THEOREM I. Let G be a semi-simple, simply-connected, almost simple linear algebraic group ...

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Japan. J. Math.
Vol. 1, No. 2, 1975
n Othe structure of semi-simple algebraic groups
over valuation fields I
By Hiroaki HIJIKATA
(Received June 11, 1969)
This paper is my doctorial thesis
, summing up my studies on the
structure of semi-simple algebraic groups over local fields during the year
1962-67, presented to The University of Tokyo in 1968 . Main results of
this paper are entirely contained in the earlier announcements of Bruhat
Tits ([3], [4], [5] and [6]} appeared in 1966 . The first part of their full paper
[23] appeared in 1972. A point to publish my paper at this time is
, if any,
it contains a proof of the existence of BN-pairs of affine type for any simply
connected p-adic groups. To be precise
,
I. THEOREM Let G be a semi-simple, simply-connected
, almost simple
linear algebraic group defined over a finite extension field k f o Q
p, and Gk b
e the group f o k-rational points f o G. Then Gk has a BN-pair with an
open bounded B and an ine of Weyl group W=N/ .BnN.
In the way of proof, we make an essential use of the following vanish
ing theorem of M. Kneser [16].
THEOREM II. Let G be as in Theorem I, then
H1(k,G)=0.
Bruhat-Tits [6] announced the validity of the above two theorems
, for
much wider classes of fields k. For Th. II, for example
, they simply as
sume k to be a complete discrete non-archimedian valuation field with the
perfect residue field of cohomological dimension 1. Unfortunately, their
thick first volume [23] does not yet contain the proof of either of the above
theorems. If I understand correctly, their idea is
, in a sense, to prove Th. I
first, using, among other things, geometry of affine weyl groups
, then get
Th. II as a consequence. Although, their idea is splendid and their results
are inclusive, one must go a long way to understand their whole theory .
Since p-adic groups call for attensions of so many people
, and since the
existence of BN-pair has such a basic importance for the theory of unitary
226 KIROAKI HIJIKATA
representation of p-adic groups (cf. Macdonald [24]) for example, it may be
worthwhile to present a short proof of Th. I, starting with well known facts
on p-adic groups. Our proof should be shorter, simply because we are try
ing much easier thing i.e. to show Th. II(cid:129)ËTh. I for p-adic field.
In fact, the proof of Theorem I can be much shorter than the one given
in this paper, if we uses at our disposal the vanishing of Galois cohomology
of the maximal compact subgroups of some maximal Lori (cf. 8.20, and
Lemma 1 of Appendix II), which we had intentionally avoided to use in
order to be applicable more general cases than p-adic fields.
Now, let me take advantage of adding two appendices. The first one
is a proof that one can lift certain symmetries of Dynkin diagrams to auto
morphisms of split simple Lie algebras over Z, which, when I had written
this paper, was planned to be published with some other results as a separate
paper.
The second one is a translation (with some more detail) of my report
[25] published in Japanese in 1968, in which the problem (3) raised as a sub
ject of my sequent study in § 0.1 of this paper was solved, i. e.
III. THEOREM Let G be as in Th. I, further assume that G has a posi
tive k-rank, then Gk coincides with its own commutator. Consequently, by
Tits [27], Gk/center is simple as an abstract group.
Platonov [28] reduces the proof of the strong approximation theorem to
the above Th. III, and gave a proof of Th. III. But I suffered some difficul
ties to follow his proof of Th. III.
Another proof of Th. III, using ordinary BN-pairs (like Platonov, in
stead of affine BN-pairs used in my proof) was orally communicated to me
by Professor J. Tits, to whom I take this oppotunity to express my hearty
gratitude for his interesting and informative comments given to me, related
or unrelated to the subject of this paper, during his stay in Kyoto in 1971.
