L c = X ◦ L c + L c ◦ Y forallX,Y ⇔ c ∂ c + c ∂ c − c ∂ c + c ∂ c − c ∂ c − c ∂ c = 0and c c = c c .1 − Z W ( c X Y ∂ c − ∂ ( c X Y ) c + ∂ ( c X Y ) c + ∂ ( c X Y ) c ) .Thenbyseeingthetermsunderlined,wehave c Y Z W ( X ( ∂ c ) − ( ∂ X ) c +( ∂ X ) c +(

(1)

(2)

(3)

(4)

(5)

(6)

(7)

Exercise 1.(b) Note

(X◦LYc(Z, W) +Y◦LXc(Z, W) −LX◦Yc(Z, W))ⁱ=cⁱ_jkX^jZ^mWⁿ(Y^l(∂_lc^k_mn) − (∂_lY^k)c^l_mn+ (∂_mY^l)c^k_ln+ (∂_nY^l)c^k_ml) +cⁱ_pqY^pZ^sW^t(X^r(∂_rc^q_st) − (∂_rX^q)c^r_st+ (∂_sX^r)c^q_rt+ (∂_tX^r)c^qsr)

−Z^mWⁿ(câ_bcX^bY^c∂_acⁱ_mn−∂_a(cⁱ_bcX^bY^c)c_mnâ +∂_m(c_bcâ X^bY^c)cⁱ_an+∂_n(c_bcâ X^bY^c)cⁱ_ma). Then by seeing the terms underlined, we have

LX◦Yc=X◦LYc+LXc◦Y for all X,Y⇔cⁱ_jk∂_lc^k_mn+cⁱ_lk∂_jc_mn^k −c^k_jl∂_kcⁱ_mn+c^k_mn∂_kcⁱ_jl−cⁱ_kn∂_mc^k_jl−c_mkⁱ ∂_nc^k_jl=0 and c^a_bcc^d_{e f} =c^a_{b f}c^d_ec.

1

(8)

Textbook C.5 We have

p(z) = ¹ z²+ ¹

20g2z²+ ¹

28g3z⁴+ ¹

1200g²₂z⁶+O(z⁷)and ζ(z) = ¹ z− ¹

60g2z³− ¹

140g3z⁵− ¹

8400g²₂z⁷+o(z⁸). Then by (C.63), which can be checked by seeing their poles, we have

η∂ p

∂w +η⁰ ∂ p

∂w⁰ +ζ∂ p

∂z = −2p²+¹ 3g2

= −3

10g3z²− ¹

84g²₂z⁴+O(z⁵) while the left hand side is

η( ¹ 20

∂g2

∂wz²+ ¹ 28

∂g3

∂w +O(z⁵)) +η⁰(¹ 20

∂g2

∂w⁰z²+ ¹ 28

∂g3

∂w⁰ +O(z⁵)). Then compare their coefficients we have

η∂g₂

∂w +η⁰∂g₂

∂w⁰ = −3

10g3×20= −6g3

and

η∂g3

∂w +η⁰∂g3

∂w⁰ = −1

84g₂²×28= −1 3 g²₂.

2

(9)

Textbook 3.5

By the Proposition 3.5, where the case for t1was done in class, we have µ_αψ_iα=

∑

k

Vikψ_kαand η_αβ=

∑

i

ψ_iαψ_iβ. Then the original problem

V_ij =

∑

α,β

η^αβµ_αψ_iαψ_jβ

is equivalent to

V_ij =

∑

α,β,k

η^αβV_ikψ_kαψ_jβ. Thus it suffices to show

∑

α,β

η^αβψ_kαψ_jβ=δ_kj. Write E := (η_ij)_i,jand P := (ψ_ij)_i,j. Then

∑

α,β

η^αβψ_kαψ_jβ =δ_kj⇔PE⁻¹P^t= I⇔E=P^tP⇔η_αβ=

∑

i

ψ_iαψ_iβ

which is given above.

3

(10)

Textbook 4.1

∂V

∂uⁱ =

∑

a

∂V

∂ξ_a

∂ξa

∂uⁱ

=

∑

a

−1

2(n+1)(2(ξ₁+ · · · +ξn) +2ξa) · −1 (ξa−qⁱ)λ⁰⁰(qⁱ)

= ¹

(n+1)λ⁰⁰(qⁱ)

∑

a

ξa−ξ₀ ξa−qⁱ

= ¹

(n+₁)λ⁰⁰(qⁱ)· (−res_p=∞(p−ξ₀)λ⁰(p) (p−qⁱ)λ(p))

= ¹

(n+1)λ⁰⁰(qⁱ)· (n+1)

= ¹

λ⁰⁰(qⁱ)

= −ηii. Hence it should be that

∂V

∂uⁱ = −η_ii.

