CLASSES OF ELEMENTS OF SEMISIMPLE ALGEBRAIC GROUPS
R O B E R T S T E I N B E R G
Given a semisimple algebraic group G, we ask: how are the ele ments of G partitioned into conjugacy classes and how do these classes fit together to form G? The answers to these questions not only throw light on the algebraic and topological structure of G, but are impor tant for the representation theory and functional analysis of G; conversely, of course, we can expect the representations and functions on G to contribute to the answers to the questions. In what follows, we discuss some aspects of these questions, present some known results, and pose some unsolved problems.
To do this, we first recall some basic facts about semisimple algeb raic groups. Our reference for all this is [7]. A linear algebraic group G is a.subgroup of some GLn (К) (n>\, К an algebraically closed field) which is the complete set of zeros of some set of polynomials over К in the n2 matric entries. The Zariski topology, in which the closed sets are the algebraic subsets of G, is used. G is semisimple if it is connected and has no nontrivial connected solvable normal subgroup; it is then the product, possibly with amalgamation of centers, of some simple groups. The simple groups have been classified, in the KilHng-Cartan tradition. Thus there are the classical groups (unimodular, symplectic, orthogonal, spin, etc.) and the five exceptional types. The rank of a semisimple group is the dimension of a maximal torus (a subgroup isomorphic to a product of GL4's). We also need the notions of simple connectedness and Lie algebra as they apply to linear algebraic groups, but we omit the definitions. Henceforth К denotes a fixed algebraically closed field, p its characteristic, G a simply conne cted semisimple algebraic group over K, and r the rank of Gf The simplest example is G = SLr+i (K)- L denotes the Lie algebra of G. If x is an element of G, we write Gx (resp. Lx) for the centralizer of x in G (resp. L). We shorten conjugacy class to class, and dimension to dim.
Now we take up the problem of surveying the conjugacy classes 3f G and finding for them suitable representative elements. Recall that an element of G is semisimple if it is diagonalizable, and unipo-tent if its characteristic values are all 1, and that every element can be decomposed uniquely x =xsxu as a product of commuting semisim-ple and unipotent elements. An element x will be called regular if dim Gx = r, or, equivalently, if dim Gx is minimal [12, p. 49]; here x need not be semisimple and may in fact be unipotent. Various chara-:terizations, hence alternate possible definitions, of regular elements
may be found in [12]. The regular (resp. semisimple) elements form a dense open set with complement of codimension 3 (resp. 1) in G [12, 1.3 and 6,8]; thus most elements are regular (resp. semisimple). In [12, 1.4] we have constructed a closed irreducible cross-section, C, isomorphic to affine r-space, for the regular classes of G. Also, we have proved [12, 1.2]:
(1) The map x-±- xs sets up a I-I correspondence between the regular and the semisimple classes of G.
Thus by choosing the elements of С and their semisimple parts we obtain a system of representatives for the regular classes and for the semisimple classes of G. If G is the group SLn, the representatives
of the regular classes chosen turn out to be in one of the Jordan cano nical forms. Thus we are led to our first problem. •
(2) P r o b l e m . Determine canonical representatives, simitar to those given by the Jordan normal forms in SLn, for all of the classes of G, not just the regular ones.
In proving the results described above, we are naturally led to consider the algebra of regular class functions, those regular functions on G that are constant on the conjugacy classes of G. This is a poly nomial algebra generated by the characters Xi> OC2, • • *> Xr of the fundamental representations of G [12, 6.1]. (If G = SLr+i, the %'s
are just the coefficients of the characteristic polynomial with the first and last terms excluded.) First of all these functions give a very useful characterization of regular elements of G: x is regular if and only if the differentials of %u %2 Xr are linearly independent
at x [12, 1.5]. And secondly they are related to our classification problem by the following result [12, 6.17]:
(3) The map x ->• (%{(x), %2 (x) %r (x)) on G induces a bijection of the regular classes, and of the semisimple classes, onto the points of affine r-space.
