are morphisms of manifolds.

An algebraic group is a group G that is also a variety such that the maps G×G G (g1,g2) g1g2 and G G g g-1 are morphisms of varieties.

A topological group is a group G that is also a topological space such that G×G G (g1,g2) g1g2 and G G g g-1 are morphisms of topological spaces.

A group scheme is a group G that is also a scheme such that G×G G (g1,g2) g1g2 and G G g g-1 are morphisms of schemes.

A complex Lie group is a group G that is also a complex manifold such that G×G G (g1,g2) g1g2 and G G g g-1 are morphisms of complex manifolds.

Remarks:

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

There are equivalences of categories

{connected reductive complex algebraic groups} G {connected compact Lie groups} K {complex semisimple Lie algebras} 𝔤 {-reflection groups} (W0,𝔥*) {Dynkin diagrams}

GLn,SLn,PGLn. GLn()= { gMn()| gis invertible } . Let V be a vector space over 𝔽. GL(V)= { gEnd(V)| gis invertible } . GLn() is a complex algebraic group.
GLn(𝔽) is an algebraic group.
GLn is ???

The group homomorphism det:GLn(𝔽) 𝔽× is a 1-dimensional representation (character) of GLn(𝔽). SLn(𝔽) = ker(det) = {gGLn(𝔽)|det(g)=1}. The center of GLn(𝔽) is 𝒵(GLn)= {c·Id|c𝔽×} PGLn= GLn(𝔽)𝒵(GLn(𝔽)) GLn() is a complex reductive algebraic group.
SLn() is a complex semisimple algebraic group.
PGLn() is a complex semisimple algebraic group.
In spite of SLn() GLn()and GLn()=SLn ()·× and 1SLn() GLn()det *1 being exact, PGLn() SLn(). 𝒵(SLn())= { nthroots of1 } =μn().

Un,On,Spn.

The unitary group U(n)= { gGLn()| ggt=1 } where g=(gij) if g=(gij).

The orthogonal group On()= { gGLn() |ggt=1 } .

The symplectic group Sp2n()= { gGLn() |gJgt=J } where J= ( 10 0 01 -10 0 0-1 ) orJ= ( 01 0 10 0-1 0 -10 ) .

Let V be a vector space over 𝔽. A symmetric bilinear form on V is a map ,: V×V 𝔽 (v1,v2) v1,v2 such that

(a) , is bilinear, i.e. v1+v2,v3 = v1,v3+ v2,v3, v1,v2+v3 = v1,v2+ v1,v3, cv1,v2 = cv1,v2, and v1,cv2 = cv1,v2, for v1,v2,v3V, c𝔽,
(b) , is symmetric, i.e. v1,v2= v2,v1 for v1,v2V.

The orthogonal group is On(𝔽) = O(V,,)= O(,) = { gGL(V)| gv1,gv2= v1,v2 forv1,v2V } , the group of invertible linear transformations "preserving the metric".

A skew symmetric form on V is a map ,: V×V𝔽 such that

(a) , is bilinear,
(b) v2,v1=-v1,v2, for v1,v2V.

The symplectic group is Spn(𝔽) = Sp(V) = Sp(V,,) = Sp(,) = { gGL(V) |gv1,gv2 =v1,v2 forv1,v2V } . Let A: 𝔽 𝔽 z z be an involution.

A sesquilinear form, or Hermitian form, is a map ,: V×V𝔽 such that

(a) , is not bilinear, instead v1+v2,v3 = v1,v3+ v2,v3, v1,v2+v3 = v1,v2+ v1,v3, cv1,v2 = cv1,v2, and v1,cv2 = cv1,v2, for v1,v2,v3V and c𝔽.
(b) v2,v1=v1,v2, for v1,v2V.

The unitary group Un= { gGL(V) | v1,v2= gv1,gv2 for allv1,v2V } .

Maximal compacts and maximal tori

{connected reductive linear algebraic groups over} {compact connected Lie groups} G K where K is the maximal compact subgroup of G.

A torus in a compact Lie group is a subgroup isomorphic to S1××S1.

A torus in an algebraic group is a subgroup isomorphic to 𝔽×××𝔽×.

GL1(𝔽)=𝔽× and GL1()=× has maximal compact subgroup U(1)= { z×| zz=1 } =S1. So the maximal compact subgroup of ×××× is S1××S1. maximalcompact GK maximaltorus TTk

SU(2)

SU(2)= { gSL2() |ggt =1 } . If g=(abcd)SU(2) then g-1= (d-b-ca) =gt= (acbd) so that g= (ab-ba) with a2+b2=1. So SU(2)= { (ab-ba) |a,b, a2+b2=1 } . Define an involution σ: SL2() SL2() g (gt)-1 Then SU(2)= SL2()σ= { gSL2() |σ(g)=g } . Let G be a complex reductive algebraic group, σ: G G xα(t) x-α(-t) hα(t) hα(t-1) an involution. Then K=Gσ={gG|σ(g)=g} is a maximal compact subgroup.

Notes and References

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

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