September (Added 1975)11,
0. § Introduction
0.1. Let k denote a field with a non-trivial discrete non-archimedian
valuation, and let G denote a semi-simple linear algebraic group defined over
k. The purpose of this paper is to give a partial conclusion to our series
[11], [12] etc. on the structure of the group of k-rational points Gk. Among
the fields we have in mind, the p-adic fields are kp the most classical and one
of our initial problems was to generalize the so called elementary divisor
theory developped by Eichler [10], Tamagawa, Bruhat, Satake [17] and
Semi-simple algebraic groups 227
others for the classical groups over kp. Namely:
(1) Find a maximal compact subgroup U of Gk, such that Gk has an
analog of the Cartan decomposition and the Iwasawa decomposition in semi
simple real Lie groups.
Then naturally arises the following question: Is the subgroup U unique
(up to ugacy) con j as in the real case? If not, are there only finitely many
conj ugacy classes? If so:
(2) Count the number of maximal compact subgroups (up to con
jugacy).
Besides the problems of the above kind, which we may call structure
theory modulo compact groups, there are other ones such as:
(3) what is the factor commutator group ?Gk/[Gk, Gk]
(4) Determine all the normal subgroups of Go.
In (4), the properties of the group of integral points G
o heavily depend
on the choice of a set of bases in the representation space and here we see the
relation to our first problem, namely the experience shows that the best set
of bases is the one which gives = Go U. For classical groups
, the first
problem (1) was solved by Bruhat and Satake ([17]), the second problem (2)
was solved by the author ([11], [14]), by using the theory of associative
algebras and forms over p-adic fields.
On the other hand, when G has a maximal torus split over k, Iwahori
Matsumoto [15] solved (1) and a part of (2) (i.e. they constructed a certain
number of maximal compact subgroups which seem to be the only ones). In
doing so, they did not make any use of locally compacity of k (replacing
`compact' in the statements by 'bounded')
. The crucial part of their results
for our problem is that if G is simply connected then:
(5) Gk has an open bounded subgroup B (now called an Iwahori sub
group) and a subgroup N such that (Gk, B, N) is a Tits system with an mine
weyl group [for definitions see 1.1 and 4.1].
Then by quite formal arguments, one gets (1) and a part of (2). Later,
Bruhat-Tits [2] showed that the system of (5) is unique if it exists, and that
even the remaining half of (2) (i.e. exaustion) is a consequence of (5). Thus
we know that (1) and (2) are corollaries of (5). The results of this paper
will imply:
(6) If k=kp, (5) is true for any simple and simply connected group G
defined over k.
However our way to (6) is not direct, and we would like to look the
situation as follows. Let G' denote a quasi-split simple group over k with
a fully ramified splitting field L (admitting the case when L=k, i.e. G' is
split over k). The results of [15] can readily be generalized for this G' [12],
228 HIROAKI HIJIKATA
and for each G' there is associated a certain diagram H. (If G' is split over
k, it is the Dynkin diagram of the union of fundamental roots l.. {c, a~}
and the lowest root -a0 of the root system of G'.) Suppose there is given
an unramified extension K of k [for the definition see 8.5], and an isomor
phism c from the Galois group Gal (K/k) into the group of symmetries
Sym H of ]. For given G', K, c and a prime ir of KL (subjecting to certain
restrictions), we can construct a 1-cocycle b of Gal (K/k) with values in the
group of K-rational automorphisms Aut such GK, that all stablize b6 a given
Iwahori subgroup B of GK. Twisting G by b, we get an algebraic group
bG' defined over k and an isomorphism p: G'bG' over K. The results of
this paper will allow us to assert:
(7) If a given simple group G over k is isomorphic over k to group bG'
of the above type, then we have good inf ormations for Gk, in particular if
G is simply connected (5) is true for Gk.
Now if k happens to be a p-adic field k, (or similarly good field), then
by Kneser [16] (or Bruhat-Tits [16]):
(8) Any simple group G is either quasi-split or isomorphic to some bG'
over k.
Hence (7) together with (8) implies (6) for any simply connected G. Now
we should restate that the purpose of this paper is to establish (7) first of
all for the proof of (5), but also in view of the study of such problems as
(3), (4), a study which is still in progress and for which we need to know
the inside structure of Iwahori subgroups.
The main part of our results are purely algebraic and independent of
the special properties of the base field k (say, completeness of k, or perf ect
ness of the residue field).
