Download - Izz - University of Oregon
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
#