Normal forms of invariant vector fields under a finite group action

Publicacions Ma emá iques, Vol 37 (1993), 75-82 . NORMAL FORMS OF INVARIANT VECTOR FIELDS UNDER A FINITE GROUP ACTION Abs ac FEDERICO SÁNCHEZ-BRINGAS Le I' be a ini e subg oup o GL(n, (C) . This subg oup ac s on he space o ge ms o holomo phic ec o ields anishing a he o igin in Cn and on he g oup o ge ms o holomo phic di eomo phisms o ((Cn, 0) . We p o e a heo em o in a ian conjugacy o a no - mal o m and linea iza ion o he subspace o in a ian ge ms o holomo phic ec o ields and we gi e a desc ip ion o his ype o no mal o ms in dimension n = 2 . In oduc ion The goal o his pape is o show ha he classic heo ems o Poinca é- Dulac [DU] and Siegel [SI] o conjugacy o a no mal o m and linea iza- ion o ge ms o holomo phic ec o ields a 0 E Cn hold o he quo ien space Cn/I', whe e I' is a ini e subg oup o GL(n, C) . In his si ua ion we conside he ge ms o holomo phic ec o ields and he ge ms o con- juga ing di eomo phism o Cl in a ian by he ac ion o he subg oup . I is well known ha Cn/I' has he s uc u e o an algeb aic a ie y and u he mo e any a ie y which is he quo ien o a ini e g oup o local di eomo phisms o Cn is o his o m (in a speci ic sys em o coo di- na es) [CA], hen we ob ain he e esul s o conjugacies o no mal o ms and linea iza ions o ge ms o holomo phic ec o ields in his kind o algeb aic a ie ies . In a di e en con ex , like bi u ca ion heo y, some imes conjugacy o ano mal o m o ge ms o holomo phic ec o ields which p ese es symme ies a e needed, his esul s can also be applied . In he i s sec ion we p o e he main heo em using he algeb aic app oach de eloped in [CH] . In he second sec ion we analyse ca e ully he case C'/I` and we gi e a desc ip ion o no mal o ms . Finally we wish o hank Xa ie Gomez-Mon o líis help ul commen s and ema ks conce ning his wo k . 7 6  F . SÁNCHEZ-BRINCAS 1 . In a ian conjugacy o a no mal o m and linea iza ion in Cn Le X(C n , 0) be he space o ge ms o holomo phic ec o ields a 0 E Cn anishing a he o igin . Le F be a ini e subg oup o GL(n, C) which ac s na u ally on X(Cn, 0), we say ha X is in a ian i i is in a ian by his ac ion, namely i o all ,y E F, d " y-1(X(1'(z)) = X(z) . Le X(C n /F, 0) be he subspace o in a ian elemen s o X(Cn, 0) . Gi en X E X(Cn,0) deno e by X 1 i s linea pa , dX(0) and suppose i belongs o GL(n, C) . Le S be he semisimple pa o X1, we say X is a no mal o m i LSX = 0, whe e L s is he Lie de i a i e o S . X1 is said o be esonan i i s eigen alues sa is y a ela ion ( esonan e) like his : A =  M i ñ2, u = 1, . . . , n, (m1, . . . , m,,) E Nn,  m i i 2 . 1  1 The ec o ield zi 1 . . . zm ~ aáu , u = 1, . . . , n is he monomial ec o ield associa ed o his esonan e . Suppose ha X E X(C', 0) is a no mal o m and he coo dina es o Cn a e gi en by a basis o eigen ec o s o he semisimple pa S o X1 . Then condi ion LSX = 0 implies ha X - X 1is a sum o esonan monomial ec o ields . Le Di (Cn, 0) be he g oup o ge ms o holomo phic di eo mo phisms which ix he o igin o Cn, F ac s by conjuga ion he e . We say ha 0 E Di (Cn,0) is in a ian i i is in a ian by his ac ion, namely i o all ,y E F, ,y-10 y = 0 . Deno e by Di (Cn/F, 0) he g oup o in a ian elemen s o Di (Cn, 0) . Fo any X, Y E X(C n , 0) we say X is conjuga e o Y i he e is a E Di (Cn, 0) such ha 0* X = Y, whe e 0* X = do- 1 XO . When Y is he linea pa o X we say X is linea izable . I X and Y a e in a ian , he conjugacy (linea iza ion) is called in a ian . Theo em  1 . Le  F  be  a ini e  subg oup  o GL(n, C)  and X E X(C n /I', 0) . Suppose he linea pa X 1 o X is in e ible . Then : 1 .1 . X is in a ian ly conjuga e, possibly o mally o a no mal o m . I X 1 is non- esonan hen his conjugacy is an in a ian linea iza- ion . 