UNIQUENESS OF REPRESENTATION THEORETIC HYPERBOLIC KAC MOODY GROUPS OVER Z. 1. Introduction

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1 UNIQUENESS OF REPRESENTATION THEORETIC HYPERBOLIC KAC MOODY GROUPS OVER Z LISA CARBONE AND FRANK WAGNER Abstract For a smply laced and hyperbolc Kac Moody group G = G(R) over a commutatve rng R wth 1, we consder a map from a fnte presentaton of G(R) obtaned by Allcock and Carbone to a representaton theoretc constructon G λ (R) correspondng to an ntegrable representaton V λ wth domnant ntegral weght λ When R = Z, we prove that ths map extends to a group homomorphsm ρ λ,z : G(Z) G λ (Z) We prove that the kernel K λ of ρ λ,z les n H(C) and f the natural group homomorphsm ϕ : G(Z) G(C) s nectve, then K λ H(Z) = (Z/2Z) rank(g) 1 Introducton Here we consder and compare two dstnct constructons of a smply laced and hyperbolc Kac Moody group G = G(R) over a commutatve rng R wth 1 One of these groups s Tts s constructon of G(R), though we work only wth a fnte presentaton of G(R) obtaned by Allcock and Carbone ([AC]) The other s a Kac Moody group G λ (R) constructed n [Ca] (followng [CG]) usng an ntegrable representaton V λ of the underlyng Kac Moody algebra wth domnant ntegral weght λ We consder a natural map between generators of G(R) and G λ (R) When R = Z, we prove that ths map extends to a group homomorphsm ρ λ,z : G(Z) G λ (Z) We prove that the kernel K λ of the map ρ λ,z : G(Z) G λ (Z) les n H(C) and f the natural group homomorphsm ϕ : G(Z) G(C) s nectve, then K λ H(Z) = (Z/2Z) rank(g) Inectvty of the natural map ϕ : G(Z) G(C) s not currently known and depends on functoral propertes of Tts group ([T]) Our noton of a Kac Moody group s the nfnte dmensonal analog of an elementary Chevalley group over the rng R, namely, a group generated by real root group generators The noton of Chevalley group over R, that s, the nfnte dmensonal analog of the Chevalley Demazure group scheme, has not yet been fully formulated, though the groundwork for establshng t was gven n [T] and dscussed further n [A] and [C] The methods for constructng Kac Moody groups over felds are numerous, though we wll only refer to constructons of [CG] (followng [G]) and [RR]) When R s a feld, Garland constructed affne Kac Moody groups as central extensons of loop groups and characterzed the dependence of a completon of G λ (R) on λ n terms of the Stenberg cocycle ([G]) Over felds, the group constructed n [RR] concdes wth the Kac Moody group G(R) of Tts ([T]) Let K λ (R) = Ker(ρ λ : G(R) G λ (R)) There are a number of known results whch determne the natural extenson of K λ to completons of G(R) and G λ (R) when R s a feld For symmetrzable Kac Moody groups over felds, [BR] showed that K λ s contaned n the center of a completon of G(R) Recent work of Rousseau ([Rou]) shows that over felds of characterstc zero, completons G(R) (as n [RR]) and G λ (R) (as n [CG]) are somorphc as Date: Aprl 2,

