Compact subgroups of <span class="mathjax-formula">$GL_n({\mathbb {C}})$</span>

Compact Subgroups of GL

(C).

JEANFRESNEL(*) - MARIUS VAN DERPUT(**)

1. Introduction.

LetGGL_n(C) be a compact subgroup. Using the Haar measure onG one obtains a positive definite Hermitian form on Cⁿ which is invariant under G. In other words, Gis conjugated, with respect to GLn(C), to a subgroup of the standard unitary groupUn(C). In particular, everyg2G is semisimple and all its eigenvalues have absolute value 1.

The inverse problem was posed by K. Millet and I. Kaplansky (see [Ba]):

Suppose that the subgroup GGL_n(C) has the property that every g2G is semisimple and all its eigenvalues have absolute value1. Is G conjugated to a subgroup of U_n(C)?

Fornˆ1;2 the answer is positive. A counterexample forn3 is given in ([Ba], Counterexample 1.10, p. 19). However, using the techniques of Burnside, it is shown in ([Ba], Corollary 1.8, p. 18), thatGis isomorphic to a subgroup ofUn(C). The aim of this paper is to present a proof of the fol- lowing positive result.

THEOREM1.1. Suppose that the subgroup GGL_n(C)satisfies:

(i) Every element of G is semisimple and all its eigenvalues have absolute value1.

(ii) G isclosedwith respect to the ordinary topology ofGL_n(C).

Then G is conjugated inGLn(C)to a subgroup of Un(C)and therefore compact.

The theorem has an almost immediate consequence.

(*) Indirizzo dell'A.: Laboratoire de TheÂorie des nombres et Algorithmique arithmeÂtique, Universite Bordeaux I, 351 cours de la LibeÂration, 33405 Talence, France. E-mail: fresnel@math.u-bordeaux1.fr

(**) Indirizzo dell'A.: Department of Mathematics, University of Groningen, P.O.Box 800, 9700 AV Groningen, The Netherlands. E-mail: mvdput@math.rug.nl

COROLLARY 1.2. Let E be an n-dimensional affine euclidean space andGa closed subgroup of the group of all isometries ofE. Suppose that each element ofGhas a fixed point. Then the groupGis compact and has a fixed point.

PROOF. The action ofg2GonRⁿis given byX2Rⁿ7!gXˆUX‡A with U2O_n(R); A2Rⁿ. One associates to g2G the matrix M(g)ˆ

ˆ U A

0 1

2GLn‡1(R). All eigenvalues of M(g) have absolute value 1.

SinceUis semisimple,M(g) is semisimple if and only if there exists a vector X2Rⁿsuch thatM(g) X

1 ˆ X

1 . This property ofXis equivalent toX is a fixed point forg. It follows thatM(g) is semisimple if and only ifghas a fixed point. The theorem implies thatfM(g)jg2Ggis compact. ThenGis

compact and has a fixed point. p

2. A result on real Lie algebras.

PROPOSITION2.1. V is a complex vector space of dimensionn1. L et gbe arealLie subalgebra ofEndC(V)satisfying:

(a) i1V 62g

(b) If VˆV1V2 with V1;V2 complex vector spaces invariant underg, then V1ˆ0or V2ˆ0.

(c) Every element ofgis semisimple and all its eigenvalues are in iR.

Then the following holds:

(1) gis a real semisimple Lie algebra,G:ˆCRgis a complex semisimple Lie algebra and the canonical mapG!EndC(V)is injective.

(2) Lethbe a Cartan subalgebra ofg. ThenH:ˆChis a Cartan subalgebra for the complex Lie algebra ofG. Let R be the set of roots for the pair(G;H). Then

(2a) a(h)2iRfor every h2handa2R,

(2b) for everya2R, the real Lie subalgebra ofg, generated by g\(GaG a)is isomorphic tosu2.

(3) There exists a positive definite Hermitian form F such that for all x;y2V and g2gone has F(gx;y)‡F(x;gy)ˆ0.

PROOF. (1). Suppose thatgis not semisimple. Thenghas a non zero solvable ideal. Leta6ˆ0 be a minimal solvable ideal, then [a;a]ˆ0. Since the elements of aare semisimple and commute there is a decomposition V :ˆCⁿˆV₁ V_rand there are distinctR-linear mapsl_j :a!iR such that the action ofaonVis given by

a X^r

jˆ1

v_j ˆX

l_j(a)v_j fora2aandv_j 2V_j for allj:

Choose an elementa2asuch that, say,l₁(a)ˆiand thel_j(a) are distinct.

Forg2gone writesb:ˆ[g;a]ˆga ag2a. Consider for a givenu2Vj

the expression g(u)ˆP

k v_k with all v_k2V_k. Now l_j(b)uˆb(u)ˆ (ga ag)(u)ˆl_j(a)P

k v_k P

k l_k(a)v_k. This implies v_kˆ0 for k6ˆj and l_j(b)uˆ0. Thus the spacesV_jare invariant underg. Condition (b) implies rˆ1. Thenaˆi1_V, which contradicts condition (a). One concludes thatg is semisimple.

According to [F-H],G:ˆCRgis semisimple, too. An element ofG can uniquely be written as 1a‡ibwitha;b2g. If the image of this element is 0 in End_C(V), thenaˆ ib. This impliesaˆbˆ0 sinceaandb have their eigenvalues iniRand are semisimple.

(2). The first statement of (2) is immediate. We recall (see [F-H]) that the Cartan decomposition (or root decomposition) GˆH(aGa) has the following properties: For any non zero linear mapa:H!Cone has

G_a:ˆ fg2Gj[h;g]ˆa(h)gfor allh2Hg :

IfGa6ˆ0 thenais called a root and in that casedimCGaˆ1. Ifais a root, thencawithc2Cis a root if and only ifcˆ 1.