4

(11)

Exercise ^:

Recall that under Type I Transf , ⇒ 22=2 _,< ^. In

So _, ( Ja ^, Jp ) k ^⇐ ( 2x ^' 2k _, Ja^.

Ip

⁾ ⁼ ⁽ ^2x ^.^Ja ^. ^2k ^.^Jp ⁾ ⁼ ⁽ ^2a ^.

_2p)=7q

⇒ £& ^is ^flat ^word ^, ^wit ^-^Ti ² ^, 7k ,

( Ja ^. Jp ,

Jr

^>.< ⁼ ⁽

2x.2x.k.JP

⁾ ^.5r⁾ ⁼ ⁽

^ax.fr

^, ^2x ^.

HIJP

⁾ ⁾

= car ,

2xnIi21

⁾ ^Ja^. ^Jp ⁼

tip ^Is

^2k ^. ¹^Ja^. ^Jp ⁾

=2xrlEx{ ^Ie

⁾

< ²ⁱⁿ ^ar.ae ^>

'Ia£r)x=CIp< ^12×-21=682

⁼

^Eipye

^, ⁼

^tipper

⁼ ^gap ^,

^=Eip'

^.

=2i2p5rEtit

^.

Exercise ^B.2

⇐

call _type I transf .

: I ^'= ^fd=

Intent

^. ⁽ ^a ^#

#

ⁱⁿ ⁾ ^I ^"=

^}

Era ⁼ #IF ⁺

IF

^'EoEo

Tap =4&p

Now , for n=2 , Hap

)=lyIp|=l ^;f

⁾ ^I

'=I⇐÷tz=%t4IfYH2-=t

^'

ii.

±

^⇒

t2=¥

^.

1.24 b ^: Fltity ⁼

,d=tI Itt

^'

ittttilogttl

Then Fltitl ⁼ Gti

'iE2l=EcE' 121¥ )+#' I

^logl

# =¥¥

^-

^¥ ^.l°gtF4

^,

Erika

²

LEE

^'s^'

¥1

^-

etplugl

^.

^E4/+IF'

^{HE 'E2I}

=÷Ey2ti

^-

^logl

^- ^El

£=z^-d⁼³

(12)

Exercises ^:

Chazy

^eqn ^: ^r^'^" ^=6rr ^" ^- ^9r'

2

The cubic eqn ^: w3 ⁺

Ezrltswttjvttiwtglr

^"Cti=o ^- ⁽ ^* ^,

Will ), WRCT ), W_} LT ) ^: hoots of ^{( * )}

W , ⁺ Wztwz ⁼

¥

^V ^if ⁼

}

⁽ ^w^, ^twztw ^{} )}

% ^"

|

^w ^,wz ⁺ ^wzwz ⁺ ^wiwz ⁼ ³³ ^V^" ^⇒ y '= g-⁽^{w ,} ^wztwrwztw ³ ^Wil

- "

wiwzwz ⁼ It j ^" ⁼ ^. 4W , Wzw

,

w, twz twz ⁼

}

^y^' ⁼ ^- ⁽ ^w ^, ^wztwzw ^} ^tWiW3 ⁾

✓

liiwztwiirztwzwz ⁺ wziy ⁺ ⁱⁿ^, ^wzt ^W ^, if =z3 ⁸ ^" ⁼ ^. ^bwiwzwz

|uYiIYwYiI'

11

wwitwwi

'hnIiMwYiYI÷r" "+Er'

^'

^2=-4

⁼ ^.

^Err +

⁽ ^w, wz + iwitnztwsiwiwzwWzwz twzw , JZ

= wirwztwiw

}tw5wT+ Zwlwrwj

^2W ,

wyiwzt

⁺ ² ^WPWLW} ^- 4 wpwzwz ^- 4W ,w5w}

- 4 Wiwzwg

= wpw } ⁺ ^W^}W } ⁺ wpwjh ^- ² Wiwzwz ^-

swiwzwj

^- ^ZWPWZW^}

⇒

time _; ^www.HI.tt

^"

: :I÷

^"

we . . .gg ^a. ^,

www.wg.wsw.p.qw.wawaw.w.wa. ^,

)

hi, ⁼ ^- W_, ( Wz tws ) + Wzw }

|

^his ⁼ ^- ^wzlwitw ^}⁾ ^t ^wiw ^} îs ^solh ôf âbove ^linear êqn 17

Wig ⁼ ^- Wg ⁽ ^w^, ^twz ⁾ ^t ^W ^, ^Wz

(13)

Ex3i1_: ^.

