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A Note on Quotients of Real Algebraic Groups by Arithmetic Subgroups

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A Note on Quotients of Real Algebraic Groups by Arithmetic Subgroups CORE Metadata, citation and similar papers at core.ac.uk Provided by Publications of the IAS Fellows Inventiones math. 4, 318--335 (1968) A Note on Quotients of Real Algebraic Groups by Arithmetic Subgroups M. S. RAGHUNATHAN* (Bombay) Introduction Let G be a connected semi-simple algebraic group defined over Q. Let F be an arithmetic subgroup of G, i.e., a subgroup of G such that for some (and therefore any) faithful rational representation p: G--, GL(N, C) defined over Q, Fc~p-I(SL(N, Z)) is of finite index in both F and p-a(SL(N, Z)). Let Kc G be a maximal compact subgroup of Ga, the set of real points of G. With this notation, we can state the main result of this note. Theorem. Let FcGR. There exists a smooth function f: GR/F~R + such that i) f-l(0, r] is compact for all r>0. ii) There exists r o > 0 such that f has no critical points outsider- 1(0, to] and iii) f is invariant under the action of K on the left. ff in addition F has no non-trivial elements of finite order, K\GR/F is a smooth manifold and f defines a smooth function fl on this manifold satisfying O) and (ii) with f replaced by f x. Corollary 1. Gs/F is homeomorphic to the interior of a smooth com- pact manifold with boundary; if F contains no element of finite order other than the identity, K\GR/F is homeomorphic to the interior of a compact smooth manifold with boundary. We now drop the hypothesis that F c G R. Corollary 2. F is finitely presentable. Corollary 3. If M is any F-module finitely generated over Z, H* (F, M) is finitely generated. Corollary 4. The functor M~,H*(F, M) on the category of F- modules commutes with the formation of inductive limits. We now deduce the corollaries from the main theorem. Corollary 1 is a consequence of elementary facts from Morse theory. For F ~ GR Corollary 2 follows from the fact that F is the quotient by * Supported by National Science Foundation grant GP 5803. Quotients of Reai Algebraic Groups 319 a finitely generated central subgroup H of the fundamental group F' of Gs/F which is finitely presented since the space Gn/F is of the same homotopy type as a finite simplicial complex. The general case follows from the fact that F/F n Gs is finite. Corollary 1 implies that it F~ G~ has no non-trivial elements of finite order the trivial F-module Z admits a free resolution 0~ C,-~ C,-1 ~ "'" --* Cl ~ Co --*Z where each Ci is a finitely generated free-module over F; in fact, in this case, K\GR/F has the homotopy type of a finite complex L and its universal covering L being of the homotopy type of K\GR is contractible. If we then take the induced triangulation of L, the associated chain- complex gives the resolution we are looking for. Corollaries 3 and 4 are then immediate consequences of this fact (when F c G~ and has no elements of finite order other than identity). The general case then follows from the Hochschild-Serre spectral sequence and the following fact due to SELBERG [4]. Any arithmetic group F admits a subgroup F' of finite index contained in G a and such that no element of F' other than the identity has finite order. w1. A Lemma on Root Systems By a root system we mean as usual a set ~ ..... cq of l linearly in- dependent vectors in R / (with the usual scalar product) such that (i) (cq, ct]> <0 for i#j and (ii) 2<cti, ~j>/(c~i, c~i> is an integer. (In the sequel we make no use of (ii).) Let 2i be the unique vector in R / such that <2i, ~])=~/j. We have then Lemma 1.1. If we set 2k= Z a~ko~,+ E b~k2] iEl ir where I is any subset of [1, ..., 1], then b~ k, ajIk are all greater than or equal to zero. Proof. Clearly, if k6L a/Ik=o for all ieI and ,,jl~I k = t~jk" Hence we can assume that k~l. Let then 2~ be the unique vector in the subspace generated by {c~i}i~I such that (2~, c9> =3kj for all jeL We then assert