haar ofeaistence - math
TRANSCRIPT
Let G be a Lie group .
x e- An TFCG) n
-- dive G-
Wg = Mtg lol g-'DI
w is left invariant in - formson Ct
⇒ corresponding positivemeasure
µ= Iwl is a left
Haar measure on G-.
This completes the proof
ofeaistence.lt- Lie group , µ
- leftHaar measure
X : f te { fight drug) a leftinvariantmeasure
2
oehifefcghldmeg = Sflgjdprlg)
an >o o : G → IRIO - continuous (exercise)
OCgigs) Safl hog,gfdµCh) == aged Sg flhgz)dµCh) =
= 0cg ,)OCg,) Saf Ch) dm(h)
⇒gig ,ga) =D Ig) .Olga)
O is a group homomorphismD is a Lie group homomorphismModular OG : a-→ IREfunction a- malt . group of positivef-G-
reals
G- is unimodalar3
-if
OG = I n
theorem .Let G- be a
compact Lie group . ThenG- is unimodalar .
Proof-
.
0cg) {11hg) d h = 01g) -MCG)"
{Hh) dh = aCat Dg
Exercise
connected
-
.G-atwo dimensional
momabelian Lie group
⇒ G is not unimodalar !
4if a Lie group A- is anine o dulled
andµ a left Haar measure onASfig) dm Cg) = f Hgh) dm Cg)
= Sf (hg) du Ig)
⇒ µ is right invariant,i.e.
, µ is bi invariant-
Haar measure on G-
Assume that a- is a compactLie group , µ a Haar measure
on G. ThemMLG)>0 .
By replacing µ with its
multiple , we can assume that
5
µ CG) = I.
Such Haar
measure is unique- normalized Haar measure-
on G-.
theorem .
Let a- be a Lie
group .Then the following
conditions are equivalent :
Cil G is compact ,Cii) play is finite .Proof . We proved Ci)⇒ Cii)
.
Assume that µCGI is finite .
Let V be a compact neighborhoodof 1 .
Thenµ (V) >O .
6Denote by 9 the familyof finite sets Igi , gas . gm }
such that g .-V n g ;V
= 0
for all , it j , it is j Em .
Then
miff giv ) = m.mu) salat .Hence
meAGIMCV)
i.e . m is bounded.
Let m be maximal possible .Then for {g . . - - n, gnn} in 'T ,
and
g c- G- we have
GV n giV t & for some i .
7-
This implies
g e giv V-'
⇒ a- = IF givv"
since V is compact , V-'
is
also compact ⇒ VV-'
is
compact ⇒ g ;VV- '
is compact⇒ G is compact . Ed
8
www.aartinuer-produetout (G) , a- compactLet a- be a compact Lie
group . Take an arbitraryinner product C. , . ) on
4G) -
g'→ ( Ad (g) 3 , Ad (g) z)
is a continuous functionou G .
Put
<3 ,z>= Sq ( Ad (g) 3, Ad (GD dyCg)
where ris the normalized
Haar measure on G .
c.,
. > : 4G) x LCf) → IR9
is a bilinear form .
Since C; . ) is symmetric ,
C;
. > is also symmetric .
<3,97 = Salad (g) S , Adly)§) fulg) == Sq HAD (g) ELFIN Cg) 70
Meant innous
positive
Cg is > = o implies that
{ HAHg)3112dm(g) =0Assume that 3 to .
Then
10
1181170 . Therefore ,there exists an open weigh .
U of 1 such that
HANg)3117 I 11511for g EV .
⇒
{ Hard lad 'sRda Cg) >7 f 4Ad Ig) 315dm Cg) 3¥ So 11515dm (g) == f. pled 1151T .
Hence < 3,3.
> so and
c.,.> is an inner product
on L (G) .
'I
A-dig) 3 , Adcgly > =
= S (Adh) Adlgl } , Adlh) Adlglz) fun)a-
= Sq ( Adlhg) 8 , Ad (ng) # dah) == SaladHas , Adhdg) drink= 28,27 .
Hence L .
,. > satisfies
( Adige , Adcg) n> =LSinsfor any geG , 3 , ZELIG) .
Hence it is an G - insouciantimmerge .
12
By differentiationwe getsadB) 3 ,n> t
( 3,ad (3)z> = o
for all 3 ,z , } c- LCG) .
⇒ ad (y) , y c- Kat, is
antisymmetric linear
map with respect to theinvariant inner product .
The existence of iuoaoiaut
inner product ou L (G) allowsto say a lot on its structure .
13
Let or be an ideal in L (A) .Let of be the orthogonalcomplement of oeSecret
, z eor ⇒ 2g , y> =o
Let S E hCG) .Then I 8 ,y] E OL
⇒
0=55,Cad 9)Cnb = - scad 9)Cg ),z>
= - C E 's is ], 27
⇒ as is ] c- oh .
at is an ideal .
((G) = or⑤ out as linear
spaces . § c- oh, ye of ⇒
[ 8 ,y] c- anof= { o } .
UG) is a direct sum of
14two ideals .
By induction
((G) is a direct sum
of minimal ideals .
M - minimal ideal in LlG).
le cM ideal in M
LlG) = M to Mt ⇒b is an ideal in L (G)
.
⇒ b --fo} or le = Me .
There are two options① dimM = I
,M is abelian
{ EM ,ad 5 Im = o , ad { Inf =0
ad } = o ⇒ § E z - cetera of L l G)
m c Z .
15
② M is not abelian.
Then dim M > I and Me
has no nontrivial ideals
- simple Lie algebra .
-
4G) is a direct sum ofits center Z and simpleideals
.
-
Example ① of two dimensional
Lie algebra with basise, , ez and Tee
, ,ez ] = e ,
Then Rie,is an ideal rn
og - of is most simple .
16
This implies that
the dimension of a simpleLie algebra =3 .
② of=L (subs)
og-- ft; Ei:) ; x.meet}
of is simple .