Algebraic groups and arithmetic groups
Armand Borel
The Institute for Advanced Study
Princeton, New Jersey
Distributed in conjunction with the Colloquium Lectures given at the Pennsylvania State University, University Park, Pennsylvania, August 31 - September 3, 1971, seventy-sixth summer meeting of
the American Mathematical Society.
(Notes typed by Abhishek Parab from an old pamphlet found at the Purdue
Math library. 17 November, 2012)
1
These lectures will attempt to describe some of the results obtained in the
last fifteen years or so and some of the present trends in the areas named in
the title. The emphasis will be on theorems of rather general scope; however,
they will often be stated precisely only in some typical special cases involving
familiar classical groups.
The following text is meant to be just a rough outline, aimed only at giving
some idea of the probable contents of the lectures.
1
Algebraic groups, Lie groups and geometry
1.1
In the last quarter of the 19th century, S. Lie introduced and studied extensively
a mathematical object which in essence is what is often called today a germ of
Lie group or a local Lie group: a neighborhood of the origin in Rn or Cn
endowed with a law of composition defined by means of convergent power series
which is associative, with inverse, for elements sufficiently close to the origin.
This notation was globalized from the twenties on to that of a real (complex)
Lie group G: a real (complex) analytic manifold endowed with a group law
such that (x, y) 7→ x.y −1 is a real (complex) abalytic function from G × G to
G. Whereas the local theory was in essence a reduction to purely algebraic
questions on the Lie algebra of G, the global point of view brought into play
differential geometry, topology and analysis.
1.2
Thus a Lie group is a “group in the category of analytic manifolds.” Similarly,
an algebraic group is a group in the category of algebraic varieties. In these
lectures, we shall be interested only in linear ones. A less intrinsic but more
down to earth definition can be given as follows: let K be an algebraically closed
field. A subgroup G of the group GL(m, K) of m × m invertible matrices with
coefficients in K is algebraic if there exists a family of polynomials Pi (i ∈ I) in
m2 indeterminates, with coefficients in K, such that
G = {g = (gij )1≤i,j≤m |Pi (g11 , g12 , · · · , gmm ) = 0, (i ∈ I)}.
Examples: GL(m, K), SL(n, K) = {g ∈ GL(n, K)|detg = 1}, the group of
upper triangular matrices, the group of automorphisms of K m leaving a bilinear
form invariant.
2
1.3
Recall that an affine variety V in K n is the set of zeros of a family of polynomials
on K n . The coordinate ring K[V ] of V is the quotient of K[X1 , · · · , Xn ] by
the ideal J(V ) of polynomials vanishing on V . In these lectures the algebraic
varieties to be encountered are either affine or projective (zeros of homogeneous
polynomials in homogeneous coordinates in projective space P(n, K)). In both
cases, the point-set topological notions will refer to the Zariski-topology, in
which the closed sets are the affine (resp. projective) subvarieties.
1.4
The group G of 1.2 is then by definition the intersection of GL(m, K) with a
closed subset of the space M(m, K) of m × m matrices over K. In fact, it is
also canonically an affine variety in its own right with coordinate ring K[G]
generated by the gij ’s and the function det−1 : g 7→ (detg)−1 (i.e., K[G] =
(K[M(m, K)/J(G))[det−1 ]).
If J(G) is generated, as an ideal, by elements having coefficients in a subfield
k of K, then G is said to be defined over k and we put G(k) = G ∩ GL(n, K).
The main purpose of the theory of algebraic groups is the study of the groups
G(k).
If G0 ⊆ GL(m0 , K) is another algebraic group, a homomorphism u : G → G0
is a morphism of algebraic groups if f 7→ f ◦ u maps K[G0 ] into K[G], i.e., if
the coefficients of u(g) are polynomials in the gij ’s and det−1 . An isomorphism
class of algebraic groups in the above sense may then be viewed as the set of
matric realizations of an affine algebraic group (i.e., a group in the category of
algebraic varieties whose underlying variety is affine).
1.5
Let K = C. Then G is a complex Lie group. Similarly, if G is defined over
R, G(R) is a real Lie group. In both cases we can avail ourselves of the standart methods of Lie group theory, in particular of the exponential map from
the Lie algebra to the group. The latter was extended to algebraic groups in
characteristic zero by C. Chevalley [10]. However, this breaks down in positive
characteristic, although the Lie algebra is still defined (as the Lie algebra of
derivations of K[G] into itself which commute with right translations of G).