:

a

}p= zcarpltisy

^- ^1*1

Show that any solin of ^the _system ^must be of the form §a=2xE ^:

pfi Jxsp ⁼ 2ps^a ⁼ ^Z

CIA

)§y^. ^⇒ 2&5p ^- 2p}a=o

Therefore , Consider S ^:⁼ Sadtd Then d3=

.2pS&dtBndtd=

0

Locally ^, ³ ^admits ^a primitive F ^sit ^. dE= S ^⇒ 2aEdt ^' ⁼ ^Sade ^' ^⇒ §&=2aF

(b ) 3£,. . ...

, 5_a^" ^: fundamental system of ^solh of ^the _sysytem ⁽ ^* ^I ^for ^a _given ^Z

The corresponding function I ^'.. ... , F ^" are flat word , for the deformed connection

Flz ,,

pf^⇒^: ⁽^The^FBy^'^,^,^. ^.^Jacobian^.^(a)^, ^F^,

=gradlEil=

^we^"⁾ ^matrix^indeed^know ^forms^that^JHIFIZ^a^Si ¹⁼system⁼

g§

(

!

⁽

^Tt÷

⁾of ⁾ local⁼ ¹^Sijword⁾⁽ ^}n¥ji^B^. ⁾^non ^.^is^singular^linearly ^indep^⇒ ^.

Denote

2xi=I

^. ^Io

:=¥gI

^Then ^2.

=2I÷

^.

^Ip

⁼³

^!

^Ip

Fa;2j ⁼

Foil #

^5k⁾

=2i3jk2~k

⁺ ^ST

;2I=ZCij3k2k+sjk%i2I

^Fa

11 =

zcitjseoktsksi _F%2I

To;2j ⁺ ^Z ^.2i .2j ⁼

Z.ci?g2e=Z.Cij3ekJk=z.cfjsek2It5jsispsF2I

⇒

sjs

^!

^FsY2I=o

^⇒ ^H ^ii. ^K ^, ^Sjrsispsrk ^=o ^- ⁱ

^ISI

⁾ ^B ^invertible

.'. Fskr ^to ^. ^⇒ lfln ^. ^. ,F ^" ) is the flat word ^. ⁱⁿ Dubwnn ^connect ^'m

?

,

Ex3= ^: n=3 , del ^. f-( t ) ⁼ tztitz ⁺

Stitz

^- ¥5 ritz ⁾^. ei.es ^. ⁼ Citfek

ckij ⁼

yklceij

^,

7=18

_.

if

⁾ ⁺ _fCTzit3^- ₎

Ci, =L CE=Gi=o

Ciz ⁼ 432=0^, Ciz ⁼¹ ^, ⁼ 423=0 ^C^,

} Ciz

^=L ^. ^C 'zz= Czzz ⁼ ^-

£-428

^'

Cz}= _Czzz ⁼ ^- z3Tzr 432=422 ⁼ ¹ Glz ⁼ ⁽ 333

5¥

^" ^r^'^"

42=(23

} ⁼ ^-

¥8

^" ^C^322=(322 ⁼

¥

^tir ^' ^c

₃₂₃₌₄₃

^, ⁼

_¥3

^r ^"

^433=433--0

(}z= ⁽ 132=0

Recall that the associatively ^condition ^: ⁽ ^ezez^)es=ez( ^eze ^} ⁾

(

^{, , ,}⁺ ^⇒

^footer ^'2=IY4r' Is:c

^"

caucus

⁺

tjtitrr

^" ^⇒ ^r ^" ^. ^brr

"t9r'2=oChYg !

