Prescribed Gauss decompositions for Kac-Moody groups over fields

R ^ENDICONTI

del

S ^EMINARIO M ^ATEMATICO

della

U NIVERSITÀ DI P ^ADOVA

J ^UN M ^ORITA

E UGENE P LOTKIN

Prescribed Gauss decompositions for Kac- Moody groups over fields

Rendiconti del Seminario Matematico della Università di Padova, tome 106 (2001), p. 153-163

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Prescribed Gauss Decompositions

for Kac-Moody Groups Over Fields.

JUN

MORITA(*) -

^EUGENE

PLOTKIN (**)

ABSTRACT - We obtain the Gauss

decomposition

^with

prescribed

^{torus elements}

for a

Kac-Moody

group ^{over a} field

containing sufficiently

many elements.

1. Introduction.

Kac-Moody

groups ^are

equipped

with canonical

decompositions

^of

different

types.

^{Let us}

note,

^for

instance,

^the

decompositions

^of

Bruhat,

Birkhoff and Gauss. As in the finite dimensional case,

they play

^{an im-}

portant

role in calculations with these _groups.

However,

in the Kac-Moo-

dy

^case each of these

decompositions

^{has its own}

special

^features

(see,

for

example, [18]

where J. Tits compares the Bruhat and the Birkhoff

decompositions

^and

^presents

^some

applications).

The aim of this _paper is to establish the so-called

prescribed

^Gauss

decomposition

^for

Kac-Moody

groups.

The

prescribed

^Gauss

decomposition appeared

ⁱⁿ

[16], [9]

^{for the} ge- neral linear _group. It was

proved

^{for all}

Chevalley

^{and twisted}

Chevalley

groups in

[3], [4], [5].

^{It turned out to} be the main tool for

proving

^the

substantial Ore and

Thompson conjectures.

(*) Indirizzo dell’A.: Institute of Mathematics,

University

^of Tsukuba, ^Tsu- kuba, 305-8571,

Japan.

(**) Indirizzo dell’A.:

Department

^of Mathematics, ^and

Computer Science,

Bar Ilan

University,

52900, ^Ramat Gan, ^Israel.

Partially

supported

by

grant ^{ISF and}

by

grant of the Centre of Excellence

«Group

theoretic methods in the

study

^of

algebraic

varieties» of ISF.

Namely,

^Ore

[12] conjectured

^that every element of a finite

simple

group is ^a

single

commutator.

The

proof

^{of this statement for all}

simple

groups of Lie

type

^is

given

in

[6].

Similar facts are also known for infinite

simple

groups. Let ^us mention the paper of Ree

[14],

^who

proved

^that every element of ^{a con-} nected

semisimple algebraic

^{group over an}

algebraically

closed field is a

commutator. A _survey on the Ore

problem ^(not including

the results of Ellers-Gordeev and

Lev)

^can be found in

[20].

This paper is the continuation of

[11],

^{where the}

prescribed

^{Gauss de-}

composition

^was established for

Kac-Moody

groups of rank 2. The gene- ral result became

possible

^{due to the} recent paper

[2]

^{where the}

elegant

idea of V. Chernousov _gave rise to a uniform

proof

^{of the}

prescribed

Gauss

decomposition

^{for all groups of Lie}

type.

^We

mostly

^{follow the}

method of this _paper,

adjusting

^{it to the}

Kac-Moody

^case.

We do not consider in this _paper Ore and

Thompson type conjectures

for

Kac-Moody

groups. These groups ^are

perfect but, generally

spea-

king,

^not

necessarily simple,

^{and their} commutator structure can be _very delicate.

2. -

Kac-Moody

groups.

Let A ⁼

(aii)

^{be an n x n}

generalized

Cartan matrix. Let _g be the

Kac-Moody

^Lie

algebra

^over the field C defined

by A according

^{to the}

choice of a realization

17, Il

^{of A with}

II = ~ a 1,

^{... ,}

a n ~,

^{the set of}

simple ^roots,

^and

11 " _ ~ a i , ... , a n ~,

^{the set of}

simple ^coroots,

^sati-

sfying

⁼ a2~

(cf. [7], [10]).