Fix an element h2End(V). The eigenvalues of the linear map End(V)!End(V), defined byg7!ad(h)(g):ˆ[h;g], are the differences of the eigenvalues ofh. In particular forh2handa2Rone hasa(h)2iR.

This proves (2a).

One writes a1; a1;. . .;ar; ar for the roots. Any g2g has a unique decompositiongˆg₀‡P^r

jˆ1(g_aj‡g _aj) withg₀2H; g_aj 2G_aj.

Choose a generic elementh₀2h, i.e., the 2relementsa_j(h₀)2iR are distinct. Form1 one has

ad(h0)^m(g)ˆX

j

aj(h0)^mgaj‡( aj(h0))^mg aj 2g:

Using this relation for mˆ2n; nˆ1;. . .;r and observing that the aj(h0)²2R; jˆ1;. . .;rare distinct, one finds that allgaj ‡g aj are ing.

Then alsog₀2g. Similarly, one finds that eachigaj ig aj 2g.

Now we study the real vector spaceT_j:ˆg\(G_aj‡G _aj). As shown above, any element of G_aj is nilpotent. Since the elements of g are semisimple one has g\Gaj ˆ0. In particular the two projections Tj!Gaj are injective. We conclude from this thatTj has a real basis of the form Xaj‡X aj;iXaj iX aj, where Xaj are non zero elements of G_aj.

The complex Lie algebra generated by X_aj is easily seen to be the complex Lie algebra sl_2;C. One easily verifies that the real Lie algebra generated byXaj‡X aj;iXaj iX ajis isomorphic tosu2. This proves (2b).

(3). One applies [F-H], Proposition 26.4. The condition (i) of that pro- position is (2a) and (2b). The equivalent condition (iii) states that the real Lie algebra associated to g is compact. This implies the existence of a positive definite Hermitian formFonV such thatF(gx;y)‡F(x;gy)ˆ0

holds for allx;y2V andg2g. p

3. Proof of the theorem.

The caseGconnected.

Putg:ˆ fA2Matr_n(C)jexp (tA)2Gfor allt2Rg. According to ([M-

T], Proposition 3.4.2 and 3.4.2.1.),gis a real Lie subalgebra of Matr_n(C) and moreoverGis generated byfexp (g)jg2gg. The elementsg2gare clearly semisimple and all their eigenvalues are iniR.

LetV:ˆCⁿˆV1 Vrdenote a maximal decomposition into (non trivial) complex subspaces invariant under g. This decomposition is also invariant under the action ofG. It suffices to prove the theorem for the restriction ofGto eachV_j. In other words we may suppose thatrˆ1. Thus gsatisfies the conditions (b) and (c) of Proposition 2.1.

Ifi1_V 2g, then one replacesgbyg:ˆ fg2g jTr(g)ˆ0g. The latter is again a real Lie algebra, satisfies (a)-(c) and moreovergˆgRi1_V. The positive definite Hermitian form of part (3) of Propostion 2.1 has clearly the propertyF(gx;gy)ˆF(x;y) for allg2Gandx;y2V.

The general case.

NowGis a closed subgroup of GLn(C) (for the ordinary topology) such that every element ofGis semisimple and such that all its eigenvalues have

absolute value 1. Let G^o denote the component of the identity of G. Ac- cording to the previous case, the groupG^ois compact.

LEMMA 3.1. G=G^o is a torsion group, i.e., all its elements have finite order.

PROOF. Letgbe an element of G. Choose a basise1;. . .;en of eigen- vectors ofg. The groupT, consisting of all elementst2GLn(C) such that te_jˆc_je_j; jc_jj ˆ1 for allj, is compact. The topological closureHGL_n(C) of the group generated bygis a closed subgroup ofTand therefore com- pact. The component of the identityH^oofHhas finite index inH, sinceHis compact. Moreover,H^oG^o. It follows that the image ofg inG=G^o has

finite order. p

The groupG^ois conjugated to a subgroup ofU_n(C) and hence compact.

One considers the real vector space Herm consisting of the Hermitian forms F on V. The group G acts linearly on Herm b y (gF)(x;y):ˆ

:ˆF(gx;gy). The real linear subspace Herm_G^o consisting of the G^o-in- variant Hermitian forms is not 0 and contains in fact a positive definite Hermitian form. The space Herm_G^o is invariant under G, since G^o is a normal subgroup of G. The action of G on Herm_G^o induces a homo- morphismG!GL(HermG^o) with kernelG^‡ and imageI. SinceG^‡G^o the groupIis a torsion group.G^‡leaves a positive definite Hermitian form invariant and is closed. ThereforeG^‡is compact.

We will need the following classical result and refer to ([Fr], p. 209, or [C-R] p. 252, or [S]) for a proof.

LEMMA3.2 (Schur's theorem). Let H be a torsion subgroup ofGLn(F), for some field F. Then:

Any finitely generated subgroup J of H is finite. As a consequence, H is the filtered union of its finite subgroups.

We apply the lemma toI. LetJIbe a finite subgroup. Its preimage JGis compact and the subspaceHerm_J of the the J-invariant ele- ments of Hermis not 0 and contains a positive definite Hermitian form.

For finite subgroups J₁J₂ of I one has Herm_J1Herm_J2. Since the spacesHerm_Jhave finite dimension andIis the filtered union of its finite subgroups, there exists a finite subgroup J0 of I such that HermJ₀ ˆ

ˆHermKfor every finite subgroupK I, containingJ0. This implies the existence of a positive definite Hermitian form invariant underG.

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Manoscritto pervenuto in redazione il 4 luglio 2005.