G- - connected compactt

lie group

q : GxT -2 G-

ie ( gatt = gt g-I

y is differentiableie (Gat) is compactsubset of Ct , i.e .

y (G-XT) is closed -

claim : yj .

It is enough to provethat y (a- xT) is open in G-since G is connected

.

I

Froot big induction in2

dim G - dimT 30

If dim G- = dim F ⇒ G-=T

y in a projection .

The statement is true .

Can assume that dim G - dimTM.

To show that q ( GxT) is open

it is enough to show that it is

a neighborhood of gt g-'= 4cgF)

for any g EG and tfT .

Since InHg ) is an automorphismof G , it is enough to showthat y l G xT) is a neig

'

bar hood

of t for anyt ←T .

3

Let It be the centralizer

of tH - { he G- l ht -- th}

11

{ he G- / tht"= h }

11

{ he a- I Int H) (h) = h }equalizer of Intlt) and id .

⇒ It is a Lie subgroupHH) = { 3 c- LlG) lad It) 151=3 }Ho - identity componentof H

.4

Clearly , TCH .SinceT is a maximal

torus in .G,it is a maximal

for us in Ho

Let Z be the center ofG-

. There are two optionsL or t¢⑦

⇒ H = G- ⇒ Ho = G .

Let T'be another maximal torus

in G.

Then T'

= hThi'

for some h EG .

Since teZ,teT implies

that t =htti'ET

'.

t is in all max .ton

.

5-

Let Ge Uf) . There ' IR q is

in a maximal abelian Lie sibalgebra ofUG) ⇒ exp } is in a maximal

torus T"

.

t . apes a- T"

.=kTk- 'c y(a-xD .

{ t . apes l -GE UG) 3 is a

neigh . oft ⇒ y (Gxt)is a neigh . of t .

'

tee then we must have

LCH) FL (G) .Otherwise

,

we would have'UH ) =L(G)

it would be open in G- .6

Since G- is connected⇒ It = ft.Hence dim Ll it) - dim L ( G) .

\

,

dim It a dimeG-

dim Ho - dim T s dim G - dint

By induction .

-

'

'

H.xT → H

.is surjective .

⇒

y (a- xT) = y (a- x Ho)y

'

n G- x Ho→ G-

41g ,h ) -- gh g-

'

.

^

T : G- x H. → G-

>

have to prove that this is a

submersion at ( lat) .

Equivalent( §,hi ) 1-5 upB) heap C- as )

is a submersion at ( O,t) .

⇐ L (G) x LtHo) z les ,z)iQ> exp tap in) exp f

- es)is a submersion at co ,o) .

( { , y ) '→ E' erupts) t.expeye.pt 's)is a submersion at lo

,O)

.

G,n)→ ap fad 3) up in) eye testis a submersion at co

,o) . I

8The differential of this map isGm) '→ ANE't est y - § =

- HADLEY -I)Gt n .

( (G) has a natural G - invariant

inner productLCG) = LI Ho) to LIHo)

-

t

ku (Ad KY - I) Im 't AdH -I)Hence

, by the formula forthe differential , we seethat it is surjective .

Hence, y is a submersion

at Kitt, image of y is

a neigh for hood of t .

This implies that9

y (G XT) is a neighborhoodof t .

⇒ eCGXT) is open and

q is surjective .

connectedtheorem .

Let G be a compact^

Lie group . Then

exp :UG)→ G-

is surjective .

theorem ..Let G be a

connected compact Lie

group .

Then away gc- G-

is in a maximal torus

T in G .