^{Let d c}

~*

^{be the root}

system

^of g with

respect

^{to the so-called Cartan}

sub algebra I).

^Let

d +

^{be the}

set of

positive (resp. negative)

^{roots defined}

by II ,

^{and L1 re the set of real}

roots. Set A = L1 Then

(root

space

decomposition)

and

(triangular decomposition),

where So-

Let C~i be a

Chevalley-Tits

^group functor from the

category

^{of com-}

mutative

rings

^{with 1 to the}

category

of groups,

corresponding

^{to g}

^(cf.

[19]).

^For any commutative

ring

^{R with 1 and for} any ^a

E .LI re,

^{there exists}

155

a group

homomorphism xa (~ ) : ~2013~CR),

^{where R + is the additive}

group of R .

Using

^a

Chevalley basis

^{for L1re}

^{(cf. [19]),}

^we

can express ⁼ exp

(tea )

^{for t E R . Let}

G(R)

^{be the}

subgroup

^of

Q)(R) generated by

for all aeL1re and ten. We call

G(R)

^a

(standard

^or

elementary) Kac-Moody

group, which

depends

^on the choice of a root da- tum and here one can choose _{any root} datum

(cf. [10], [13], [15], [17], [19]).

Sometimes

G(R )

^{is said to be of}

type

^{A . From now} on, ^{we assume} that R ⁼ K is a field and shorten

G(K)

to G. We also _suppose 1 for all aEL1re and t E K

(t ~ 0 ).

^Let

-.. -- u I , - 1 ’B u

For i ⁼

1,

^{... , n we define}

homomorphisms g

for all a e K and t e 7~ . Let

Ui, Vi

^{be the root}

subgroups corresponding

to the roots ^a

i and - a i , respectively.

^The

subgroups Ui and Vi

^{are defi-}

ned as follows:

Then

( G , U, T , Y, ~ ~ 1,

^{... ,}

~ n ~ )

^{is a}

triangular system ^{(see [11]}

^{for the}

definition).

^Hence

(cf. [11]),

every

Kac-Moody

group G over a field has a

Gauss

decomposition, i.e.,

Set

then

TN ,

^and N/T ^is

isomorphic

^{to the}

Weyl

group W

(cf. [7], [10]).

^We

sometimes

identify

^{W with}

^{N/T .}

^For

w E W,

^{we write}

with w ⁼ w T

and w E N,

^{where w can} also be identified with an inverse

image 10

of w under cannonical

epimorphism

^Then

and

(cf.[8], [13]). Therefore,

^{if 1 ~ u E U}

(resp.

^{1 ~ v E}

V),

then there exists w E N such that

for some i ⁼

1,2, ..., n, where ui

^E

~/

^e

Ui’ (resp. Vi E Via, ~/

^e

Otherwise

(resp. -W v -W -’ E Vi’)

for all WE Wand all

i

=1,

... , n , which

automatically

^{me: , -1}

(resp.

v E

f l

and u = 1

(resp. v

⁼

^1).

^{The above}

property

^is

important

for

~7and

^{we will use it later.}

For a subset X of

II,

^{we denote}

by L1 x

^{the subroot}

system

^of L1 gene- rated

by ^X,

^and

by L1 x

the subset of real roots

generated by

^{X . We defi-}

11C LliC luiiuwilig

Let ary.

in

W = N/T ,

^and

identify a, i With

^{if necess-}

PROOF.

(1): Let g

^e

Gx

^{n U.}

Then, by

the Bruhat

decomposition

^of

Gx,

^we

write g

^{= uhum with} u, ve

Tx

^{and w e}

W,

^where

Wx

is the

parabolic subgroup

^{of W}

corresponding

^{to X.}

Using

the Bruhat de-

composition

^of

G ,

^we

get w

⁼

1,

^which

implies that 9

^can be written as

g = uh for some u E

UX

^{and h E} Therefore h ⁼ 1 since U n T = 1.

Hence, g

^{= u E}

Ux. Clearly Ux c Gx

^{n U.}

Therefore, Gx

^{n U =}

Ux.

(2):

^{Let z E}

Ux

ⁿ

Ui’ .