Progress in the theory of linear algebraic groups has been due to the introduction of completely different global methods leaning on algebraic geometry, and
valid in arbitrary characteristic, which have, in fact, brought new results also in
3
the classical case. In order to give an idea of those, we shall sketch the proof of
two extremely simple, nevertheless basic, facts pertaining to an algebraic group
G operating on an algebraic variety V :
1. G always has a closed orbit;
2. if G is connected, solvable, and V is projective (or complete), then G has
a fixed point.
Let G ⊆ GL(m, K) be solvable, connected. Then (2), applied to the natural
action of G on P(m − 1, K), or better, on the projective variety of maximal
flags in K m (a flag is a strictly increasing sequence of vector subspaces) implies
immediately that G can be put in triangular form (Theorem of Lie-Kolchin). For
a general G, one sees in a similar manner that the maximal connected solvable
subgroups B of G are conjugate and that the quotient G/B of G is a projective
variety [1, 2]. Thus, in a way, the study of G is reduced to that of connected
solvable groups (which is rather easy), and of G/B (which is hard).
1.6
The connected algebraic group G is almost simple if it has no infinite proper
closed normal subgroup, semi-simple if it has no infinite closed normal commutative subgroup. As in the classical case, a semi-simple group is finitely covered
by a direct product of almost simple groups. A main result of the theory (due to
C. Chevalley [12]) is the fact that the classification of semi-simple groups over
K, up to isomorphism, is “the same” as over C. Thus it is given by the KillingCartan classification of complex semi-simple Lie algebras, plus some data to
distinguish between locally isomorphic groups. More precisely, given a complex
semi-simple Lie group G0 ⊆ GL(m, C) there is a procedure (to be outlined), to
construct the analogue G of G0 over K, as a subgroup of GL(m, K), defined over
the prime field K0 of K. If G0 is almost simple, simply connected, then G(k),
as an abstract group, is simply modulo its center for every k ⊆ K (barring a
few exceptions in which k has two or three elements) [11, 24]. The G(k)’s are
the Chevalley groups or split groups. For finite k, they yield a number of series
of finite simple groups, several of which were new at the time. Later on, variations of this construction gave further finite simple groups (groups of Steinberg,
Suzuki, Ree, Tits, see [21] for a survey).
4
1.7
Some of the main properties of G(k), or of G/B, are expressed by the existence
of a “Bruhat decomposition.” The latter in turn follows from the existence of
a “Tits system” or “BN-pair” [9] on G(k). Furthermore, to a Tits system T
there is associated an incident geometry and a simplicial complex now called the
building of T [9, 25]. The geometries thus obtained are far reaching generalizations of projective geometry. Let us try to describe these objects for SL(n, k).
Let B be the subgroup of upper triangular matrices in SL(n, K), D the group
of diagonal matrices, N the normalizer of D. Then W = W (G) = N/D is
the symmetric group in n letters, realized as a group of permutations of the n
coordinate lines. It is generated by the set S = {s1 , · · · , sn−1 } where sj permutes K.ej and K.ej+1 and leaves the other lines fixed. We have N ∩ B = D.
The system T = {SL(n, k), N (k), B(k), S} is then a Tits system or BN -pair.
This implies in particular: SL(n, k) is the disjoint union of the double cosets
B(k).w.B(k), w ∈ W (G) (Bruhat decomposition); the subgroups which contain
B(k) are the subgroups PI (k) = B(k)WI B(k), where I runs through the subsets
of S, and WI is the subgroup of W generated by I; the conjugate of these subgroups are by definition the parabolic subgroups of T . They are the stability
groups of the flags in k n . To T let us associate a simplicial complex X in the
following manner: the vertices are the maximal proper parabolic subgroups; a
set of distinct vertices spans a simplex if and only if the intersections of the corresponding parabolic subgroups is also parabolic. This is a simplicial complex
of dimension n − 1; every simplex is contained in one of dimension n − 1. This
is the building associated to the given BN -pair.
On the other hand, we can attach to P(n − 1, k) a simplicial complex X 0 , the
flag complex of P(n − 1, k): its vertices correspond to the projective subspaces
of P(n − 1, k); distinct vertices (x0 , · · · , xs ) span an s-simplex if and only if
the corresponding subspaces form a flag. It is immediate that by assigning
to a simplex of X 0 the stability group of the flag it represents, we define an
isomorphism of X 0 onto X. Moreover, the automorphisms of X 0 correspond to
the bijections of P(n − 1, k) which send collinear points onto collinear points.
By the fundamental theorem of projective geometry, these are the projective
transformations combined with automorphisms of k. Thus the Tits system of
SL(n, k) leads to a geometry which is essentially equivalent to the projective
geometry.