0.2. We will say a few words for each section. In the first section,
we give a definition of a Tits system (G, B, N), various related objects, and
state some known properties for the sake of references. In the second
section, we consider a group P of automorphisms of a Tits system, and
give an elementary criterion (Theorem 2.9) for the groups of P-fixed points
G" and Br again to form a Tits system with some N*. There we assume
the existence of a certain P-invariant semi-direct product decomposition B
= UZV [in 2.8] which will turn out to be automatically satisfied by any
~roun G which we have in mind.
In the third section, we introduce an affine root system b, which is in
essence nothing more than the direct product of an ordinary root system r
and the set of rational integers Z. We give an axiomatic definition for it,
mainly because its complete analogy with an ordinary root system amuses
us. Thus the practical meaning of this section should be understood as only
Semi simple algebraic groups 229
to provide languages for later sections, hence the given proofs of statements
are cut short and not intrinsic at all. In the forth section we consider a
Tits system whose Weyl group is isomorphic to the Weyl group Wey ti of
an affine root system b. Under a mild assumption about the existence of
a system of fundamental subgroups [defined in 1.5], we give a definition of
a standard subgroup Go (which will have the properties of U in (1) (0.1).
Then we prove the Cartan decomposition, Iwasawa decomposition, and give
a substitute (cf. 4.7 (D.1) and (D.2)) of Satake's regularity condition between
these two decompositions.
In the fifth section, we take a field le with a non-trivial discrete non
archimedian valuation, and a split simple group G defined over lc . Picking
up the properties of Gk which are essential for our arguments, we take them
axioms, and call a system satisfying the axioms, of type S(Zi) (in 5.8), then
construct a Tits system in Gk reproving [15]. Where the product decom
position in 5.4 (which strengthen the product decomposition of an Iwahori
subgroup in [15]) is a key lemma not only for this paper but also for our sub
sequent studies. In the sixth section, we consider a subgroup P of the group
of symmetries Sym 11 of a fundamental system 17 of an affine root system
. ƒÕ Then P naturally operates on the Weyl group Wey D and we give one
(not canonical) way to construct an affine root system 'b1 such that the
group of fixed points Wey cDr is realized as Wey Q1. The complete list of
Dynkin diagrams, the symmetry groups, its subgroups P and correspond
ing c r are given in 6.6.
In the seventh section, we consider an abstruct group with a system of
type S(ƒÕ) of (cid:129)˜ 5, and a group of automorphism P of G. On the one hand,
combining the results of (cid:129)˜ 5 and (cid:129)˜ 6, we can reduce the criterion of (cid:129)˜ 2 to a
simpler form in Theorem 7.6. On the other hand, we pay a special atten
tion to the case where the induced action of P on is in Sym 11, and show
that G~ has a system 'almost' of type S(ir) independently of the criterion
of § 2 [in Proposition 7.8].
In the final section, we substitute two kinds of concrete groups for
(G, f') of the previous section. The first kind is the following; G is a split
simple group over k, L is a fully ramified extension of k, a: ƒÐƒÐ is a 1
cocycle of Gal (L/K) with values in Aut GL such that the twisted group aG
is quasi-split G =GL and P=aGal (L/K)={aa o a; 6 e Gal (L/K)}. The second
kind is the ones described in (7) 0.1, G =Gk and P = bGal (K/k). For the
fi rst kind of groups, we can apply 7.8, and reproving [12], we show that
the system associated to Gr is really of type S(Z*) with a certain ƒÕ*(cid:129)¼ƒÕ„C
Thus we know the structure of Gk, then starting from it, we show that the
criterion of (cid:129)˜ 2 (simplified in (cid:129)˜ 6) is aplicable for our second (G, P). In 8.19,
we state the possibility to construct a cocycle b from given materials based
230 HIROAKI HIJIKATA
on the results of Appendix I.
The section closes with a brief observation about the isomorphisms be
tween the groups of type bG'.
0.3. Notation. Let the lefthand side of each of the following equations
(1)(cid:129)`(12) denote the righthand side of it.
(1) Z=the ring of rational integers.
(2) Q=the field of rational numbers.
For a ring R with 1,
(3) R=the multiplicative group of the inversible elements of R.