^Then

...~ ... ¿,.J,. ¿,.J,. , v v m Bh ~

invariant since

Gx= (Ux, Tx,

^Let

aieX, ~d X ,

^{t E}

E K,

^and

YEUX.

^{We write with}

zi E U2

^{and s E K.}

(a)

^In the ^case when s ⁼

s

(b)

^{In the case when s # 0 and}

0,

^{where means the}

value of the linear function at

a’( E lj,

where x’

= xp(t)

^{or x’}

= xp (t)

for some t’ E K.

(c)

^{In the case when s ~ 0 and}

~3( a i ) ^0,

for s

(4)

^{can be}

proved

^{in the same} way ^as in

(3),

^and

(5)

^{is ob-} vious.

Q.E.D.

Let

Z(G)

be the center of the group G. Note that

Z(G)

^c

T ,

^{and we}

can

explicitly

describe it ^as follows

(cf. [15], [19]):

159

We make a short review here. Let

Using

the Bruhat

decompo-

sition G =

UNU,

^{one can}

write g = uhwx

E U n

w -1 Vw,

^{which has a}

unique expression.

^{If w #}

1,

^{then we} choose

fl

^E

L1 +

^{such that E L1 - . Then =}

x,~ ( 1 )

^{g , and we obtain a con-}

tradiction

by

^the

uniqueness

^of

expressions.

^In

particular, w

^{= 1}

and g

^E

E UT .

Similarly

^{we see g E VT .}

Hence,

^{we obtain}

Z( G )

^{c T = UT n VT.}

Considering

the action of T on the

subgroups Ui ,

^{... , I}

Un , VI ,

^{... ,}

Vn ,

^it

is now easy to ^see that the center

Z(G)

^{can be}

expressed

^{as above. We}

notice that we need not assume

«simply

connectedness» here.

3. - Theorems.

We shall _say that a

Chevalley

^or

Kac-Moody

group G has the

prescri-

bed Gauss

decomposition

^if

given

^an

arbitrary

^element

h * E T,

^{we have}

- .. I I I’TTf oL. T’7"’ _1 1

wnere is tne center ot u kL6j, lllJ).

The result on the

prescribed

^Gauss

decomposition

^for

Kac-Moody

groups of rank two is as follows.

THEOREM A

([11]).

^{Let A} ₌ ²

a

^{be a}

generalized

^{Cartan ma-}

-b 2

trix with ab ~ 4. Let m ⁼

max {a, 6}.

^{Let K be a field with}

K ( &#x3E;

^{m + 3.}

Then every

Kac-Moody

group G over K of

type

^{A has a Gauss}

decomposi-

tion with

prescribed

^{elements in T.}

A similar result holds for 0 ~ ab ~ 3 in the rank two case without _any restrictions on the

cardinality

^{of K.}

Moreover,

^{in the case of}

Chevalley

groups, there is a

general

^{result on the Gauss}

decomposition

^with pre- scribed

semisimple

^elements.

THEOREM B

([2]).

^{Let A be a}

(finite)

^Cartan

matrix,

^{and let K be a} field. Then _every

Chevalley

group G over K of

type

^{A has a} Gauss decom-

position

^with

prescribed

^{elements in T.}

In the

remaining part

^{of this} paper, ^we will establish the

following

main result on

Kac-Moody

groups G. For ^a

generalized

^{Cartan matrix}

A ⁼ we set

I l 1

160

MAIN THEOREM.

Suppose K ~ &#x3E; m + 3.

Then every

Kac-Moody

group G over K of

type

^{A has the Gauss}

decomposition

^with

prescribed

elements in T.

COROLLARY.

Every

noncentral element of a

Kac-Moody

group G can

be

expressed

^{as a}

product

^{of two}

unipotent

elements in G.

4. - An inductive method.

Here we will show the

following proposition.

^{Denote 1=}

={1~...~}.

PROPOSITION. Let 7" =

(r, G)

^{be a group with G} normal in r and such that the

conjugation

^{is a}

diagonal automorphism

^{of G. Let}

be the center of r.