5
1.8
In 1.7 we have considered implicitly only split groups. There is also a Tits
system for any group G(k), where G is semi-simple, defined over k. Its parabolic
subgroups are the groups P (k) where P runs through the closed subgroups of
G, defined over k, which are parabolic in the sense that G/P is a projective
variety. T is not empty if and only if there is at least one P 6= G (if not, G
is said to be anisotropic over k). If, say, G = SO(F ) is the special orthogonal
group of a non-degenerate quadratic form over k n , then the parabolic subgroups
are the stability groups of the flags in k n which consist of isotropic subspaces.
The dimension of T is s − 1, where s is the Witt index of F . In general, dim T
= rkk G - 1, where rkk G is the rank of G over k; by definition, the rank rkk G
of G over k is the (common) dimension of the maximal k-split tori of G, i.e., of
the maximal subgroups which are isomorphic over k to a product of GL(1, K).
1.9
Let now M be either G if K = C or G(R) if K = C, and G is defined over R.
Then M is a semi-simple Lie group, and its Lie group topology is finer than the
Zariski topology. A fundamental role in the study of M as a Lie group is played
by the maximal compact subgroups of G, which are all conjugate, and by the
quotient X = G/K of G by one of them. The latter is a simply connected, complete Riemannian symmetric space with negative curvature. Moreover, every
such space occurs in this manner (provided its de Rham decomposition displays
no flat factor).
Assume now that k is a non-archimedean and non-discrete local field, say
the field Qp of p-adic numbers (p prime) to fix the ideas, and assume G to
be semi-simple, defined over k. Then G(k) inherits from k a topology, finer
than the Zariski topology, with respect to which it is a locally compact, totally
disconnected group, in fact a p-adic Lie group. The importance of the above
facts in the real or complex cases has led to a search for some sort of analogues
in the p-adic case. They cannot be straightforward, since maximal compact
subgroups are not always conjugate and are open, so that the quotient of G(k)
by one of them is a discrete set. However, Bruhat and Tits showed that G(k)
has a Tits system, quite different from the one alluded to above, for which the
maximal parabolic subgroups are the maximal compact subgroups [8]. They
form finitely many conjugacy classes. The building X associated to it is an
infinite (unless G(k) is compact), locally finite complex which has some geometric properties akin to that of the above symmetric spaces: in particular, it is
6
contractible and carries a complete metric, invariant under all automorphisms;
the distance between two points is realized by a unique geodesic segment joining
them. Recent results on harmonic functions, harmonic forms, cohomology and
unitary representations of G(k), due to P. Cartier, H. Garland, J-P. Serre and
the speaker, although still quite fragmentary, indicate strongly that X will be in
many more respects a substitute for the symmetric space of non-compact type
of the classical theory. In these lectures, some brief indications on these results
and a description of the building for SL(2, Qp ) will be given.
2
Arithmetic groups
2.1
Let G ⊆ GL(m, C) be an algebraic group defined over the field Q of rational
numbers. A subgroup Γ of G(Q) is arithmetic if it is commensurable with
the group G(Z) = G ∩ GL(m, Z) of elements in G which have integral entries
and determinant ±1. (We recall that two subgroups A, B of a group L are
commensurable if A ∩ B has finite index in both A and B.)
Examples: SL(n, Z), GL(n, Z), the group of units of an integral quadratic
form, Siegel’s modular group Sp(2m, Z) or rather Sp(2m, Z)/{±1}.
In the above definition, Q and Z may be replaced by a number field k and
the ring of integers of k, but in fact this does not lead to more general groups.
Via the theory of automorphic forms, arithmetic groups occur in many questions in number theory, moduli, abelian varieties, etc., which form a powerful
incentive to their study. Here, however, we shall probably limit ourselves largely
to the discussion of some properties of the arithmetic groups themselves. Much
of the information gathered on them so far, even of a purely algebraic nature, has
been obtained by highly transcendental methods. A basic tool is the so-called
reduction theory, which we shall discuss first.
2.2
Although this assumption can be relaxed, from now on we assume G to be
connected, semi-simple, and let X = G(R)/K denote the quotient of G(R),
hence acts properly on X. The quotient X/Γ has finite invariant measure, and
is compact if and only if G is anisotropic over Q (cf. 1.8) [5]. In the non-compact
case, it is important to have some information on the shape of X/Γ outside large
compact sets. To get this is a main goal of reduction theory [3].
7
Assume that Γ is torsion free, a rather harmless restriction since any arithmetic group has torsion free subgroups of finite index. Then X/Γ is a manifold.
