Prescribed Gauss Decompositions for Kac-Moody Groups Over Fields

RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀDI PADOVA

JUN MORITA EUGENE PLOTKIN 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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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ 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 important 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 general 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 decomposition was established for Kac-Moody groups of rank 2. The general 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 a choice of 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, satisfying = 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 commutative 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 datum 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

= n we For i 1, ... , 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 defined as follows:

a Then ( G , U, T , Y, ~ ~ 1, ... , ~ n ~ ) is 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 -1 me: , i =1, ... , n , which automatically (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 generated by X, and by L1 x the subset of real roots generated by X . We defi- 11C LliC luiiuwilig

Let in W = N/T , and identify a, i With if necess- ary.

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 decomposition 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 =

(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 obvious. 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 decomposition 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. on Considering the action of T 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 prescribed Gauss decomposition if given an arbitrary element h * E T, we have

- .. I I I’TTf oL. T’7"’ _11 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.

2 THEOREM A ([11]). Let A be a generalized Cartan ma- =-b a2 trix with ab ~ 4. Let m = max {a, 6}. Let K be a field with K ( > m + 3. Then every Kac-Moody group G over K of type A has a Gauss decomposition 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 prescribed 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 decomposition 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 ~ > 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 > m + 3. Then for every element zg e l, with g e G and rg g Zen, and every elen ~ E T, there exists z E G such that

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 holds 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. X = Y= write Set a2, ..., an_1~, and la2, a3, ..., an~. We

L L, ~ ~ B ~ l f 1

F° i > 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 noncentral 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

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 respectively, 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 already considered case i > 1. The main theorem follows from the proposition in an obvious way.

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