1 .2 . I X is holomo phically conjuga e o a no mal o m, hen i can be conjuga e in an in a ian holomo phic way . The p oo o his heo em is a consequence o he ollowing lemma . Le OC .,o be he algeb a o ge ms o holomo phic unc ions a 0 E C" , and NORMAL FORMS OF INVARIANT VECTOR FIELDS  77 m = { E 0c_,o ; (0) = 0} i s maximal ideal . Fo each non-nega i e in ege k deno e ,7c- o = 0cn,o/m k he algeb a o ini e dimension o k-je s o holomo phic unc ions . The elemen X E X(Cn, 0) de ines a de i a ion X* o Oc~,o, X* = LX and in a na u al way he k-je o X de e mines a de i a ion X * o ,7c_ ,o, hen X * has a canonical decomposi ion : Xk = Sk + Nk , whe e Sk is he semisimple pa and Nk is he nilpo en pa . A ema kable ac p o ed in [CH] is ha S* is a de i a ion . In a simila way, we deno e by Di k(Cn, 0) he g oup o k-je s o ge ms o holomo phic di eomo phisms o Cn . The de ini ion o conjugacy o a no mal o m (linea iza ion) is ex ended in a na u al way o he space o k-je s o ge ms o ec o ields, Xk(Cn,o) . Lemma 2 . Le F be a ini e subg oup o GL(n, C), k a non-nega i e in ege and Xk E X k (C n /I', 0) hen 2 .1 .  The semisimple pa S* o X * is in a ian . 2 .2 . S* is in a ian linea izable i X 1 is in e ible . P oo . . 2 .1 . On one handwe ha e he ollowing ac [Hu] : Le V be a C- ec o space o ini e dimension and T an endomo phism o V . Then he semisimple pa o T has a polynomial exp ession in T, p(T) wi h coe icien s in C . On he o he hand, as X is in a ian and y E F is linea we ha e d-y - 'Xy= y -1 X-y = X hen o any non- nega i e in ege k, y - 1 Xky =X* and -y-1X% o . . . oXky = X* o . . . oX* hen any polynomial exp ession in X* wi h coe iicien s in C is in a ian . 2 .2 .  Le Ok E Di k(Cn, 0) be he Poinca é-Dulac di eomo phism which exis s because X 1 is in e ible . Ok is angen ((k - 1)-o de ) o he iden i y di eomo phism, and linea izes he semisimple pa o Xk . De ine he a e age ~k = 1 FI -1 1 :,, E , y -1 Oky . Ok is in a ian and angen ((k - 1)-o de ) o he iden i y di eomo - phism . Besides y-1 0ky I (S*) = ¡FI-1 57 (y -1 0k 1 y)*S*(y -1 0ky) 7EF  yE = IFI -1  ySiy_ 1 -yEF whe e Si is he linea pa o Sk which is in a ian because o Sk . 7 8  F . SÁNCHEZ-BRINGAS P oo o he heo em : 1 .1 . Ls,Xk = 0 i and only i 0 = (Ls,Xk)* = Si X¡* -Xk Si , hen he canonical decomposi ion o X¡ * implies ha conjuga ing o a no mal o m in xk(C , ,0) is equi alen o linea izing he semisimple pa Sk* . Now le 4 be like in lemma 2 . Rema k ha i we w i e 010 . . . 0 w2 = Id +02 + . . . + 01 + . . . hen 01+1 0 01 0 . . . o 02 = (Id+02+ . . . +~~+~ 1 + 1 + .. . )-}-~ +1(Id+ . . . )+ . . . =Id+02+ . . .+01+ .. . so his wo composi ions lla e he same 1-je and Ok o . . . o 02 conjuga es in a ian ly Xk o a no mal o m because as we showed, his di eomo - phism linea izes S¡ . Finally he limi limk -w (& o . . . o w2) de ines a di eomo phism 0, e en ually o mal which conjuga es in a ian ly X o a no mal o m . 1 .2 . Le 0 be he holomo phic conjugacy (in any sys em o coo di- na es) hen a simila a gumen as in 1 implies ha ¡F¡ -1 E~, EI , " y - 'Oy E Di (C', 0) conjuga es X holomo phically and in a ian ly o he espec- i e no mal o m . 