It is known to be isomorphic to the interior of a manifold with boundary [18].
More precisely, one can construct a contractible manifold “with corners” X, of
which X is the interior on which Γ operates freely and properly with a compact
fundamental set [6](1). The corners correspond to the parabolic subgroups defined over Q of G, and in fact X − X has the homotopy type of the building
associated to G(Q) (1.7). The construction generalizes one given by C. L. Siegel
[21] in the case of SL(n, Z); it allows one to put reduction theory in a much more
pregnant form. Some indications on it, in the case of SL(n, Z), and on other
compactifications of X/Γ will be given.
2.3
Since X and X are acyclic, the cohomology of Γ with coefficients in a module
M , in the sense of Eilenberg-MacLane, is canonically isomorphic to that of X/Γ
or X/Γ, with coefficients in a local system naturally associated to M . A first
application of the above construction is a duality theorem relating homology and
cohomology of [6](1). It implies in particular that the cohomological dimension
of Γ is dim X − d, where d is the rank of Q over G 1.8. If d = 0, which is the
case if and only if X/Γ is comact, this reduces to Poincaré duality.
2.4
Using the corners, it is possible to study the boundary behavior of differential
forms. This allows one to prove for the Betti numbers of X/Γ where X/Γ is
compact [17], by methods in part similar to those of H. Garland’s [14]. This has
applications to the stable real cohomology of classical arithmetic groups, e.g. one
proves that limn→∞ H ∗ (SL(n, Z), R) is an exterior algebra over infinitely many
←−
generators, one for each dimension of the form 4k + 1(k ≥ 1); in particular,
H 2 (SL(n, Z), R) = 0 for n big enough (in fact, n ≥ 7) [14].
2.5
The notion of arithmetic group can be generalized to that of an S-arithmetic
group, where S is a finite set of primes. Let ZS be the ring of rational numbers
whose denominators are power products of elements in S and GL(n, ZS ) the
group of n × n matrices with coefficients in ZS , determinant invertible in ZS .
A subgroup ΓS of G(Q) is S-arithmetic if it is commensurable to G(ZS ) =
G ∩ GL(m, ZS ). Let then XS (resp. X S ) be the product of X (resp. X) by the
8
Bruhat-Tits buildings (1.8) attached to the p-adic groups G(Qp ), p ∈ S. Then
ΓS acts in a natural way properly on XS and X S , freely if it is torsion free, and
X S /ΓS is compact. Moreover, XS and X S are contractible, so they play the
same role in the study of ΓS as X and X in the study of Γ [6] (b), [19, 20].
2.6
This also shows that if ΓS is torsion free, then Z has a Z[ΓS ]-free resolution
of finite rank, hence the Euler-Poincaré characteristic χ(ΓS ) is well-defined. In
the general case χ(ΓS ) is defined, following C. T. C. Wall, as χ(Γ0 ).[ΓS : Γ0 ]−1
where Γ0 is a torsion free subgroup of finite index and [ΓS : Γ0 ] the index of
Γ0 in ΓS . If Γ = SL(2, Z), then χ(Γ) = −1/12, which happens to be the value
ζ(−1) of Riemann’s zeta function at −1. This is no accident however, and
such relationships have been established by G. Harder [15] in much greater
generality. They involve the Euler characteristic of arithmetic subgroups of
Chevalley groups and the values of zeta-functions of totally real number fields
at certain negative odd integers (essentially the “exponents” of the ambient
group). The main result of [15] is an extension of the Gauss-Bonnet formula to
X/Γ, which, for split groups, had been computed by R. Langlands [16].
It is clear from the above definition of χ(ΓS ) that if ΓS has a torsion free subgroup of index N then N.χ(ΓS ) is an integer. Together with Harder’s formula,
this, applied to SL2 , yields some information on the denominator of ζk (−1) when
k is a totally real number field [19, 20].
9
References
[1] A. Borel, Groupes algébriques linéaires, Annals of Math. (2) 64 (1956), 2082.
[2] A. Borel, Linear algebraic groups, Notes by H. Bass, Benjamin, New York,
1969.
[3] A. Borel, Introduction aux groupes arithmétiques, Hermann, Paris, (1969).
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groups, Springer Lecture Notes 131 (1969), Part A.
[5] A. Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups,
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[6] A. Borel et J-P. Serre:
1. Adjonction de coins aux espaces symétriques; application à la
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[8] F. Bruhat et J. Tits:
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10
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[20] J-P. Serre, Cohomologie de groupes discrets, Sém. Bourbaki, Juin 1971.
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11