For a set S,
(4) |S|=the cardinality of S.
If a group P is operating from left on S (we write the transform of s (cid:129)¸ S by
r (cid:129)¸ P as r(s) or Ts indifferently, i.e. r: s!rs=r(s)), then
(5) Sr=the set of P-invariant points of S.
For a vector space V over a field k with a finite dimension n,
(6) End V=the algebra of all the linear endomorphisms of V,
(7) GL(V)=the group of all the linear automorphisms of V,
(8) SL (V) =the subgroup of GL (V) consisting of elements with
determinant 1.
If k is a universal domain, we sometimes write GL (n) (resp. SL (n)) instead
of GL (V) (resp. SL (V)), and we understand that it has the natural structure
of algebraic group defined over a prime field.
For a group G and its subset A2(2eA),
(9) <A2;2eA>=the subgroup of G generated by the union U2 A2.
For x, y in G,
(10) [x, y]=xyx-1y-1
For subgroups A, B of G,
(11) [A, B]=<[a, b]; aeA, beB>.
Suppose there is given a homomorphism cp from A into a certain group G'
such that Kernel of cpcA(1B, for a subset X of cp(A),
(12) BX (resp. XB)=BƒÕp-1(X) (resp. ƒÕ-1(X)B).
For a finite family of subgroups {B2; 2eA}, if for any arrangement A=
{2(1),...,2(N)} of A, any element b of B can be uniquely written as the
product of b2()eB2(), b=b2(l)...b2~), we call B is a semi direct product of
B2 and write as B=flBz (s. d. ).
1. § BN-pairs and Coxeter systems
1.1 Tits systems. Let us recall the definitions of so called BN-pairs
after Tits [21]. A triple (G, B, N) consisting of a group G and its subgroups
Semi-simple algebraic groups 231
B and N is called a Tits system (or (B, N) is a BN-pair of G), if it satisfies
the following four conditions.
(T.0) : G is generated by B and N, the intersection H = B n N is a
normal subgroup of N.
(T.1): The factor group W=N/H has a set of involutive generators
R={r1;ieI}(rir;ifij, and ri (cid:129)‚1).
(T.2): For any rieR and any weW, r1BwCBriwBUBwB.
(T.3): For any rieR, riBri(cid:129)‚B.
The last two conditions are written under the convention of 0.3. The group
W=N/H is called the Weyl group of the Tits system with a set of distin
guished generators R. A subgroup of G which is conjugate to B (i.e. of
the form gBg-1, geG) is called a Borel subgroup, and a subgroup which
contains some Borel subgroup is called a parabolic subgroup of the Tits
system. When the Weyl group of the Tits system is an affine Weyl group
[a definition of an affine Weyl group will be given in the section 3], the names
'Iwahori' subgroup and 'parahoric' subgroup are used instead of 'Borel' and
' parabolic'.
1.2 Coxeter groups. Let consider any group W satisfying the above
condition (T.1) (disregarding N and H). Let m(i, j) denote the order of the
product rirj in W jeI). (i, Since W is generated by R, any element w of
W admits an expression like w=ri(1)...ri(l). If it has the smallest length
among all such expressions of w by R, the expression (or the sequence of
indeus (i(1),...,i(l))) is called reduced. Let 1(w) denote the length of a
reduced expression of w, and will 1(w) be called the length of w with respect
to R. It can be proved that the following three conditions are mutually
equivalent.
(C.1): If (i(1),...,i(l)) is reduced but (i(0),i(1),...,i(l)) is not reduced,
then there exists an integer m(1(cid:129)…m(cid:129)…l) such that ri(1)ri(2)...ri(m)=ri(0)ri(1)
...r i(m-1).
(C.2): If (i(1),...,i(l)) is not reduced, the there exists a pair of in
tegers p, q (1(cid:129)…p(cid:129)…q(cid:129)…l) such that
ri(l)...ri(1)=ri(1)...ri(p-l)ri(p+1)...ri(q-1)r2(q+l)...ri(l).
(C.3): Any non-trivial relation among ri (ieI) is a consequences of
the relations (rirj)m(i,j)=1 (i, 3eI).