2 . Desc ip ion in C 2 : In a ian no mal o ms Le Xl be a linea ec o ield in C 2 wi h eigen alues >11, 1 2 . Choose a base o C 2 , {el, e2} o eigen ec o s o S, he semisimple pa o Xl . We say ha X l belongs o he Poinca é domain i 0 is no in he seg- men [Al, A2] . O he wise we say Xl belongs o he Siegel domain . The eigen alues A1, A2 a e o ype (C, ), C, >_ 0 i : ¡w - mlal - MA21> Qml1 + Im2j) - " o all (ml, m2) E (N')*, u = 1, 2 . Be o e applying heo em 1 in his con ex we poin ou ha condi ions o X be conjuga e holomo phically o a no mal o m lla e been es ab- lished in [DU] i X l is in he Poinca é domainand in [SI] i X l belongs o he Siegel domain . Theo em 3 . Le Xl be he linea pa o X E x(Cn/F, 0) . Le A1, A2 be he eigen alues o X l . 3 .1 . I X l is no esonan , hen X is linea izable in a holomo phic in a ian way in he ollowing cases : i) X l belongs o he Poinca é domain . ii) Xl belongs o he Siegel domain and Al, A2 a e o ype (C, ) o some C, > 0 . NORMAL FORMS OF INVARIANT VECTOR FIELDS  79 3 .2 . I X l is esonan , hen X is conjuga e in a holomo phic in a ian way o a no mal o m in he ollowing cases : i) Xl belongs o he Poánca é domain . ii) X l belongs o he Siegel domain and he no mal o m is col- inea o Xl . Rema k . The e exis cases whe e he in a ian conjugacy o a no mal o m is only o mal . Fo example i F is a diagonal g oup (Le . each o i s elemen s a e diagonal) we a e going o show he e a e no mal oms wi h linea pa in he Siegel domain which do no e i y condi ion 2,ii) . In his case he conjuga ing di eomo phism 0 may be di e gen because one o i s coo dina e unc ions can ha e coe icien s which g ow like he Eule unc ion, [B ] : 00 We exp ess he in a iance condi ion in X(C 2 , 0) wi h an a e age mo - phism o he g oup ac ion . Le II : X(C 2 , 0) --> X(C 2 /F, 0) be he mo - phism o C- ec o ial spaces de ined by II(X) _ ¡I7¡-1 ~ y E y*X . Then X is in a ian i and only i i(X) = X . The e a e wo di e en cases o he amily o ini e subg oups o GL(2, C) : i) I F is diagonal, he monomial ec o ields a e eigen ec o s o II and II(X) = 0 i X is no in a ian . ii) I F is no diagonalizable he eigen ec o s o II a e no monomials and II does no anish monomials . Le us ega d i s he case o diagonalizable g oups . P oposi ion 4 . I F is no diagonalizable and Xl is a linea ec o ield, hen II (X 1) = Xi i and only i Xl = 1 (Z1, z2) . P oo£ Suppose X l is gi en in i s Jo dan canonical o m . I Xl has di e en eigen alues he condi ion X 1 -y = yXl implies y is diagonal, he e o e F mus be diagonal . I X l y = -yX, implies bu F is ini e hen yn = Id and b mus be 0 . 8 0 F . SÁNCHEZ-BRINGAS Rema k . This p oposi ion implies ha o non diagonalizable g oups ou heo em is a linea izing heo em illus a ed by he ollowing example : Le F be he bina y dihed al g oup gene a ed by _ a0l and _ (0  1l 10 -i/  11 Ól The ec o ield X(zl, z2) = (zl, z2) + (zi ; z2) is no a mul iple o he adial ec o ield' which belongs o x(C 2 /F, 0) . Suppose now he g oup is diagónal . In o de o simpli y he desc ip ion o in a ian no mal o ms we will Suppose F is cyclic and gene a ed by e (27 li '-1)/ni 0 e(27 l2 ~---1-)/n2 n  , E N, 1,, E Z, (l i ,, n i ,) = 1, u = 1, 2 . I X(zl, z2) = Eu=1,2(E¡~_o CI'j .ziz2)eu, he equi a iance condi ion is imposed independen ly o monomial ec o ields . The nex p oposi ion desc ibes in a ian ec o ields . P oposi ion 5 . Le n be he leas common mul iple o n1, n2 and lu = nlu /n u . The monomial ec o ield zu(zúz )eu , u = 1, 2, u =,A , i >_ -1, j > 0', i + j >_ 0 is in a ian 'i and only i lui = -?7 j (mod n) . P oo . . Suppose belongs o F hen II(z1 2e u ) = ( 1 / n Z :7E wl,yi72)zizjeu, he e o e zizieu is in a ian i and only i n-1 7EP 7u 1 i7z = 1 .  