Recall : E is a hie
group
,
representation of G :
f-.
G → GLLV) hie
group
homomorph: Im .
.
Adjoint representation of G . Ad : E Elly)
.
/
T
Adlg) : =
econjg
Def.1.25-supposeg.is
a We
algebra ouervlk R
, Q .
A- representation of you a IK Vector
spocevis a
ke
algebra homomorph isn
y :
g →
gllv) = { linear uops
✓→ v } -
( i - e .
4 is linear and YLEXMI ) =
[ 4k) , yly )]
=
YKIOYLY) -
414).
4K)
V-X.ee
g) .
Equivaleutly ,
a bi linear
cuop y :
gxv → V s t
.
YEXMI ,
v ) =
4k , 4M, r ) ) -
414, ykiu) )
H X. Y
EY and tv e- V .
By Prop . 1.12
, any representation .
y
: G- → GLIV)
of a we
group G in auce) n
representation y
'
Iy
:
g.
→
gfk)
of its hie
algebra g .
For E- = G- Llu ,
R) ,
the standard representation 4 of G- Uni IR)
gives tue
Haidar) on R
"
:
y
'
:
gllu.IR) × IR
"
→ IR
"
( Xiv ) n xv
S
imitiertes , for my
matrix
group and its standard representation .
For the ad joint regeres .
of a we
group
G
,
Ad , G- →
Gag!
the iuduced representation of g. ,
the so called
odjoint representation of g ,
is
givee.by
ad :
g- gllg) .
Ad
)
=
adk) (y) =
IX. 4] KAYE
g. .
=
Te Ad
as the
fdlowing proposition shows :
Prop.1.IE hie
group with hie
algebra (
g ,
E , ] ) .
① For XE
of and
GEE :
↳ (9) =
Rad!;) ②
For
Kitty , odk ) M) =
IX. 4] HEY e-
g
.
③ For
XEJ , GEG we love
explt Adlg) k )) =
gexpltx) g-
^
.
④ For X. YE
of we love :
Adlexpk )) (4) = öd
"
(y ) =
!!# idk
[X. Es
,
EA ..
. IX. 4]])-
*
= y + [ die] t
IV. EX :D] +
E. HE
,
,
y]]]+ .
.
.
Proof①
↳ Lg) =
Radio) ) .
Jg
-
99 .
conjgTeig
×
=
Tel 9.
Teconjg X = RLG) .
" Adlg)
k )
↳
Lg)
Adlg) X
② ( house a has - s X
, . . .
, Xu of the necker
spacey .
Then Adlg ) :
g. →
g wrresräuds to a nen matrix
Kaijlg) ) .
f- er
eng ge G. -
Note that
aij : G- → IR eine smarte
, since Ad is sueoote .
Matrix
representation of adk ) :
g- g equds
( X.
aij ) =
In X =
( Li aij ) ( e ) .
-
Any
YEJ Laune written es Y €1;
Xi and
/
G)
Ring,
=
Yiaiilg )
Rxilg)
- ¥
ig ;
[ 4. Ly] =
.
#Sig
=
=
aijELx.R.IT/Lx-aij)Rx ;
Fo b)
Pvp.
1-14
=
↳ yjllieij)
Rxi .
Evalnahe
de eG :
ELx.ly] (e)
=
;?
ys-lx-a.rs/Xi=YY).
③ Since
Adlg ) =
Teconjg ,
tue nesulr tolles
ehiectkg
frau ① of Them .
1.23 .
( y : E- → H
Lf o ex
PG =
exptt.
y
'
↳
ujglexpltx ) ) =
exp ( Ad
G) ( tx)) =
explt Adlg ) k) ) .
④ App } ① of Then .
1.23 to Ad : G- →
Gllg) .
Adlexplx)) (Y) =
exp (adlx))(y) = öd y)
-
"
¥4.
od KFV) .
o .
Prop.1.27-surr.se C- is a we
group
with hie
algebra
(
J ,
[ ,
]) .
