Bilinear forms for SL(2, q), Àn and similar groups

Publicacions Ma emá iques, Vol 36 (1992), 1001-1010 . A bs ac BILINEAR FORMS FOR SL(2, q), AND SIMILAR GROUPS ALEXANDR,F TURULL * A icle dedica a la memó ia del bon amic Pe e Menal The se o in a ian symme ic bilinea o os on i educible mod- ules o e ields o cha ac e is ic ce o o ce ain g oups is s udied . Resul s a e ob ained unde he p esen e in a ini e g oup o ele- nen s o o de ou whose squa e is cen al . In pa icula , we ind ha he ele an modules o he g oups men ioned in he i le always accep an in a ian symme ic bilinea o m unde which he module admi s an o hono mal basis . In oduc ion Le G be a ini e g oup and X some complex i educible cha ac e o G wi h eal alues . l F is any eal numbe ield con aining Q(X), (he e Q(X) is Q ex ended by all he alues o X), hen he e is a unique (up o iso no phism) FG-module M which a o ds he cha ac e 7nF(X)X, whe e nzF(X) is he Schu inclex o X wi h espec o F . A basic p oblem in ep esen a ion heo y o ini a g oups is o desc ibe hese modules . Since F is a eal ield, he s anda d a e a .ging a gumen shows ha AI will a o d some (posi i e de ini e) symme ic G-in a ian bilinea o m . Wha can be said abou ? Synime ic bilinea o os o M a e classi ied up o iso no phism in GL(AI) by he signa u a o unde each embedding o F in o R, he de e minan o (de ined up o squa es in F*) and he 1 - lasse in a ian o , sea o example Co olla y 3 .3 in p . 168 o [1] . Fo e e y AE .F*, A will also be a non-degene a . e G-in a ian symme ic bilinea o m on M and i s de e minan will be de (A ) = ~ din'F (11!) de ( ) up o squa es in F* . I ollows ha he p oblem will be mo e ac able i dimp(M) is *Pa ially suppo ed by a g an om he NSA . 100 2  A . TURULL e en, o hen, a leas and A will hen ha e he same de e minan . The e a e a ious condi ions ha o ce diMF(M) o be e en . l he Schu index o X wi h espec o F is no one, o example, hen dimF(M) is e en . This case is analyzed in [3J . Ano he example is ha i Ad' is ai h ul and G' l Z(G) is o e en o de hen X(1) mus be e en . The p esen pape analyses si ua ions ha occu equen ly in his case, and in pa icula ou esul s yield he answe o G = SL(2, q), A,, o S  and X a ai h ul cha ac e o G . Be o e s a ing ou esul s ; we desc ibe ou no a ion . NVe deno e by B (F) he B aue g oup o F . I a ; b E F* we deno e by (a, b) he elemen o B (F) which has as a ep esen a i e he qua e nion algeb a -, ~ -2 -2 o dimension 4 o e F gene a ed by i and j sa is ying i  = a ;  j = b and i j = - j i . I is asymme ic non-singula o m on M we deno e by Hasse( ) i s Hasse in a ian . Hasse( ) is an ele nen o B (F) and is calcula ed as ollows . Le el, . . . ; e,, be an o hogonal basis o M and se al = (e¡, ej . Then Hasse( ) = .H . (a2, aj) . %<j He e he p oduc is in he B aue g oup o F and Hasse( ) does no depend on he o hogonal basis chosen . Theo em A . Le G be a ini e g oup and x E G be an elemen o o de 4 such ha x 2 E Z(G) . Le F be a eal ield and X an i educible cha ac e o G wi h alues in F . Le Al be an FG-7nodule a o ding mp(X)X and assume ha x 2 ac s non- i ially on Al . Le be a G- in a ian non-ze o symme ic bilinea o m on M . Then he ollowing hold . 