2 topologcal groups (see also [Mar]) The complete Kac Moody group of [CG] s constructed wth respect to a choce of domnant ntegral weght λ Thus Rousseau s work also shows that over felds of characterstc zero, the complete Kac Moody groups of [CG] are ndependent of the choce of λ up to somorphsm, as topologcal groups Here our nterest les n mnmal or ncomplete Kac Moody groups over rngs, where the methods for determnng K λ are less transparent Thus we work only over R = Z and study the kernel of ρ λ,z when G s smply laced and hyperbolc We hope to eventually have the technques to extend these results to a wder class of Kac Moody groups over more general commutatve rngs The authors wsh to thank Danel Allcock for hs comments whch helped to clarfy the results n the last secton 2 Tts Kac Moody group Let g be a Kac Moody algebra wth Cartan subalgebra h and root space decomposton: g = g + h g The roots h are the egenvalues of the smultaneous adont acton of h on g, and g ± = α ±g α where the root spaces g α = {x g [h, x] = α(h)x, h h} are the correspondng egenspaces When g s nfnte dmensonal, = In ths case, g has 2 types of roots: real roots wth postve norm and magnary roots wth negatve or zero norm The magnary roots wll not play a role n what follows, as we wll work wth Tts presentaton for Kac Moody groups, whch uses only the real root groups as generators Tts showed how Kac Moody groups can be presented by generators and relatons, generalzng the Stenberg presentaton for fnte dmensonal Le groups ([T]) In the fnte dmensonal case, there s a Chevalley type commutaton relaton of the form [χ α (u), χ β (v)] = m,n χ mα+nβ (C mnαβ u m v n ) between every par of root groups U α, U β Here u, v R, C mnαβ are ntegers (structure constants) and the χ α are vewed as formal symbols n U α = {χ α (u) α, u R} = (R, +) However, n the nfnte dmensonal case, Tts presentaton of Kac Moody groups has nfntely many Chevalley commutaton relatons To descrbe these, we gve the followng defnton Let (α, β) be a par of real roots and let W be the Weyl group Then (α, β) s called a prenlpotent par, f there exst w, w W such that wα, wβ re + and w α, w β re A par of roots {α, β} s prenlpotent f and only f α β and (Z >0 α + Z >0 β) re + s a fnte set For every prenlpotent par of roots {α, β}, Tts defned the Chevalley commutaton relaton [χ α (u), χ β (v)] = χ mα+nβ (C mnαβ u m v n ) mα+nβ (Z >0 α+z >0 β) re + where u, v R and C mnαβ are ntegers (structure constants) 2

3 3 Tts presentaton Tts defned the Stenberg group St(R) = α re U α /Chev comm relns on prenlpotent pars Tts Kac Moody group G(R) s a quotent of the Stenberg group St(R) by some addtonal relatons whch are easy to descrbe Tts Kac Moody group has an nfnte set of generators and an nfnte set of defnng relatons The sets of generators and relatons are redundant and can be reduced sgnfcantly, as shown n [A] and [AC] In [A], Allcock defned a new functor, the pre Stenberg group PSt(R) and showed that n many cases PSt(R) = St(R) The presentaton of PSt(R) s defned n terms of the Dynkn dagram rather than the full nfnte root system 4 Smply laced hyperbolc type Carbone and Allcock obtaned a further smplfcaton of the somorphsm PSt(R) = St(R) for root systems that are smply laced and hyperbolc ([AC]) A Dynkn dagram s smply laced f t conssts only of sngle bonds between nodes It s hyperbolc f t s nether of affne nor or fnte dmensonal type, but every proper connected subdadram s ether of affne or fnte dmensonal type rank 4 rank 5 rank 6 rank 7 rank 8 rank 9 rank 10 Table 1 The smply-laced hyperbolc Dynkn dagrams number of nodes The rank equals the 3

