9 Semisuple Lie algebra ,t maximal tonal subalgebra ←

could be K .

C. , .) : ogxoy→E some non-degenerate symmetric invariant

bilinear fo -n .

Cartan decomposition cry= Z ④ ④ %

YHER

F- go -

- Cglt) = go # he 't't l ya# 03

G.) Izz is non-degenerate g- ExesI [ h ,xI= Nh) x

thet}

Example g - Sen CG) .

We've shown this is semiapciGuen sup le

'

) .

Let t -- subalgebra of tracezero diagonal matrices .

Toral ✓

hi -- diag (o, - - so, I , -1,0 , - -yo)Basis for 2- : hi , hz, - - thn- i ith Citi)

Let E. c. Z't

,s:(drag Ctu . .tn) to ti . These spar 2- *

subject to one their relation E. tszt - - t En = O -

Then cry= Z to ¥0.

Key .

←Carton decarsosihoioggwrt Z

matrix mutt .

t n.

If h -- diag Ct , . - .tn) e 2- then heij -- tieij , eijh-

- tjeij

⇒ [ h,eijT= (ti- tj)eij= Gi-g)Ch) eij

Shows :

• go-

- Z,hence,

2- is maximal tonal subalgebra

• R = f si -g . I Ifisjfn , i#j }

•each of for

HER is l -D Gpanned by Ccj cfi A -

- Ei -Ej)

• Say G.) = 2, the form. Ther :

I nausea- f÷÷÷÷÷j-÷:÷:I 0

Gram matrix of Ga )

Back to general case ! 9 , Z , cos . )

Properties of the root system

C. , .) : Zx 2-→ E is non- degenerate

Use it to cdeitjy 2- with 2-* . . . soXEZ

'tis iieeihpid wth ↳ c- 2-

where (t, ,h) -- X Ch)the Z .

Note R spanst't

.

TP=of Else , youcould fid OF he 2- so Nhl- O the R

.

Thee [ h , ya ]-

-O taek ,

so he 3cg ) -O # t

Take her .

Cos . )lga=0, Goo >↳+g-a

" non-degenerate,

here,-LER

,and - - r

Rice Oteaeoyg , F fae g- as - t - tea

, fa ) # o .

(choice here ! ! ! )

Lemma I ① Eea , fa ] = Cea, fa ) ta FO

→ ② ( ta , ta) #O,here

, after rescaling fa qi necessary,

for her . - -we can assume Haifa) = Fata) .

③ Let ha =

¥. Then

Cta,ta )

[ex , fat-

- ha,[ha , eh ]

-

- Zea,

[ ha , fate -2£ .

⇒ Cea,ha,fa ) span a subalgebra of g I

slack) .

⑨ g -Senas above,X = Ei -Sj

Take ex-

- eij , fa = eji , ha - diag co - - id- - -

ji- - o)

.

Proof ① Know Cea, fa) ta to

.

Take he 2- . Ther ( h , Eea , f-a ] )= ( Eh , eat , fa )

→

= Nh ) Cea , fa )

the Z = (ta , h) Cea , fa)

have [ex , f-a] = Cea , fa) ta= Clea , fa) ta ,

h )

Whig non-degeneracy of formon7

② Suppose Ctx,ta ) - O .

Then Eta , eat = O= Eta

, fa ]

So by= Cl eat Q ta t Q fan

is a solvable tie.

subalg . q og .

Lie 's theorem ⇒ F a basis for yWnt ya I )

But ther th E Tal which consult of mlpoliat matrices ii adjoint repeater

Shows ad ta is mepoteit , also semisup-6 , here ,ad ta = 0

So ta :O # .

③ Eea,fat = ha ✓ ha :

ZtaCta

,ta )

[ ha,ex ] -

- fat, Eta ,eat -- ftp.YHea-zea

[ha,fat = -2 f-a savior

Lemma 2 For HER, only multiples of L that belong to R are In .

Moreover , dem ga= I

,so dim of =

dui 2- t IR l .

Proof Let ex , ha , fabe as a Lemma I

,so ya = Ehatocfa

Let M = 9,

Ese. .

Nok ya Cs Mvia ad

, ha acts on Day as

Rep . theory of sea ⇒ ch Cha ) -- f÷¥C ⇒ a .

yay-_ 0 unless ZCE 27 . (eigenvalues q ha ) .

g. ×acts timidly or be@ : 2-→ Cl ) ←

codcni . I subspace q Z .

←

So M = ftp.z-0 to ④ ⑤ 0¥ for 2C being odd .

gaiterza ut

Ze-2Space

shows :

large:L:L'" 'a''am

"

÷÷,2- 2C

We can 't have any Dca # O for 2e odd -2C

As if one did , we'dhave I -eegeespaie for ha ,

so Catz . . - Dga #O as if

'

{her tree LER#

So now we've shown :

yeabasis

, Ffa C- 9-a

D=Z to ④ ok

ha -

- Eea,fat

so ex , ha , fa sfitnsieher t -D

ha -

- TELL '

ha 's Her) spanZ

ed's HER) givea bani for the rest

deni g :dui Z t Irle

rankles )

Lemma 3 413 c- R, B # IL

Let r,qE INbe maximal so B -ra and ptqt he' ii R -

Then all Pti x C -r Ei Eq) belong to R .

and p cha )= r- q

Hua, p cha ) C- 27

and p - p cha) Lc- R .

Proof Let % be as above .

I -Doro

Let M -

-

,Dp,

bsknnaz % ↳M na

' ad .

ha act on spanas Gtia) Cha) =p cha) +

i*c*Hatta)

Ff"pfha) even . . . all are even,O wt space is I-D =p Cha)t2i

If B Cha) odd - r - all are odd, I wt space is I -D)⇒ M is an irreducible

Da -module

So mis imid,highest ha -weight is p cha) t2q

lowest ' ' " B Cha )-Zr

We have every pcha)t2i ,-r Ei Ece (justone sloshing)

fuatly p Cha- Zr= -( B Cha)t2E )

:. 2p(ha) -

-

Zr- 2g

-

'

. pcha ) : r- q

#