1) de ( ) = 1 up o squa es in F* . 2) Hasse( ) = (A o , -1) o some A,, E F*, wi h A o > 0 i is posi i o de ini e . Fu he mo e, i dimF(M) - 2 (mod 4), hen o e e y AE F* he e is some p, E F* wi h Hasse(p, ) _ (A, -1) . Ou heo em has a consequence abou local Schu indices which we now p oceed o desc ibe . Recall ha , i X is an i educible cha ac e ; he local Schu indices o X a e he posi i o in ege s m,,(X) and m (X) ( o p a a ional p ime), whe e m, > , > (X) = mR(X) and m p (X) = mQ,(X) (Qn being he ield o p-adic numbe s) . The F obenius-Schu indica o gi es a s aigh o wa d (and well known) o mula o n ,,,(X) . Namely, i X has eal alues, m,,,,(X) = 1 i 1  E X(g2) = 1, and m .(X) = 2 DGI -qEG BILINEAR FORMS  1003 and ~ 1  X (g 2 ) = -1, o he wise .  The e is no known simila o - gEG mula o mp(x) . Fu he mo e, knowing m,, (X) p o ides in gene al e y li le in o ma ion abou m,(x) . Fo example, o e e y ini e subse S o {oo, 2, 3, 5, 7, 11, . . .} o e en ca dinali y he e is a a ional alued i - educible cha ac e X o a double co e o some al e na ing g oup such ha mp(X) = 2 o pE S and mp(X) = 1 o p « S, see [2] . Howe e , some u he condi ions o e X and G do imply some u he ela ionship be ween m,,(X) and he mp(X) o p a a ional p ime . I F is a ield con aining he alues o X we deno e by [X] he elemen o B (F) ep e- sen ed by EndFG(M) whe e M is a e i educible FG-module a o ding he cha ac e MF(X)X . Co o111 y 1 . Le G be a ini e g oup and x E G be a e elemen o o de 4 such ha x 2 E Z(G) . Le x be a e i educible chaT ce e o G which does no con ain x 2 in i s ke nel and such ha X( 1 ) - 2 (mod 4) andm,, (X) = 2 . Le F be a eal ield ha con ains Q (X) . Then [X] = (A,, -1)  in  B (F)  o some nega i o  A o E F* . In pa icula , i X is a ional alued hen m , p(X) = 1  o e e y p ime  p - 1(mod 4) . Al hough Theo em A ully desc ibes Hasse( ) whe e dim (M) - 2 (mod 4), i diMF (M) - 0 (mod 4) hen, as we shall see, Hasse(M ) = Hasse( ) o all p E F* . Hence, Hasse( ) will be de e minad by G and X a leas i X( 1 ) - 0 (mod 4) and M is absolu ely i educible . Howe e , he conclusion o Theo em A can no be made mo e p ecise e en in his case, as ou nex esul shows . Theo em B .  Gi en A, E Q*, A o > 0, hen he e exis G, x, F = Q ; X, M and sa is yi7 - ¿g he hypo heses o Theo em A wi h he ollowing p ope ies : a) 777,Q (X) = 1 and X(1) - 0 (mod 4) . b) Hasse( ) = (A,, -1) in B (Q) . c) E e y G-in a ian synáme ic non-singula bilinea o m g o e M sa is ies Hasse(g) = (A,-1) in B (Q) . l in addi ion o assuming ha G has a ce ain elemen o o de 4 we ass une u he ha G con ains a ce ain copy o he qua e nion g oup o o de 8, hen all Hasse in a ian s a e i ial . 100 4  A . TuRULL Theo em C . Le G be a ini e g oup and le Q be a subg oup o G isomo phic o he qua e nion g7-oup o o de 8 and such ha Z(Q) C Z(G) . Le F be a eal ield and X be an i educible cha ac e o G wi h alues in F . Le M be an FG-module a o ding MF(X)X and assume ha Z(Q) ac s non- i ially on M . Le be a G-in a ian non-ze o symme ic bilinea o m on UVI . Then he ollowing hold . 