I u = 1, -^YE ^Yi -1 Y2 - k=1 e2~ ~k((li/ni)(a-1)+(12/n2) .Í)  This sum is n i (n/n1)lli + (n/n2)l2j - (n/n1)l1(modn) and anishes o he wise . Simi- la ly o u = 2 . When X 1 belongs o he Poinca é domain and i s eigen alues a e eso- nan , hen he o igin, A1 and A2 a e colinea , besides 0 1 [A1, a2], he e- o e he e is only one possible ype o esonance : Au = m w, u =,/= . When X 1 belongs o he Siegel domain he esonance Au = muAu + m A, gene a es an in ini e amily o esonan es o ype : Au = k((mu - 1)Au + m w) + ñuñ = k((mu - 1)Au + m ñ ) + w, keN,u7~ . Finally le us make he - ollowing classi ica ion : NORMAL FORMS OF INVARIANT VECTOR FIELDS  8 1 P oposi ion 6 . Le I' be a ini e diagonal cyclic subg oup o GL(2, C) . The in a ian no mal o ms a e : 1 . I X1 belongs o he Poinca é domain . X (z1, z2) = X1( 2 1, z2) + a¡z eu whe e 17 i - 7 u (modn), u qÉ and u, = 1, 2 . 2 . I X1 belongs o he Siegel domain . X(z1, z2) = X1(z1, z2) + Cz1 (É akz1iz2j/ , z2 (E bkz1 i z2 j k=1  k=1 whe e 71i - -772 j (mod n) o k - 0(mod n) and i, j a e like in p oposi ion 5 . Fo all cases i l1/n1 - l2/n2 0 Z, hen X1 is diagonal . This sec ion can be applied o ob ain conjugacies o no mal o ms in su aces o ype C 2 /P whe e I' is a ini e subg oup o SU(2) . These su aces a e embedded in C 3 wi h an isola ed singula i y a he o igin [KLE] . I I' is diagonal, hen i is cyclic and gene a ed by e 27 V I'--- 1 /n  0 0 e -27 - -l/n P oposi ion 6 applies in his case : 971 = -7 72 = 1, l1/n1 - l2/n2 = 2/n1Zi n7~ 2 . When I' is non-diagonalizable, we ha e he g oups which a e he in- e se image o he co e ing su jec ion p : SU(2) ---> SO(3) o he g oups o index 2 (p ese ing o ien a ion) o iangula sphe ical g oups [MIL] . Re e ences [BR] BRJUNO, A . D ., Analy ic o ms o Di e en ial Equa ions, 7yans . MoscowMa h . Soc . 25 (1971), 131-282 . [CA] CARTAN, H ., "Quo ien d'un espace analy ique pa un g oupe d'au omo phismes," Algeb aic Geome y and Topology, P ince on U .P ., 1957 . [CHI CHAPERON, M ., "In a ian mani olds and a p epa a ion lemma o local holomo phic lows and ac ions," Holomo phic dynamics, P oceedings o Sp inge Ve lag, Lec u e No es in Ma h . 1345, 1986 . 8 2  F . SÁNCHEZ-BRINCAS [DU] DULAC, H ., Solu ions d'un sys eme d'equa ions di e en ielle's dans le oisinage des aleu s singulié es, Bull . Soc . Ma h . F ance 40 (1904) . [HU] HUMPHREYS, J ., "In oduc ion o Lie algeb as and ep esen a- ion heo y," Sp inge Ve lag . [KLE] KLEIN, F ., "Lec o es on he Icosahed on and he solu ion o he equa ions o i h deg ee," Teubne 1884, Do e , 1956 . [MIL] MILNOR, J ., "On he 3-dimensional B iesko n mani olds M(p, q, )," Ann . o Ma h . S udies 84, P ince on U .P ., 1975 . [SI] SIEGEL, C . L ., "be die no mal o m analy ische di e en ial- gleinchungen in de Nhe eine gleichgewich slosung," Nch . Akad Wiss ., Go ingen Ma h-Phys . kl, Ma h .-Phis- Chem . Ab , 1952, pp . 21-30 . Ins i u o de Ma emá icas Uni e sidad Nacional Au ónoma de México Ciudad Uni e si a ia México 04510 D .F . MÉXICO P ime a e sió ebuda el 30 de Gene de 1992, da e a e sió ebuda el 13 d'Oc ub e de 1992