(4) P r o b l e m . . Assume that x and y in G are such that for every rational representation (p, V) of G the elements p (x) and p (y) are con jugate in GL (V). Prove that x and y are conjugate in G.
By (3) this holds if x and y are both regular or both semisimple, and it can presumably de checked when G is a classical group, but we have not done this in all cases.
To continue our discussion, we will use the following result. (5) If y is a semisimple element of G, then Gy is a connected reductive group (i.e. the product of a semisimple group and a central torus).
More generally one can show: The group of fixed points of every semisimple automorphism of G is connected. Such connectedness the orems are important in problems concerning conjugacy classes, as we shall see, and also in problems about cohomology [1, p. 224].
Assume now that x and x' in G have the same semisimple part y. Then x and x' are conjugate in G if and only if xu and x'u are conjugate
in Gy, in fact, by (5), in the semisimple part of Gy. The latter group
may not be simply connected, but this is immaterial for the study of unipotent elements. Thus the study of arbitrary classes is reduced to the study of unipotent classes, and these are the clases we now consider:
At present not much is known about the classification of the uni potent classes of G. Observe, though, that (1) implies that regular unipotent elements exist and that they in fact form a single conjugacy class.
(6) P r o b l e m . Classify the unipotent classes and determine cano nical representatives for them.
Up =0, the study of unipotent classes in G is equivalent to that of nilpotent classes in L, and a classification of these latter classes together with representative elements may be found in [3; 4]. The result, however, is in the form of a list, which, though finite, is very long, thus subject to error and inconvenient for applications. The main proce dure in the construction of the list, that of imbedding each nilpotent element in an algebra of type s/2 and then using the representation
theory of this algebra, or the corresponding procedure with G in place of L, is not available if p фО [12, p. 60]. In studying the nilpotent classes of L (still for p =0) one is naturally led to study the class H of G-harmonic polynomials on L: indeed these polynomials are in natural correspondence with the regular functions on the variety of all nilpo-tent elements of L, by results in [61. To our knowledge, no one has succeeded in effectively relating H to the classification problem at hand.
(7) P г о b 1 e m. Do this.
(8) P г о b 1 e m. Do the same for the unipotent classes of G (for p arbitrary), having first found an analogue for H.
Now it follows from (5) that a unipotent element centralized by a semisimple element not in the center of G is contained in a proper semisimple subgroup of G. Thus an important special case of (6) is: (9) P г о b 1 e m. Same as (6) for the classes of unipotent elements x such that Gx is the product of the center of G and a unipotent subgroup.
As is easily seen, the class of regular unipotent elements always satisfies this condition. If p = 0, then relatively few other classes do [3, Th. 10.6].
. A more modest problem is as follows.
(10) P r o b l e m . Prove that the number of unipotent classes of G is finite.
Here only classes which satisfy (9) need be considered. Observe that a proof of (4) would yield the finiteness together with a bound on the number of classes. From the work of Kostant one obtains a rather lengthy proof of (10) together with the bound 3r, in case / 7 = 0 .
Richardson [6] has found a simple, short proof that works not only if p = 0 , but, more generally, if p does not divide any coefficient of
the highest root of any simple component of G, this root being expressed in terms of the simple ones. In the sequel such a value of p will be called good. Richardson's method depends eventually on the fact that an algebraic set has only a finite number ,of irreducible components; thus it yields no bound on the number of unipotent classes, but it does yield the following useful result.
(11) If p is good and G has no simple component of type An, then Lx is the Lie algebra of Gx for every x in G.
(12) P г о b 1 e m. Extend (11) to all p and G, with the conclusion replaced by: Lx is the sum of the Lie algebra of Gx and the center of L.
This is easy if x is semisimple and known if x is regular (cf. [ 12,4.3] ). Finally let us mention an old problem which seems to be closely related to (6).