1) de ( ) = 1 up lo squa es in F* . 2) Hasse( ) = 1 in B (F) . This heo em has a consequence ha is analogous o Co olla y 1, namely ha wi h he hypo hesis o Theo em C i m,>,(X) = 2 and X(1) - 2 (mod 4) hen 7np(X) = 1 o all odd p io es p . Howe e , his can also be p o ed easily by no ing ha mp(X)j2 and he i educible ai h ul cha ac e o Q has odd mul iplici y in X1Q . Mo e impo an ly, i should be no ed ha Theo em C applies o ai h- ul X, whene e G- SL(2, q), he special linea g oup o dimension 2 o e he ield o q elemen s, G - A n (n >_ 4), he double co e o an al e na ing g oup, o G- 5,, (n >_ 4) some double co e o a symme ic g oup . In pa icula , we ha e he ollowing . Co olla y 2 . Le G be isomo phic o SL(2, q) o q odd, o some double co e o S n o A n o n >_ 4 . Le M be an FG-i educible ai h ul G-module, whe e F is a eal ield and he cha ac e o M is a mul iple o some i educible cha ac e o G . Then he e is a symme ic G-in a ian bilinea o m on 1V1 unde which M has an o hono mal basis . 1 . P elimina y Lemmas In his sec ion we e iew some esul s ha we need o ou p oo s . Unless o he wise s a ed ou ec o spaces and o ms a e o e a ixed bu a bi a y eal numbe ield F . Recall ha a hype bolic plane is a wo dimensional ec o space wi h a non-singula symme ic bilinea o m ha has a non-ze o ec o whose p oduc wi h i sel is ze o . Lemma 1 .1 . Le , V a ad, W be ec o spaces and le and g be non- singula symme ic bilinea o ms on V and W espec i ely . Then he ollowing hold . a) Hasse( L g) = Hasse( ) Hasse(g) (de ( ), de (g) ), whe e 1 g is he o hogonal suin o and g . b) Fo e e y A  e  F,  Hasse(A )  =  Hasse( )(A, (-1) n 2 1 d"-1) whe e n= di nF(M) and d = de ( ) . P oo . See Theo e n B o [3] . BILINGAR ORNIS  1005 c) I W is a hype bolic plan( hen de (g) = -1 and Hasse(g) = :1 . P oo :: These ac s a e easily e i ied . They can also be ound in [1] ; o example b) appea s as Fxe cise 8 on page 140 . Le nma 1 .2 . Le C be a cyclic g oup o o de 4, and M be an i e- ducible FC-module ai h ul o7 - C . Le be a non-singula C-in aHan sy7nnie ic bilinea o o on Al . Then de ( ) = 1 and Hasse( ) o sume AE F* . P oo . Since F is eal, he cha ac e a o ded by Al will be he sum o he wo ai h ul i educible cha ac e s o C . Any C-in a ian bilinea o m g en Al ® O sa is ies g(eL, el) = g(iei, ¡el) =- g(ei ; el) = 0 = g(e2, e2), whe e e l , e2 E M ® O a e eigen ec o s o a gene a o o C co espond- ing o i and -i espec i ely . I ollows ha he O-space o C-in a ian symme ic bilinea o os on AJO O is one dimensional . This, in u n, implies ha he F-space o C-in a ian symme ic bilinea o os en 111 is one dimensional . Hence, by Le n na 1 .1, b), and he well known ac s ha (u, 1) = 1 a ld (a, 0) (c ', 3= (CYLY~,,~) in 13 (F) o cY, c ', E F * , i is enough o show ha Lenuna 1 .2 holds o so ne . Le W be he one dimensional ai h ul module o he cyclic sub- g oup o o de 2 o C o e F . Then W a o ds a symme ic in a i- an bilinea o m b a ad a basis ec o e such ha b(e, e) = 1 . AY is iso no pliic o he C-module induced om W ; and i ollows ha M a o ds a C-in a ian synune ic bilinea o o and a oasis el, e2 such ha (el, eI) = (e2, e2) = 1 and (el, e2) = 0 .  