that 2'k = ~ mi ~i with m i>>.O. iel If not, in fact, let Y Z nj j i~II VeI-It 320 M.S. RAGHUNATHAN: with nj>0 for allj~[-I1 and mi>=0 for all i~I1. We then have, 0< ~". nj(2~, otj>= ~. m i nj<~i, ~j> -- 1[~ nj ~jll 2 tJ a contradiction, since m~nj>O and <~i, ~j>=<0 for i#j. Hence 2~=~mi~i with ms>0. i~.l Now consider ~k--2~. Clearly <2k--2~, ~i> =0 if i~I and for ir jzI Since if ir i#j for any jzI, in particular for j=k. It follows that 2k--2~,= ~, bj2j where bj=<2k--2'k,~j>>O. jCz It follows that 2k= Z mio~i+ Z bj2j i~l ir where ml>O, by>=O. Hence the lemma. w2. A Lemma on Siegel Domains Let G be a connected semisimple algebraic group defined over Q. Let T be a maximal Q split torus of G. For a subgroup H of G we denote by Ha, the group Hc~Ga where Gn is the set of real points of G. Let A be the connected component of the identity of TR. Let X(T) denote the lattice of rational characters on T. Then for aEA and x~X(T), x(a)>0. Let g be the Lie algebra G and for xeX(T), let gx= {v/v ~ g, Ad t (v) = X(t) v for all t ~ T} and let be the system of roots of G with respect to Ti.e. ={ZIx~X(T), Z#0, gX#0}. We introduce a lexicographic order on X(T) and denote by ~+, r and A the system of positive negative and simple roots of G with respect to this order. Let n= LI g~; then n is a Lie subalgebra and the Lie subgroup N corresponding to it is a unipotent algebraic subgroup of G defined over Q (it is moreover maximal with respect to this property). Let Z(T) be the centralizer of T; then Z(T) is reductive and can be written in the form M. T where M is a reductive algebraic group defined and anisotropic over Q. More- over M normalizes N so that MN=P ~ is a subgroup of G. Finally let K be a maximal compact subgroup of G a so chosen that its Lie algebra Quotients of Real Algebraic Groups 321 f is orthogonal to that of A with respect to the Killing form on g. (Lie algebras of Lie subgroups of G are identified with the corresponding Lie subalgebras.) Definition2.1. For a relatively compact open subset ~lcP ~ and a map t: A~R + (following BOREL [2]), we call the set St_.= K . At. tl where A!={alaeA , o~(a)<=t(s) for all seA} a Siegel-domain. The following fundamental theorem is due to BOREL [1] (see also [2]). For a subgroup H of G we denote by H e its intersection with G e the set of Q-rational points of G. Then we have Theorem (BOREL). (i) The set of double coset classes Pe\Gt~/F is finite. (ii) For any relatively compact set tl in Pa and t: A--rR + and any pair q, q'EGQ, the set {~]KAttlq~c~KAttlq' 4:0 and ~eF} is finite. (iii) If ql, ..., qm are representatives in Gafor the double coset classes PQ\Ge/F, then there exists a relatively compact open subset ~h cP~ and a map tl: A~R + such that if ~IcP ~ contains th and t: A~R + is such that _t(s)>tt(s) for all seA, ra [3 KAttlqiF=G. i=1 Now it is known that the Lie algebra g of G admits a basis el ..... eN such that a) the structural constants of g with respect to this basis are rational b) each g",~e~, as also 3 the Lie subalgebra corresponding to Z(T) is spanned by those elements of the basis which belong to it c) F is commensurable with the subgroup of G which under the adjoint action fixes the lattice S a generated by et ..... eN in g. In the sequel when we speak of the entries of Ad g (or simply g) we mean the entries of the matrix of Adg referred to the basis et .... , eN. We note then that the denominators of the entries of 7eF when reduced to the minimal form remain bounded. For seA, we denote by #~ the set {/~l/~e § ~= Y. mB(O)O,m#(~)>o}. OeA Then U~ = LI g~ 322 M.S. RAGHUNATHAN" is a Lie subalgebra of G. Its normalizer p, in g is easily seen to be n~380 H g-e. We denote the corresponding Lie subgroup by P=. Then P= is a parabolic subgroup of G defined over Q and is maximal with respect to this pro- perty. With this notation, we have the following crucial Lemma2.1. Let qcP ~ be any relatively compact open subset, t: A-+R + any map and p be any integer. We fix a root c~eA.
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