Representations of linear Lie groups and applications

Description

Niveau: Supérieur, Doctorat, Bac+8
Representations of linear Lie groups and applications Salem Ben Said (i) This course requires knowledge of the courses done by J.-P. Anker and L. Kamoun. (ii) These lecture notes are written in english for the students to get use to this language. As one can see from the works of Euler and Gauss, mathematicians of previous centuries did not only publish general theorems but also examples which they might enjoyed and which might have educated the reader. From Siegel [Gesammelte Abhandlungen IV, 1979]

• gesammelte abhandlungen


• publish general


• might enjoyed


• linear lie


• can see


• might


• finite dimensional


• salem ben


• lie group


• basic representation

Sujets

• Dimension (vector space)
• Lie group
• Tempered representation

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 24
• Langue: English

Extrait

RepresentationsoflinearLie
groupsandapplications

SalemBenSaid

(i)
ThiscourserequiresknowledgeofthecoursesdonebyJ.-P.AnkerandL.
Kamoun.
(ii)Theselecturenotesarewritteninenglishforthestudentstogetuseto
thislanguage.

AsonecanseefromtheworksofEulerandGauss,mathematiciansofprevious
centuriesdidnotonlypublishgeneraltheoremsbutalsoexampleswhichthey
mightenjoyedandwhichmighthaveeducatedthereader.

FromSiegel[
GesammelteAbhandlungenIV,1979]

PContents

art1.BasicRepresentationTheoryofLinearL
1.LiealgebrasoflinearLiegroups:Abriefreview
2.RepresentationsoflinearLiegroups
3.AnapplicationofNelson’stheorem

Part2.Representationsof
SL
(2
,
R
)
,
andBeyond
4.Basicpropertiesof
SL
(2
,
R
)and
SU
(1
,
1)
5.Finitedimensionalrepresentationof
SL
(2
,
R
)
6.Anapplication:Thewaveequation
7.Thediscreteseriesrepresentationof
SL
(2
,
R
)
Bibliography

3

eirGuosp561022

920383248475

cisaB

traP

1

Representation

Linear

eiL

Theory

Groups

fo

1.LiealgebrasoflinearLiegroups:Abriefreview
Definition
1.1
.
Atopologicalgroup
G
isasetthatissimultaneouslya
groupandatopologicalspacesuchthat:
(i)
themap
(
a,b
)
!"
ab
from
G
#
G
to
G
iscontinuous,
(ii)
themap
a
!"
a
!
1
from
G
to
G
iscontinuous.
Let
H
beasubgroupof
G.
Byinheritingthetopologyfrom
G,H
is
atopologicalgroup.If
H
isanopenin
G,
then
H
isclosed.Indeed,the
complementof
H
in
G
istheunionoftheorbits
xH,
with
x
$
G
and
x
%$
H.
Thisisduetothefactthatif
x
%$
H
then
xH
&
H
=
!
and,therefore
G
\
H
=
'
x
"#
H
xH.
Ontheotherhand,thetranslation
L
x
:
y
!"
xy
is
continuousfrom
G
to
G.
Thus
xH
isanopensetin
G,
andtheunionof
opensetsisanopen.Hencethecomplementof
H
in
G
isanopenand,
therefore
H
isaclosedsubgroupin
G.
On
M
(
n,
R
)weconsiderthenorm
(
A
(
:=
)
A,A
*
,
)
A,B
*
:=tr(
t
AB
)
.
!Clearlywehave
(
AB
(+(
A
((
B
(
.
Thegroup
GL
(
n,
R
):=
A
$
M
(
n,
R
)
|
det(
A
)
%
=0
,
#"inheritingthetopologyfrom
M
(
n,
R
)
,
isatopologicalgroup.Moreover,
GL
(
n,
R
)isanopensetin
M
(
n,
R
)asdet:
M
(
n,
R
)
,"
R
isacontinuous
mapand
GL
(
n,
R
)isthecomplementoftheinverseimageoftheclosed
subset
{
0
}
-
R
.
Definition
1.2
.
Agroup
G
iscalledalinearLiegroupifthereisan
n
$
N
suchthat
G
isisomorphismtoaclosedsubgroupof
GL
(
n,
R
)
.
Example
1.3
.
(i)
Themap
det:
GL
(
n,
R
)
,"
R
$
iscontinuous.
Itskernel
SL
(
n,
R
)=
g
$
GL
(
n,
R
)
|
det(
g
)=1
#"isaclosedsubgroupof
GL
(
n,
R
)
.
Itisanoncompactgroupunless
0Inn
=1
.
$%
(ii)
Let
J
=
,
I
n
0
.
Thesymplecticgroupisbydefinition
Sp
(2
n,
R
)=
g
$
GL
(2
n,
R
)
|
t
gJg
=
J.
#"Inparticular
Sp
(2
,
R
)=
SL
(2
,
R
)
.
Wecanshowthatforevery
g
$
Sp
(2
n,
R
)
wehave
det(
g
)=1
.
Thus
Sp
(2
n,
R
)
isaclosed
subgroupof
GL
(2
n,
R
)
.
(iii)
Weset
GL
(
n,
C
)“=”
g
$
GL
(2
n,
R
)
|
g
!
1
Jg
=
J.
#"