(13) P r o b l e m . Determine, within a general framework, the conjugacy classes of the Weyl group W of G, and assuming that W acts, as usual, on a real r-dimensional space V, relate them to the W-harmonic polynomials on V.
Next we consider some results and problems about centralizers of elements of G. We have already stated, in (5) and (11) above, two such results. Springer [10] has proved that for any x in G the group Gx contains an Abelian subgroup of dimension r. It follows that if x
is regular, then the identity component Gx of Gx is Abelian.
(14) P г о b 1 e m. // x is regular, prove that Gx is Abelian. (15) P r o b l e m . Prove conversely that if Gx is Abelian, then x is regular.
These results would yield an abstract characterization of the regular elements, but other such characterizations are known [ 12, 3.14]. Both problems may easily be reduced to the unipotent case. If x is a regular unipotent element, then, as already mentioned, Gx is the
product of the center of G and a unipotent group Ux. Springer [11,4.Ill
has proved:
(16) If p is good, then Ux is connected (hence Abelian).
Thus (14) holds in this case. If p is bad, (16) is false, in fact x $ U%. (17) P r o b l e m . For p bad determine the structure of Ux/Ux and whether Ux is Abelian.
A solution to (16) would complete the proof of (14), and quite likely would also have cohomological applications (cf. [11, § 3]). We do not know whether (15) is true even if p = 0 .
(18) P r o b l e m . Prove that dim Gx— r is always even. In other word, the dimension of each conjugacy class is even.
This has been proved in the following cases: if p = 0 [5, Prop. 15], if p is good (because of (11) the same type of proof as for p = 0 can be given), if x is semisimple (this is easy); and presumably can be checked when G is a classical group.
All of the above problems are special cases of one big final problem about central izers.
(19) P T о b 1 e m. For every element x of G determine the structure >fGx.
Next we discuss briefly the individual classes, and their closures. Each class is, of course, an irreducible subvariety of G (because G is connected). A class is closed if and only if it is semisimple [12, 6.13]. Vlore generally, the closure of a class consists of a number of classes )f which exactly one is semisimple [12, 6.11]. Whether the number s always finite or not depends on (10). The most important case >ccurs when we start with the class UQ of regular unipotent
ele-nents; then the closure U is just the variety of all unipotent elements )f G. Although some results about U are known, e.g. U is a complete
ntersection specified by the equations %t(x) = %t(l) ( l < t < r ) (the
t's are still the fundamental characters), and the codimension >f U — UQ in U is at least 2 [12, 6.11], we feel that U should be
tudied thoroughly.
(20) P r o b l e m . Study U thoroughly. Of course this problem is related to (8).
Using the properties of U just mentioned and also (16), Springer las proved the following result.
(21) If p is good, then the variety U (of unipotent elements of G) is amorphic, as a G-space, to the variety N of nilpotent elements
fL.
If p is bad, this is false, because (16) is. If p = 0, Kostant [5] ias obtained results about the structure and cohomology of N, in articular that N and any Cartan subalgebra of L intersect at 0 with multiplicity equal to the order of the Weyl group W.
(22) P r o b l e m . Find a natural action of W on affine dim G — r)-space so that the quotient variety is isomorphic to N.