⁾

(14)

sail

.it?IEtztEIY:t=zkIiIidgYMsi2i5=z(!.i:.)s

make :# :

III. ^tins ^I '⇒f¥⇒I" Effi

;iM$

^⇒

Htt¥firt¥¥rH' to

"

→

f¥farf

^"

#

tin

^- ¹ ^au ^becomes ⁽²¹

^:p

^1-^-^-

^)§=o

^2-^ir^I^" ^=V

^'s

^"

The system ^223ps ^-

Zcarptl

^Sr ^- ^HI

|@z+zu

⁾ ⁵⁼⁰ ^- ⁽ ^* ^I ^'

123⁺ ^z ^V⁾ 5=0 If § is a Solis of (* 1

'

, then (22 ^t ^ZU )

5=0

, ( 2stZV )

5=0

⇒ [

2z+zU

,

2z*zU 2ztzV 2stzV

^]

^§=o

^~> ^[ ^, ^] ^=o

Also, note That [ 2ztU , 23 TV ] ⁼ ⁽²²⁺ ^U ⁾ l2stV ⁾ ^- last VI ( 22+01

= 0.223+22^V ⁺ ^U 23 ⁺ ^UV ^- 2322 ^- 2sU ^- VJZ ^- VU ^- ^L** ⁾

ant ^'÷ir ^¥r

^"^" ^'

:.ln=t¥l;YIIII II : :o)

Tier^"

Fair

^'

UV=

tfgtlbrr

^"^. ⁱ^"

lagticrr :*

^"⁾

^Ehr

^,

⁾

^° ^° ^°

seen

"

eater

"

tier

^" _, ^ur ^-

w¥¥"

^" ^" ^" ⁺⁹^" ^"

qq.nl ;)

^"+µ ^,

¥r

^"

Eitir

^o ^o

VU ⁼

ttirv

^tern^"

IYlr" ^agtzlrr µ

^'^- ^'^.^brr^zv^"l

^"l¥tir jeter

^'^"

^|

(15)

Plug 5 ^into ^C** ⁾^,

ask.jo/5+U?#v2_

^-

zuvs

⁺ zVU§ ⁺ ^[ ^v.v ^]§

^-2¥

=L^. ^zti ) [ V.V ] ^} ⁼⁰ ^⇒ [ UIV ]=o _and _[ _V.V ]=o gives Chazy Eqn^.

E×3= ^: ^For ^Wi ^, ^wz ^e R ^'⁽ M ) _, define ( ^w ,, wrl ^* ⁼ Le ( ^w _, _.wzj Then for hive PITM ), ( Eau ^.^v ) ⁼ ( uiv >

pf ^: ^2a ^:=

¥

^a ^22.2ps ⁼

CIpki2r

^→ ^dt ^'^. ^dtf ^:= ^CF ^dtr

Then CY ⁼ y Mci,

Therefore^, ^GDP ⁼ ^{( dt} ^' ^, ^dt ^'⁾^'t ⁼ Leldttidtfl ⁼ ^Er ⁽ ^{dt ?dt} ^'¹ ⁽ ^2,1

= ErCjP= ^Eryosecear

Suppose

JH

^is ^invertible , Sf=g[

agdf

⁼ ^Sea^Ery ^BE ⁽ ^Ir

⇒

yopsf

⁼

nopgcaery

t ⁼ ( 20.2 -o7 ^'^' ^Cir ⁼

:

^Staff ^Erceg ⁼ ^Ercoi ^Gca

⇒ (^E. 20, 2e ) ⁼ ¹ Er2r .2o , at )

±

⁽ ^Et ^Cio ^2a ^, ^k ⁾ ⁼ ^Etch ^Gate

⇒ ( 20, 2t7 ⁼ ( ^E. ^{20,2T )}

Denote # ^: dual via < , > _, ^* ^: dual via ( , ) ^:

Then for we PITM )^, 4dtB⁾^#^, ^w ⁾ ⁼

dttscwldtf

⁾ ^# ⁼ ^At ^2T

⇒ ( HH)^#, 2r7 ⁼ At yer ⁼ ^SB, ⇒ Atneyyr ^' ⁼ sfyre ⁼ ^y ^Be

11

⇒ (dtp) ^# ⁼ yPez{

At

fqE=A

^'

(

^12,2ps^⇒ ^Write⁾

^Begipgpe

^TBTSE⁼ ⁿ^last^2d*=⁼

TIIIIIIIIIIIII

^,2p*j*^Be ^Be⁼ ⁸⁼

^!Ip{

^dtt^gay ⁼^Gp⁽

:/

^fine^2£^,

:I" III :: ::

^,

ldtr.d.ir/*=9arfpign=8jlgpo=gpa=gap

^{del )*=}^⇒ ^Begets^2*= ^gaedte⁼ ^Sf

(16)