1.LIEALGEBRASOFLINEARLIEGROUPS:ABRIEFREVIEW7
Itistheinverseimageoftheclosedsubset
{
0
}
-
M
(2
n,
R
)
bythe
continuousmap
g
!"
g
!
1
Jg
,
J
from
GL
(2
n,
R
)
to
M
(2
n,
R
)
.
Thus
GL
(
n,
C
)
isaclosedsubgroupof
GL
(2
n,
R
)
.
(iv)
Theorthogonalgroup
O
(
n
)
isthesetof
g
$
GL
(
n,
R
)
suchthat
t
gg
=
g
t
g
=
I
n
.
Thatis
|
det(
g
)
|
=1
forall
g
$
O
(
n
)
.
Thisisan
exampleofacompactlineargroupsince
(
g
(
=
(
g
!
1
(
=1
.
The
subgroup
SO
(
n
)
of
O
(
n
)
definedby
SO
(
n
)=
SL
(
n,
R
)
&
O
(
n
)
isa
closedsubgroupof
GL
(
n,
R
)
.
Let
K
be
R
or
C
.
A
K
-vectorspace
g
providedwitha
K
-bilinearmap
g
#
g
,"
g
,
(
x,y
)
!"
[
x,y
]
iscalledaLiealgebraif
[
x,y
]=
,
[
y,x
]
,
.
x,y
$
g
,
andtheJacobiidentity
[
x,
[
y,z
]]+[
y,
[
z,x
]]+[
z,
[
x,y
]]=0
holds.Thedimensionof
g
asa
K
-vectorspaceiscalledthedimensionof
theLiealgebra
g
.
HenceforthwearemainlyinterestedinLiealgebrasover
=K.RExample
1.4
.
(i)
Everyassociativealgebra
g
isaLiealgebrawith
[
x,y
]:=
xy
,
yx
forall
x,y
$
g
.
(ii)
Theset
M
(
n,
K
)
isanassociativenon-commutativealgebrawith
composition
XY
oftwoelements
X,Y
$
M
(
n,
K
)
givenbyma-
trixmultiplication.Inviewof(i),
M
(
n,
K
)
isaLiealgebrawith
[
X,Y
]:=
XY
,
YX,
for
X,Y
$
M
(
n,
K
)
.
Henceforthwewillwrite
gl
(
n,
K
)=
M
(
n,
K
)
.
(iii)
If
g
isaLiealgebra,and
g
0
isasubspacewith
[
x,y
]
$
g
0
forall
x,y
$
g
0
,
then
g
0
isaLiealgebra.Forinstance
sl
(
n,
R
):=
X
$
gl
(
n,
R
)
|
tr
X
=0
,
#"so
(
n
):=
X
$
gl
(
n,
R
)
|
X
=
,
t
X
#"areexamplesofsuchLiealgebras.ThedimensionoftheLiealgebra
sl
(
n,
R
)
is
n
2
,
1
,
andthedimensionof
so
(
n
)
is
n
(
n
,
1)
/
2
.
Exercise
1.5
.
Verifythatthematrices
$
01
%$
00
%$
10
%
E
=00
,F
=10
,H
=0
,
1
formabasisof
sl
(2
,
R
)
withtherelations
[
E,H
]=
,
2
E,
[
F,H
]=2
F,
[
E,F
]=
H.

8Exercise
1.6
.
Showthat
g
=
R
3
providedwiththecomposition
((
p,q,r
)
,
(
p
%
,q
%
,r
%
))
!"
(0
,
0
,
(
pq
%
,
p
%
q
))
isaLiealgebra,calledtheHeisenbergalgebra
heis
(
R
)
,
withbasis
P
=(1
,
0
,
0)
,Q
=(0
,
1
,
0)
,R
=(0
,
0
,
1)
andrelations
[
P,R
]=[
Q,R
]=0
,
[
P,Q
]=
R.
Definition
1.7
.
Let
G
bealinearLiegroupcontainedin
GL
(
n,
R
)
.
Henceforthwewillwrite
g
=Lie(
G
):=
X
$
gl
(
n,
R
)
|
exp(
tX
)
$
G
forall
t
$
R
.
#"Theorem
1.8
.
Theset
g
providedwiththebilinearmap
[
∙
,
∙
]:
g
#
g
,"
g
,
(
X,Y
)
!"
[
X,Y
]=
XY
,
YX
isarealLiealgebra,calledtheLiealgebraof
G.
Remark
1.9
.
(i)
Clearlywehave
Lie(
GL
(
n,
R
))=
M
(
n,
R
)
,