So far we have been considering G over an algebraically closed eld K- From now on we shall assume that G is defined over a perfect eld k which has К as an algebraic closure; thus the polynomials rtiich define G as an algebraic group can be chosen so that their oefficients are in k. The problem is to study the classes of Gh, the group
f elements of G which are defined over k, i.e. which have their coor-inates in k. The main idea is to relate the classes of Gh to those of ! with the help of the Galois theory. We will discuss mainly the
pro-lem of surveying, and finding canonical representatives for, the lasses of GÄ. Recalling our solution to this problem for the regular
md semisimple) classes of G, we naturally try to adapt our cross-action С of the regular classes to the present situation. Consider a regu-ir class of G which meets GÄ. This class, call it R, is necessarily defined
is a variety) over k. How can we ensure that the representative ele-îent С П R of R is in GÄ? The obvious way is to construct С so that
it also is defined over k. For this to be possible, the unique unipotent element of С must be in GA, and then the unique Borel (i.e. maximal connected solvable) subgroup of G containing this element [12, 3.2 pnd 3.3] must de defined over k; i.e. G must contain a Borel subgroup defined over k. This last condition is a natural one which arises in other classification problems, and, although a bit restrictive, holds for a significant class of groups, e.g. for the so-called Chevalley groups 12], of which one is the group SLn. Conversely, if this condition holds, then it turns out that with a few exceptions (certain groups of type Am (but not SLm+i) are forbidden as simple components of G) С can be constructed over k [12, § 9]. Thus if we form С (] Gk, we obtain a set of canonical representatives for those regular classes of G which meet Gk; but not necessarily for the regular classes of Gk since a single class of G may intersect Gk in several classes. In other words, elements of GA conjugate in G may not be conjugate in Gk. Looking for an idea to remedy this situation, we consider the case G = SLn,. We have the Jordan normal forms for all the classes of SLn (K), no! just the regular ones, and, though this fails in SLn (K), it does hold in GLn(k). Now the action of GLn(k) by conjugation is that oi PGLn (k), i.e. of öh, if we let ù denote the adjoint group of G, i.e. the quotient of G over its center, which is PSLn in the present case. Thus we are led to the following problem.
(23) P r o b l e m . Asume that G contains a Borel subgroup definec
over k (and perhaps that G has no simple component of type Am)
(a) Prove that every class of G defined over k meets Gk, i.e. contains an eh
ment defined over k. (b) Prove that two elements of Gk which are conjugatt
in G are conjugate under Gh.
As we have seen, (a) holds for regular classes. It holds also foi semisimple classes, without any restriction on the components o: type Am [12, 1.7].
(24) Assume that G contains a Borel subgroup defined over k. Thet
every semisimple class of G defined over k meets Gh. (And conversely*.]
This result has a number of applications to classification problem: [12, p. 51-2], especially if (cohomological) dim fe<l. We recal [8, p. 11-8] that dim fe<l is equivalent to: every finite dimensiona division algebra over k is commutative. This holds in several casei of interest in number theory, in particular if k is finite [8, p. 11-10] As a consequence of (24), we haveJI2, 1.9]:
(25) / / dim &< 1 and H is any connected linear group defined over k
then the Galois cohomology H1 (k, H) is trivial.
This result and an extension due to Springer [8, p. 111-16] lea| to the classification of semisimple groups defined over k, if dim &< '. (we get [12, 10.2] that there is always a Borel subgroup defined ove
k), and answer most of the questions raised above. Concerning (23a), w<
(26) J/ k and H are as in (25), then every class of H defined over
k meets Hh.
Concerning (23b), we get [21, 10.3]:
(27) / / d i m £ < l , then two semisimple elements of Gk which are
conjugate in G are conjugate in Gh.
This result is false for arbitrary elements, e.g. for regular ones. To show a typical kind of argument, we give the proof of (27). Let
y and z be semisimple elements of Gk conjugate in G; thus ayar1 = z with a in G. Combining this equation with the one obtained by applying a general element y of the Galois group of К over k, we
see that arxy (a), call it xy, is in Gy. As we check at once, x satisfies the cocycle condition ху& = xyy (x6), hence represents an element of Нг(к, Gy), which is trivial, by (25), because G„ is connected, by (5), Thus xy = by (b"1) for all у and some b in Gy. We conclude that у and z are conjugate in Gh, by the element ab in fact. The same argument, with (16) in place of (5), shows that regular unipotent elements of ôk which are conjugate in ô (the adjoint group) are con-jugate in ôh, if p is good (here d i m f e < l is not necessary, but p good is).