Claim ^; ⁽ ^, ⁷ ^* ⁼ ^he ⁽ ⁱ ⁾ ^* ⁽ Proof of 13.391 in Dubrovin 's hook ₎

Proved n the flat word ^, HT

e=2

^, ^E=E' ²^, ^< ^dt ^'^, ^dtp ^> ^=ydP

gH=(

dt ^'_, dtp ) ^* ⁼ LE ( de^'^. dtp ) ⁼ Er CY ^{=Er YPECER}

(Leg ^{) I} ^dt ^'^, dtf ) ⁼ Lelgldtdidtpil ^- ^gl ^Ledtt ^, ^dtB⁾ ^- ^gldt ^'^, ^Ledtf ⁾

tedt ^' ⁼ dlledttl =D 18,4=0

⇒ ( Leg ^{) (}^de ^'^, dtts ) ⁼ Lele ( dtt .dtB) ⁼ e ( He^'^. dtp ) ( EI ⁾

= 2. ( Ercdfl ⁼ ^2. Î Êrl CY ⁺ Êra , 149 ^=D

,CP+

^Era ^, lcipl

= di

ypecij

^Era ^ICP⁾ ^=D

e.

^, 74 ⁺ ^Era ^, ICP)

Er 2. (Cip⁾ ⁼ ^Er ^2. ⁽YPECIY ⁾ =Er7PE2 ^, lcir ⁾ ⁼ Eryp ^' ^2,19 ^" Ger )

=

Eryp

^' ^g. ^" ^2. ¹ ^Cter ⁾ ⁼ ^ERYPEY

"2tl4er

) ⁼ ERYPEY ^" 2tl7ev ¹⁼⁰

Set deli (Leg⁾ ⁽ ^dtd^, ^dtp ⁾ ⁼ 74

(17)

Exercise 4.3

.

- i

F ⁼ Itpts ⁺ I Eitz^' ^-

Eftz4+tzet3

^E ^:= ^t.at ^Ita

²²⁺²²³

fctz -1

,tsi=

- tz4ttzet3

24

When n=3 , WDVV ^a _, ⁼ fz

£222

}} ⁺ fzzzfazz fzzz ⁼⁰ fz}} ⁼ tzet ³ fzzz ⁼ ^- tz fzzz ⁼ et^}

→ F satisfies WDVV

2. f ⁼ tits ^.^, Its 22 F ⁼ ^titz ^-

zltzeets

^asf ⁼ ^ftp.tzet3

Left ^ti2 ^, ^F ⁺ It ^> ^{22 F} ⁺

z3%F=

^2ft zptp

M F _is indeed a soin of ^WDVV ^with ^Euler ^v. ^f^. ^E

C,,^, ⁼ 0 = C ₁₁₂ C₁₁₃=/ C 122=1 Clz } ⁼ C ₁₃₃₌₀ C ₁₁₃ ⁼ ¹

C zzz ⁼⁰ Cz _{} }} ⁼ Et^} Czzz ⁼ ^- tz Czzz ⁼ fzet ^}

Iii ⁼

(

o

; :O !

⁾

^niitiniji

^' ^. ^/ ^,

^:O ^:}

⁾

C i. ⁽, ,} =L CI =CI=o

Ciz=C3z= 0=432--43

Cz's ⁼ Czz }=et3 ^C }z=Cz3z=o ^C}z= 423=0 ⁽ Ci

'f= Yklcije

⁾

Cjz ⁼ ⁽ 223=0 Cz2z=Czzz= ^-^Tz ^{C z3z} ⁼ ^C ^zz , ⁼ I

C³¹³ ⁼ ^C_{} } }} =tzet3 ^C}z ⁼ ^{C 233} ⁼ ^et3 ^C }z=C|z}=o

CiI=

^Ciek

74=4*747

^"^'

CY ⁼ ⁽ 331=0 C 's ⁼ C_} }z=et3 ^C's ⁼ C_z }3=tzet3

C^'? ⁼ C 321=0 ⁽ ^l} ⁼ ^C 322=0 C ^'} ⁼ ⁽ 323 ⁼ et3

CY ⁼ 11=1^C↳^l^}= ^C 2,12=0 ^C^'} ⁼ ^Cziz ⁼ ⁰

( 2,3 ⁼ C _,z _, ⁼ ⁰ C²²³ ⁼ C 221=1 ^C²³³ ⁼ Cizz ⁼ ⁰

CY ⁼ ⁽ ¹²²⁼¹ ^C²²² ⁼ ^C ^zzz ⁼ ^- ^tz ^C²³² ⁼ ^C ^zzz ⁼⁰

C

3,3=41

^, ^=o ^C³³² ⁼ ^Can ^=o ^C³

^}

⁼ ⁽ ¹¹³⁼¹

g^" ⁼ tic _'t ⁼

zltzets

⁺

z3tzet3=2tzet3

^g ^"=EiC^'^f ⁼

}et3

gB=Eid3=t

^,

g22=EicY=t

^,^-

Its g23=EiCY=Itz

g33=

}

⇒

g=2tzet3dti+3et3dt,dtz+ ztidtidtztltisttijdtzttzdtzdtztzsdts

^'

(18)

or

g.