Ob iously ; o his , de ( ) = 1 and Hasse( ) = 1 . Hence, he Lemma holds . Theo em 1 .3 . Le G be a ini e g oup, F some eal ield and X some i7- educible cha ac e o G wi h alu,es in F such ha m" .(x)=?~ l . . Le M be an FG-module o o- ding he cha ac e mF(X)h and be so ne G-in a ian non-singula symme ic bilinea o na on Al . Then he ol- loiaing hold . 1) de ( ) = 1 (up o squa es in F*) . 2) Hasse( ) = (-1,-1)[Y] i 4 ~ ;Y( 1 ), and Hasse( ) = 1 i 4X(1) . 100 6  A . TURULL Lemma 1 .4 . Le X be an i educible cha ac e o some ini e g oup G and assume ha he alues o X a e in F . Suppose [XI = (A, -1) in B (F) o some A E F* . Then m,,,,(X) = 2 i and only i A is nega i e . Fu he mo e, i F= Q, hen mp(X) = 1 o e e y a ional ini e p ime p such ha p - 1 (mod 4) . P oo : The local Schu indices o X a e 1 o 2 depending on whe he o no he enso o (A, -1) wi h he comple ion o F spli s . Fo example, m < , (x) = 1 i and only i (A ;-1) = 1 in B (IF8) . Howe e , he la e condi ion holds i and only i A is posi i e, so he i s asse ion o he lemma holds . I p is any ini e p ime and F =Q hen 7 np(X) = 1 i and only i (A, -1) = 1 in B (Q,), whe e Q, is he ield o p-adic numbe s . Now (A, -1) = 1 in B (Q p ) i and only i Ax 2 - y 2 = z 2 has a solu ion wi h x, y, z E Q 7 , and z 7~ 0 .  Suppose p is a ini e a ional p ime and p - 1 (mod 4) . Then p is he sum o wo a ional squa es . So, in his case we may assume ha p is no in ol ed in he p ime ac o iza ion o A . Bu hen (A, -1) = 1 in B (Q p ) by, o example, Exe cise 10 in p . 186 o [1] . This comple es he p oo o he lemma . Lemma 1 .5 . Le n > 1 be an in ege , le G= S n be he symme ic g oup o deg ee n . Then he e is an absolu ely i educible QG-module W o dimension n - 1 and a G-in a ian symme ic bilinea o m on W such ha de ( ) =n up lo squa es in Q* . P oo . Le N be he na u al pe mu a ion module o S n o e Q . Then dimQ(N) = n and he .pe mu a ion basis el, . . ., e n o N can be aken o be an o hono mal basis o an S n -in a ian symme ic bilinea o a g on N . Then de (g) = 1 . The ec o = el + ... + e n is S n -in a ian and g( , ) =n . The space 1 o ec o s in N which a e o hogonal o is an S n -submodule o N o dimension n - 1 . We se W = 1 . Tüen N = W 1G > . I ollows ha i we se o be he es ic ion o g o W hen de ( ) = n, since de ( )n = 1 up o squa es in Q* . Since S n ac s doubly ansi i ely on e l , . . ., e n , W is absolu ely i educible, and he lemma holds . Lemma 1 .6 . le G be a ini e g oup and M an FG-module . Suppose M is endowed wi h a non-singula G-in a ian symme ic bilinea oTin . Then we can w i e M=M .®M I ®M 2 as FG-modules whe e he ollowing hold : 1) M o is an o hogonal sum o i educible submodules on which is non-singula . BILINGAR FORMS  1007 2) is o ally iso opic on bo h M l and M 2 . 