Let
y : G → ELW ) he o hie
grau representation
of G w.hn
iuduced representation y
'
:
g.→
ghlv)
of g .
① ylexpltx) ) Lu) =
explty
'
k ) ) 4)
-
HI
=
rtte #
*
EG ,
ve V
,
+ ER .
+
¥4 4
'
4) v
+ . - r
② X. v
: =
y
'
lu) off
exrltxk-dd-feqltx.tv
Pri
① fellows from ① of Then . 1.23 and ② follow)
iuuuediahely freu ① .
We will toter
develop tue basic representation theory
of
hie
group, and hie
degeneres .
1.3 Li e sub
group
and virtual hie Sub
group >
-
We
already de find what a hie Ins
group of a
hie
group
is .
Prop.1.LT Suppen H is a hie
Sungrap of a hie
group G .
Then It is Closed as a Sunset of the Toro logical space G .
ProofAny
submfd . N
of a mfd .
M is
loudly Closed ,
i. e . N is
open in its done RT ( emery x EN Los
neighbhd U in M
s .
t .
Un N is closed in U ) .
→
For any subgrap H of a
homolog Iced
group
G
,
Ä is also
•
sungar.tt ( ! I.¥9
#
!;]
"
"
F)
If A is a we Sun
gap of a hie
gap
G
,
H, is
open end
denn Ä .
Heuce ,
liegt ,
bg ( H) EÄ is
open in Ä ( sg : ÄÄ
hauaeuogeei)since It is denn in IF
,
dgl H) n HF ¢ ,
Wi whichim
rlies
g E H .
/ -
7h, h
'
EH s .
l
.
g. h
'
=
h →
gt #).
D
Converse ly ,
one los :
That Suppe H is a sub
group of a we
group E that
is Closed as a sunset of the
hopolog . Erde G .
Then It is a we
group .
Proof we wie
J for the hie
dlg .
of G and
G : =
{ c
'
( o) : c : IR - E is Smooth ,
do) -
e
and c ho> value, in H } -
cg
.
Chief '
G is a linear sub snece .
If cnn.cz : IR - HE GEOCuries
ovalen lo) =
G (o ) =
e
,
then ( (t ) : =
GH ) Ghat ) is a
8- arme witnvaues in H
and CLO) =
e .
( ae R) .
Then
,
c
'
6) EG
" -
¥!!
? H) ,
GH ) ) =
%! (Ilo) , acico)
=
c) lo) to
ello )
Leuna 1.7
-
-
Check:
Sepp one ( Xu ! e. „vis
a
Sequence ing
with
n
= × C-
7 and ter ( tm )
ne! le
sequence in IR>•
" t .
1in tu = 0 .
n →so
Then if expltn ) EH the IN
,
then expltx ) EH HTEIR .
- Fix
TER . Er ne IN let an
he the
hargest integer E
.
Then , ante Et and t antu
<tu
,
so
-
.
limontu = t
"
a
"
II.ed
=) Ii
!
( expltnxn)
"
) =
line expluntuxn ) I
my continuity
- " →
°
=];) ÄHof expiaff
H
by b) sumpfige t H
Sungraep Sie
Claires
G- { XEJ
:
expltx ) EH tt ER} .
-1
RHS E G .
by definition e) § .
To show
GE RHS
,
kt c. IRA HEG abe a A- Gerne
wih do) = e ( c
'
lo) EG ) .
Then 7C > 0 and a 0are
v :LE.
c) →
g.
sie . alt) =
explvlt)) V-tel-E.ae) .
( vlo ) - 0
Eeg) .
=) und 3¥
ilo ) ¥
rum) ilo)
-
" 'ok!!:
Set tu : -
I
and Xn : =
nv
(E) ,
tue
expltnxu ) =
explvl?)) = < ( E ) e H
T
for
high
By Claire 2
, exp Hello)) EH THEIR .
Clair Wrie
g- G ⑦ k os a
never space ( k is
und ueupliueut of Ging .
)
Then 7 an open
neigend . W
of Oek in k st .
lxp (W ) n H =
{ es .
To BE WNFINUED . . .