⁼ ¹ ^tl

Fitts III ^I

^zltz

! ^;)

t_, ⁼ 25^' EFZ ( ext ⁹ ⁺ e- ^× + I 4)

tz ⁼

zFeJ'

^Z ( ^I ^'×⇒ ^' text e " ₁ A i az

Tz ⁼ ^Z ^- ^-

dt^dtz=_, ⁼ ^,^-^- ^{25 EIZ}

^th 2%5425×+54.1 ¢

^IeY+zet*"⁾^dx

Hdye

⁺

Ydx

⁺⁽^It^tz^,

E

^-^- ^2555£²⁵ ÊFZ^l2e→e⁽ âÊÊ^×^'Î ^dy^* ^" ⁺

^It

^,

jtzdz

^dz

-

^Bz

dtz ⁼ dz

ztzets ^.

ftp.3et3.ge titzt

²⁴ ^.

^It

^, ⁺ ^, ^-

^It ^Its

⁾

^.gl

^Tit

^site

⁺

^I

=

Ig

^titzets ⁺

÷

^titz ^et³ ⁺

^Fti

⁺

^al

^tie ^, ^-

^ylgt

^it ^t

Stitt

(19)

Ref^: ^Dubrovm ^, Zhang ^- Êxtended Âffine Weyl ^Groups ând ^Fwbenius ^Manifolds

( 19981 ^arxiv ^: hep ^- ^th 19611200

§ ^: irred , reduced _root system ⁱⁿ ^Vi=E^" ⁽ ^Euclidean space ⁾ ^Kc ^B) ^=p ^- ^*

' 2

( 2,2 )

filled

.se?arIdeesItEinaeE.eaeIo

^⇒ ^⇐ ^{± '}

⁾

iiij fd ^, BEE , 241 B ⁾ V) § ^Cannot decompose ^into orthogonal

# ^€ ^K proper ^subset

Fix a simple ^roots of Ê ^: ^4. ^. ^- ^. ^. ⁱ ^du ÊE ⁽ î.e. fpei ^,

p=§n

,

kidi , where KI âre âll Êo

LY ⁼ LLi

d-

, Li ) ^: ^wroot î=l ^. ^. ^. ^, ⁿ . ôr âll ^Zo ⁾

→ Aij ⁼ ¹ ^Li ^, ^2T¹ ê ² ⁽ î.ê^. ^The êntries ôf ^Cartan ^matrix ⁾

WCIE ) ^: The Weyl _gp âssociated Î î.ê ^,

W=( rain ^, ^... ^... ^, by ^>

Then affine Weyl _gp ^Walt ⁾ ^~ ^V by ^: ^x ^- ^wcx ⁾ ⁺

§

,

Midi ^, ^we ^WIEI

mi EZ

Introduce fundamental weight ^wi ^.^. ^.^.^. ^, ^wn EV sit . ( ^w ; , LY) ⁼ Sij So _, pick ^WK ^for ^k€51 ^.^. ^i. ^ill ^,

we define extended affine

Weylgp

^W~== ^Nhk^'l ^E) ^{- '} ^T ⁼ ^VAIR

to be The gp generated ^by ^:

11) ^,Rl^Xeti ¹ ^1-

XILX ×€

^lwcxlt

^§

^,

^Midi

^, ^{Xeti )} ^we ^WII ⁾ ^, ^MJEZ

and (2) I ⁽X_, Xeti ⁾ ^- ( Xtwk , Xeti

=

^- ¹ )

22 21

Az ^:

÷X÷

^Dynkin ^diagram ^• ^- ^Weylgp ⁼ ⁵³

21

Bz^:

•

;;

.

:&

^, ^Dynkin ^diagram ^• ^⇒^•

(20)

Exercised ^: ^F ⁼

ftp.yttzt.ti.at#+zitiet3.stze2t3

with Euler uf ^. E=tc2i+Itz2t ₂₃ ,

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(22)

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