3) As quad a ic spaces M = M,, 1 (A11 + M2) and A1 1 + M2 is he o lioyonal sum o dime, (M ) = dim (Nl2) copies o he hype bolic plane . P oo :: Suppose he le nma is also, and pick a coun e example wi h dim (M) as small as possible . Le N be an i educible submodule o M . Suppose is non-singula on N1 . Then 111 = N 1N¡ and Nl is a G-submodule on which is non-singula . Hence, by he minimali y o ou coun e example, he le ima holds o NI L, and i ollows ha i also holds o A4l, a con adic ion . The e o e, we mus assu ne ha is singula on N 1 and on e e y o he i educible G-sub odule o M . Since is G-in a ian , i ollows ha is o ally iso opic on N and on e e y o he i educible G-submodule o A1 . In his case ; N C_ N iL . By Maschke's Theo em, hese is al] i educible sub odule N 2 o NI such ha N2 n Ni = 0 . Now is o ally iso opic on bo h N 1 and N2, and (Ni +N2)n(Ni+N2) 1 =(Ni+N2)nNi n N2 =Ni nN2=0, so is non-singula on N + N 2 . The bilinea o a p o ides an iso- mo phis n be ween he dual o N and N2 - N + N2/Ni . Hence ; i we choose e l , . . ., e,, o be a basis o NI, we can hen choose o N 2 he co esponding "dual" basis e*, . . ., e*, Le .  dim1 : (N1) = dimp(N 2 ) and (e ;, e~) = S i s whe e b2j is K onecke 's del a . Hence, as quad a ic spaces , he < e¡, e? > a, e iso no phic o he hype bolic plano and N + N2 =< e1, el >l< e2 : e2 >1 . ... 1< e, e* > . Hence, induc ion applied o < N + N2 >1 comple es he p oo o he lemma . 2 . P oo s o he main esul s P oo o Theo em A : Le C=< x > be he subg oup o G gene - a ed by x . Then C is a cyclic g oup o o de 4 and x 2 ixes no non- ze o ec o in A1 . I ollows ha , as an FC-module, M is he di ec sum o ai h ul i educible FC-modules . Apply Lemma 1.6 o M un- de he ac ion o C . Then, as a quad a ic space, A1 is he o hogonal sum o non-singula quad a ic spaces on i educible C-submodules o M and diMF (AJII) copies o he hype bolic plano, whe e M 1 is some FC- sub odule o M . Since F is eal, dimF(1M 1 ) is o en and i ollows om Le nma 1 .1 ha he sum o all hese hype bolic planes o ms a subspace wi h de e minan 1 and Hasse in a ian ei he 1 o (-1, -1) in B (F) . The o he o hogonal summands o M all na e de e minan 1 and Hasse in a ian (A, -1) o a ious A E F* by Le nma 1 .2 . Hence, by Lemma 100 8  A . TURULL 1 .1, de ( ) = 1 up o squa es in F* and Hasse( ) is a p oduc o elemen s o he o m (A, -1) o a ious A E F* . Since (A, -1) (p, -1) = (AM, ; -1) in B (F), i ollows ha Hasse( ) = (A, -1) o some A o E F* . I is posi i o de ini e, hen Hasse( ) is i ial o e R, so A o > 0 in his case . Hence 1) and 2) o he heo em hold . Suppose now ha dimF(M) - 2(mod 4) . Then by Lemma l .l, b), Hasse(p ) = (A ., -1)(p, -1) = (Aop, -1) . I ollows ha Hasse(p ) _ (A, -1) o e e y A E F* i we se p, = AoA . P oo o Co olla y 1 : Since F D_ Q(X), we can ake M o be an i - educible FG-module a o ding he cha ac e MF(X)X . Since F is eal, he e is a posi i e de ini e G-in a ian symme ic bilinea o o on M . The Hasse in a ian o can be calcula ed in wo ways . On he one hand, Theo em A ells us ha Hasse( ) = (A, -1)  in B (F) o some A o E F* . On he o he