If R satisfies one of the above conditions, (W, R) is called a Coxeter sys
tem (or W is a Coxeter group with a good system of involutive generators
R). For a subset K of I, let K denote W the totality of the elements of W
which admit a reduced expression by elements of RK={r1; ieK}. Then WK
is exactly the subgroup of W generated by RK and (WK, RK) is again a
232 HIROAKI HIJIKATA
Coxeter system.
1.3 Properties of a Tits system. Let us recall some of the fundamental
properties of a Tits system which will be used in this paper. In the follow
ing, (i) can be found in [22], all the rest are in [21].
(i) If W is the eyl group of a Tits system, and R is a set of distin
guished generators of W, then (W, R) is a Coxeter system.
(ii) Assuming the condition (T.0) and (T.1), the condition (T.2) is
equivalent to the following two.
(T.2'): For any rieR, RUBr2B is a group
(T.2"): If l(riw)>l(w), r2eR and weW, then riBwcBr,wB.
(iii) w(eW) is in R if and only if RUBwB is a group.
(iv) For any subset K of 1, let PK denote a parabolic subgroup gener
ated by W, B then PK=BW KB. Furthermore wBwB (weWK) gives a
bijection from the group to WK the B-B double cosies B\FK/B of FE . In
particular we have G=BWB (disjoint union), and this double costes deco
position is often referred to as a Bruhat decomposition of G by B.
(v) Any parabolic subgroup is its own normalizer. Any parabolic
subgroup containing B has the form PK with KcI. PK is conjugete to FK,
if and only if K=K'.
1.4 Fixed points set of a Coxeter group. Let (W, R) be a Coxeter system .
Let Aut (W, R) denote the group consisting of all automorphisms of W which
stabilize R. Let P be a group operating on W as automorphisms which stabi
lize R, i.e. there is given a homomorphism P--~Aut 1k: (W, R) such that rw=
for iJi(Y)w weW, reP. Let Rj (jeJ') be P-orbits (=minimal P-invariant
subsets) of R, R=UjJ,Rj (disjoint union). For any subset A of W, let r A
denote the set of P-fixed points of A. The following properties are known
[13].
(i) If is WRj an infinite group, then (WR3)r is trivial (={1}). If WRj
is finite, then (WRj)r is a cyclic group of order two generated by the element
sj of the maximal length in WRj.
(ii) Let J denote the subset of J' consisting of j such that is finite, WRj
and let S={sj jeJ}. ; Then (W", S) is again a Coxeter system.
(iii) For any weW and sjeS, the following four conditions are mutu
ally equivalent.
(1) l(sjw)>1(w) [resp, l(sjw)<1(w)] for any sjeS,
(2) l(sjw)=l(sj)+l(w) [resp. l(sjw)=1(w)-l(sj)] for any sjeS,
(3) l(sw)>1(w) [resp. l(sw)<1(w)] for any seWE,,
(4) l(rw)>1(w) [resp. l(rw)<1(w)] for some reR1.
1.5 Fundamental subgroups. Let (G, B, N) be a Tits system and let W
,
Semi-simple algebraic groups 233
R and I be as in 1.1. For any i e I, let Bi denote the subgroup of B defined
as the intersection flw (Bf1wBw-') where we let w run over all weW satis
fying the condition l(r1w)>1(w). A set of subgroups {X1; ie1} indexed by
the same set I as R will be called a system of fundamental subgroups of a
Tits system (G, B, N), if it satisfied the following two conditions for any
iel.
(X.1): (X.0):
If {Xi; ieI} is a system of fundamental subgroups, then by (T.2") 1.3
(resp. (T.3)1.1) the following condition (X.2) (resp. (X.3)) is also satisfied
for any ieI;
(X.2):
(X.3):
Now it is easy to establish the following.
(i) A Tits system (G, B, N) has a system of fundamental subgroups if
and only if B=B2(Bnr2Br2) for any ieI.