By combining (26) and (27), we get a complete survey of the semi-simple classes of GA, if d i m £ < l ; in particular, these classes
corres-pond to the rational points of affine r-space, suitably defined over k (cf. [12, 10.3]). Thus, if A is a finite field of q elements, so that Gk is a simply"connected version of one of the simple finite Chevalley groups or their twisted analogues, the number of such classes is qT, a useful fact in the representation theory of these groups [ 11]. A related result, whose proof, unfortunately, does not use the same ideas, states that the number of unipotent elements of Gh is (/d,mG-r [13].
Finally we conclude our paper with another problem.
(28) P r o b l e m . If G is defined over k (any perfect field), prove
that every unipotent class of G is defined over k.
This would yield a proof of (10). In fact, assuming p Ф 0, as we
may, choosing k as the field of p elements, applying (28) with G suitably defined, and then using (26), we would get (10) with the number boun ded by \Gk\.
Dept. of Mathematics,
University of California, Los Angeles, USA
R E F E R E N C E S
[1] B o r e l A., Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, Tôhoku Math. J., 13 (1961), 216-240.
[2] С h e v a 1 1 e y C , Sur certains groupes simples, Tôhoku Math. J., 7 (1955), 14-66.
[3] Д ы н к и н E. Б., Полупростые подалгебры полупростых алгебр Ли,
[4] K o s t a n t В., The principal three-dimensional subgroup and theBetti numbers of a complex simple Lie group, Amer. J. Math., 81 (1959), 973-1032.
[5] К о s t a n t В., Lie group representations on polynomial rings, Amer. J. Math., 86 (1963), 327-404.
[6] R i c h a r d s o n R. W., Conjugacy classes in Lie algebras and algeb
raic groups, to appear. * [7] Séminaire С. Chevalley, Classification des Groupes de Lie Algébriques
(two volumes), Paris (1956-8).
[8] S e r r e J.-P., Cohomologie Galoisienne, Lecture Notes, Springer-Verlag, Berlin.
[9] S p r i n g e r T. A., Some arithmetical results on semisimple Lie algeb-ras, Publ. Math. I.H.E.S., 30 (1966), 115-141.
[10] S p r i n g e r T. A., A note Чэп centralizers in semisimple groups, to appear.
[11] S t e i n b e r g R., Representations of algebraic groups, Nagoya Math. J., 22 (1963), 33-56.
[12] S t e i n b e r g R., Regular elements of semisimple algebraic groups, Publ. Math. I.H.E.S., No. 25 (1965), 48-80.
[13] S t e i n b e r g R., Endomorpisms of linear algebraic groups, Memoirs Amer. Math. Soc, to appear.
О НЕКОТОРЫХ ПРОБЛЕМАХ БЕРНСАЙДОВСКОГО ТИПА Е. С. Г О Л О Д Целью доклада является описание приема, который позволяет строить контрпримеры для некоторых проблем бернсайдовского типа [1] в случае «неограниченного» показателя. Как известно, в случае «ограниченного» показателя большинство из этих проблем имеет положительное решение [2, 3, 4], за исключением собственно проблемы Бернсайда о периодических группах [5]. Далее будет доказана следующая Т е о р е м а . Пусть k — произвольное поле. Существует А-алгеб-ра А (без единицы) с d > 2 обА-алгеб-разующими, котоА-алгеб-рая бесконечно мерна как векторное пространство над k, в которой всякая подал гебра с числом образующих < d нильпотентна и которая такова, что П Ап = (0). n=i Частными случаями этой теоремы являются результаты, получен ные тем же методом в [6], а также следующие утверждения: С л е д с т в и е 1. Существует финитно-аппроксимируемая бес конечная р-группа с d > 2 образующими, в которой всякая под группа с числом образующих <С ^.койечна *). х) Как доказал С. П. Струнков [7], если в группе с числом образующих d > 3 всякая собственная подгруппа конечна, то и сама группа также конечна.