Suppose ] K

&#x3E; m + 3. Then for every element zg e l, with _g ^e G and rg g

Zen,

^and every elen

exists z E G such that

~ E

T,

^there

for some v e V and u e U.

This

proposition

^{can be}

proved

ⁱⁿ

exactly

^{the same} way ^as in

[2].

^We

proceed by

^{induction on n . It is}

already

known that the

Proposition

^hol-

ds for

n = 1, 2 (cf. [2], [111),

^{which is}

precisely

described in Theorem A and Theorem B. Now we

suppose n %

^{3. To make our induction}

complete,

we have to assume that the

cardinality

^{of K is}

greater

^{than m +}

3,

^since

we

essentially

^use the information in ^case of rank two.

Since rU = Ui and G =

UVT U,

^{we have rG =}

= UW7Y/.

Hence,

^{for our} purpose, we can assume that _Tg is of the form zg = ivhu with v e

V, h

^e

T , u

^{E U.}

Then,

^{we fall into one of the}

following

three cases,

using

^{the fact}

^{f l w Uw -1= f 1}

and the defini-

weW wEW

tion that r induces a

diagonal automorphism

^{of G.}

(Case 1)

Let u # 1 or v ~ 1. Then there exists w ^E W

satisfying

161

(Case 2)

^{Let u = v = 1} and the element _ig be of the form

_ _ ._ 2013T~

.. - ....-- . v - ~

---- f - - - -

(Case 3)

Let u ^{= v =} 1 and the element _rg be of the form

vviv,, "I - ’L G4111A. r - - B,.. 1.

In

(Case 2),

^{we can find an with t # 0 for}

some j

^{E I}

such that

I., I " , 1 .1 11 1 1

with

1 ~ u~

^E

Uj.

^Thus

^{(Case 2)}

^{can be reduced to}

^{(Case 1).}

^{Our assum-}

ption

^on the form of _ig

implies

^{that we can}

skip (Case 3).

So let us assume

that we are in

(Case 1)

and fix an i e I

appearing

^{in the formula for} y.

Set X = _a2, ...,

an_1~,

^{and Y=}

la2,

^{a3, ...,}

an~.

^{We write}

L L, ~ ~ B ~ l f 1

F° i &#x3E;

1,

^we

put

^Ty=

rh,,, (tl)

^and

^ry= (Ty, Gy).

^Then

--- - . 1 ~ .._1 _ ~ 1 ~ __1 _ ~ 1 ~ 7 --- -.1 --. ^- --- - - - ----

1. Let us choose K " such that ih ^*

ha2 (t2 )

^{is non-}

central in

Gx ~,

^{where rx= This is}

actually possible

since there are

enough

elements in K. In

fact,

^{we can choose ± 1 if}

centralizes both of

U2

^and

V2,

^{otherwise we}

just put t~’ =1.

In _{any case,} does not centralize

Ga2 = ~ U2, ^V2),

which

implies

^{that is not} central in

Applying

induction to non-central element

....

W 11111 1 ~ we l:au lUlU au eleUleUl1 uy JlAl;il Lll’G6L ;4

with and

.L i ~ i .A-

where Then we 1

with

v" E V, U" e U ; ( t2 ) . ^Next,

^{we write}

I

Then we

apply

^{induction to}

Since _{y x} is noncentral in

T X,

^{we can find an element} ZX ^E

Gx

^{such that}

with

vX E VX, UIE Ux

^{and rh *}

= Z x hX . Therefore,

we have

with v"’ E

V,

^{u"’ E}

U,

^{and we are done.}

If i

= 1,

^we take first X and then

Y,

^and

repeat

^{the same} process ^as above. Then

by choosing

^{a n, and a 1} instead of _{a 2} and _{a n} re-

spectively,

^we

complete

^the

proof

^of

Proposition.

^{We can also}

explain

^the

matter in the

following

way. The order ^on II is irrelevant for the

proof.

Therefore,

^{if in our} initial order i turns out to be _one, we can reorder II in such a way, that

taking

^the

corresponding X

^{and Y we come to the al-}

ready

considered case i &#x3E; 1.

The main theorem follows from the

proposition

^{in an obvious}

way.

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