(ii) When there are given a group G and its subgroups B, N and X2
(i e I) satisfying the conditions (T.0), (T.1), (X.0) and (X.1), (G, B, N) is a
Tits system if and only if the conditions (X.2) and (X.3) are satisfied by
them. If that is so, {X1; ieI} is a system of fundamental subgroups of
(G, B, N). Indeed there are the following obvious implications: (X.3)=
(T.3), (X.1) and (X.2)~(T.2'), (X.o) and (X.1)~(T.2").
1.6 Normal extensions. Let (G, B, N) be a Tits system. Suppose that
G is contained in some bigger group G. We will consider the case when G
has a subgroups M satisfying the following two conditions.
(M.0): G is generated by G and M.
(M.1): M normalizes B and N, M(1B is normalized by N.
Set N=MN, H=BflN. We will call G (or (G, B, N)) a normal exten
sion of (G, B, N). The following properties are readily derived from that
of (G, B, N).
(i) H=(BnM)H and H is a normal subgroup of N.
(ii) Set W=N/II and Q=M/B n M. Then W=QW (s.d).
(iii) Since M normalizes N and H=B(1N, M operates on W by the in
ner action. By (iii) 1.3, M stabilizes R, hence we have a homomorphism
MAut (W, R) via inner actions. Similarly we have a homomorphism Q
Aut (W, R) via inner actions. If this QAut (W, R) is injective we call
G a reduced normal extension of (G, B, N).
(iv) Let Ma be the kernel of the above homomorphism M-Aut (W, R).
Then M0N) M0B, (M0G, is a Tits system and its Weyl group can be canoni
cally identified with the Weyl group (W, R) of (G, B, N). G is a reduced
234 HIROAKI HIJIKATA
normal extension of M0N).M0B, (M0G,
(v) Let (~, K) be a subset of QXI such that is a subgroup of Q and
RK={ri; ieK} is stable under the action (induced by Q-~Aut (W, R)) of ~.
Let P(~ ,K)=P=B2WKB. (a e ~'W~) Then gives o-H-*BciB a bijection be
tween the group ~'WK and the B-B double cosets B`P(z,K)/B of P(I,K). In
particular, C=BWB (disjoint union). Let 2 be the normalizes in Q of (~',
K) i.e. 2 is maximal among the subgroups of Q which stabilizes both of ~'
and PK by inner actions. Then P(i,K) is the normalizes in G of P(~ ,K).
Furthermore P(~, is conjugate ,K') in G to P(~,K), if and only if (s', K') is con
jugate to (', K) under the action of Q. Finally any subgroup of G contain
ing B has the form P(I ,K).
2. § Fixed points of BN-pairs
2.1. Let (G, B, N) be a Tits system. Let Aut (G, B, N) denote the
group of automorphisms of G which stabilize B and N. Let P be a subgroup
of Aut (G, B, N). For any P-set A, let Ar denote the set of P-fixed points
of A. Since P stabilizes N and H=B n N, it operates on W=N/H, and by
(iii) 1.3, it stabilizes R. The results of 1.4 are applicable and in the same
notations as there, (W", S) is a Coxeter system. For any weW, let B(W)
denote the group B n w-1Bw. Fix an element w of W", and let consider the
following conditions.
(w.1): The projection: Br-*(B/B(w-1))r is surjective.
(w.2): (wBw~1)r c Br.
(w.3): (BwB)r is not empty.
(w.4): (BwB)r is a non-empty double coset of Br.
(w,5): There exists a subgroup Z of B containing H and normalized
by w, such that (wZ)" is not empty.
(w.6): The projection: (BwB)r->(BwB/B)r is surjective.
Let (W.i) and (S.i) i= 1, 2,...,6, denote the following conditions.
(W.i): (w.i) is true for any weW".
(S.i): (w.i) is true for any s~eS.
In (W.5) (resp. (S.5)), we understand that (w.5) is satisfied by one fixed
Z for any weWr (resp. s~eS).
2.2. LEMMA. Let s~eS and weW", then
PROOF. Taking P-fixed points of Bruhat decomposition, we have G"=
(BW"B)r. By repeating application of (T.2) 1.1, Bs~BwB BW C R~wB.
Since r={w, R;w) (W sw}, taking P-fixed points of the both sides, we get