hand, Theo em 1 .3 ells us ha Hasse( ) = (-1,-1)[X] . lld=V®W . Sol ing o [X] we ob ain [X] = ( -A, -1) . By Lemma 1 .4, -A, is neg- a i e, Since moo(X) = 2 . Fu he mo e, i F = Q, hen m (X) = 1 o e e y a ional ini e p ime p such ha p - 1 (mod 4), by Lemma 1 .4 . Hence Co olla y 1 holds . P oo o Theo em B : Since o e e y a E Q*, (A, -1) = (a 2 ~Xa, -1), we assume wi hou loss ha a o is a posi i e in ege di isible by 9 . Fu - he mo e, (2, -1) = 1 in B (Q), so we u he assume wi hou loss ha A o is odd . Now A o > 1 and we se n = a o and G= D8 xS, o be he di ec p oduc o he dihed al g oup o o de 8 and he sym ne - ic g oup o deg ee n . We ake x o be any elemen o o de 4 in D8 . Then 1 0 x 2 E Z(G) . Le V be a quad a ic F-space o dimension 2 wi h o hono mal basis el, e2 . We endow V wi h he s uc u e o a D8- module by le ing wo gene a o s o o de 2 o D8 ac on V as linea ans o ma ions which ha e he ollowing ma ices, wi h espec o he basis el ; e2 . I is clea ha D8 s abilizes he quad a ic o m on V . Le W be he S,, module gi en by Lemma 1 .5 . Se BILINEAI2 . FORMS  1009 Then AI is an absolu ely i educible G-module and x 2 does no ac i - ially on AI . Le x be he cha ac e a, o ded by M and be he sym- me ic bilinea o n ob ained by enso ing hose o V and 14 7 . Clea ly ITIQ(X) = 7 . and X(1) = 2(A, - 1) - 0 (mod 4) . As a quad a ic space AI - W -L 1 , 17, so ha Hasse( ) = (de (W), de (W)) = (, o ,, o ) by Lemma 1 .1 and Lemma 1 .5 . Since (, o , A j = -1) in B (Q), his shows a) and b) o Theo em B . Since M is absolu ely i educible all G-in a ian bilinea o ms on AI a e mul iples o . Since de ( ) = 1 ; c) ollows om Le nma 1 .1, b) . P oo o Theo em C : No ice ha as F is eal, Q has only one isomo - phisin class o ai h ul i educible .FQ-modules and hey ha e dimension 4 . Since Z(Q) ixes no non-ze o ec o o M, M is a su i o i educible ai h ul FQ-modules . Applying Le n na 1 . .6, i ollows ha as quad a ic Q-modules M = M, 1 (11 1 + 112) whe e M,, is he o hogonal sum o non-singula i educible FQ-modules and All, +M2 is an FQ-module and as o hogonal space i is he o hogonal su i o a mul iple o 4 copies o a hype bolic plane . I ollows ha de (Al + AI2) = 1 and Hasse(M I + M 2 ) = 1, by .Lemma 1 .1 . Le 0be he ai h ul i educible complex cha ac e o Q . I is well known ha [0] = (-1, -1) in B (F) . I hen ollows om Theo em 1 .3 ha i N is any o he o hogonal summands o AI,,, hen de (N) = 1 and Hasse(N) = (-1, -1)(-1, -1) = 1 . Hence, by Lemma 1 .1, de ( ) = 1 and Hasse( ) = 1, as desi ed . a P oo o Co olla y 2 : Le o be a sym ne ic bilinea o n on M unde which M admi s an o hono mal basis . Since F is a eal ield o is posi i e de ini e, a.nd in ac , i is posi i e de ini e unde each imbedding o F in o R . De ine :AllxA/1-->F ( , w) = 1 ~ sé o(9 ,9w) o , .w E M . Then is a G-in a ian symme ic bilinea o m . Fu he - mo e ; is posi i e de ini e unde e e y imbedding o F in o R . Fu he - mo e ; by Theo ein C, de ( ) = 1 and Hasse( ) = 1 . I ollows ha AI admi s an o hono mal basis unde . This comple es he p oo o he Co olla y .