4 5 Fntely many defnng relatons parametrzed over R The Kac Moody group G(R) s generated by elements X (t), t R, S, {1,, l} elements h (a) n the frst column of Table 1 are The h (a) := s (a) s ( 1), s (a) := X (a)s X (a 1 )S 1 X (a) Any, all a, b R Each, t, u R not adacent adacent t, u R t, u R h(a) h (b) = h (ab) X (t)x (u) = X (t + u) S S = S S S S S = S S S S 2 SS 2 = S 1 [S 2, X(t)] = 1 [S, X(t)] = 1 X(t)SS = SSX(t) S 2 X(t)S 2 = X (t) 1 S = X (1)S X (1)S 1 X (1) [X (t), X (u)] = 1 [X (t), X (u)] = S X (tu)s 1 [X (t), S X (u)s 1 ] = 1 Table 2 The defnng relatons for G(R), G smply laced and hyperbolc, R a commutatve rng wth 1 Over R = Z, the generators X (u) are obtaned from X = X (1) va X (u) = X u Thus we can rewrte the presentaton wthout the scalars from the underlyng rng and obtan a fnte presentaton for G(Z) as n Table 2 Any Each = {1,, l} not adacent adacent (1) S 4 = 1 (2) [S2, X ] = 1 (4) S S = S S (7) S S S = S S S (8) S 2S S 2 = S 1 (3) S = X S X S 1 X (5) [S, X ] = 1 (9) X S S = S S X (10) S 2 X S 2 = X 1 (6) [X, X ] = 1 (11) [X, X ] = S X S 1 (12) [X, S X S 1 ] = 1 Table 3 The defnng relatons for G(Z), G smply laced and hyperbolc 6 Representaton theoretc Kac Moody groups over rngs Here we descrbe the constructon of a representaton theoretc Kac Moody group G λ (R), over any commutatve rng R wth 1, constructed usng an ntegrable hghest weght module V λ for any symmetrzable Kac Moody algebra and a Z form U Z of the unversal envelopng algebra U R Ths constructon was developed n [Ca] followng the methods of [CG] and s a natural generalzaton of the theory of elementary Chevalley groups over commutatve rngs (see for example [VP]) Let g = g(a) be Kac Moody algebra correspondng to a symmetrzable generalzed Cartan matrx A = (a ), I Let V λ be the unque rreducble hghest weght module for g correspondng to domnant ntegral weght λ 4

5 We let Λ h be the lnear span of the smple roots α, for I, and let Λ h be the lnear span of the smple coroots α, for I Let e and f be the Chevalley generators of g Let U C be the unversal envelopng algebra of g Let U Z U C be the Z subalgebra generated by em m!, f m ( ) h m! for I and, for h Λ m and m 0, U + Z be the Z-subalgebra generated by em m! for I and m 0, U Z be the Z subalgebra generated by f m m! U 0 Z U C(h) be the Z subalgebra generated by for I and m 0, ( h m), for h Λ and m 0 We set g Z = g C U Z, g ± Z = g± C U Z, h Z = h C U Z ( h Then g C = g Z Z C and h Z s generated by, for h Λ m) and m 0 For R a commutatve rng wth 1, set g R = g Z Z R Then g R s the nfnte dmensonal analog of the Chevalley algebra over R Set h R = h Z R and for α, let g α R = gα Z R Then g R = n R h R n + R s the root space decomposton of g R relatve to h R, where n R = n α R, n + R = n α R α α + We now consder the orbt of our hghest weght vector v λ V under U Z We have snce all elements of U + Z U + Z v λ = Zv λ except for 1 annhlate v λ Also U 0 Z v λ = Zv λ snce U 0 Z acts as scalar multplcaton on v λ by a Z valued scalar Thus U Z v λ = U Z (Zv λ) = U Z (v λ) Defne U(g) R = U(g) Z R Ths s the nfnte dmensonal analog of a hyperalgebra as n [C] (see also [Hu]) We set V λ Z = U Z v λ = U Z (v λ) Then VZ λ s a lattce n V R λ = R Z VZ λ and a U Z module For each weght µ of V, let Vµ λ be the correspondng weght space Set VR λ = R Z VZ λ and Vµ,R λ = R Z Vµ,Z λ so that = µ Vµ,R λ V λ R 5

6 Here, V R = V Z Z R free R module wth bass v λ 1 and a module over g R called the Weyl module For s, t R and I, set χ α (t) = exp(ρ(se )), χ α (t) = exp(ρ(tf )), where ρ s the defnng representaton for V, and we set w α = χ α (t)χ α ( t 1 )χ α (t), h α (t) = w α (t) w α (1) 1 Then these are elements of Aut(V λ R), thanks to the local nlpotence of e, f The group W s known as the extended Weyl group We wll use a presentaton of W gven n [KP] Let H λ G λ be the subgroup generated by the elements h α (t), t R, I We let G λ (R) Aut(VR λ ) be the group: G λ (R) = χ α (s), χ α (t) I, s, t R We may refer to G λ (R) as a representaton theoretc Kac Moody group We summarze the constructon n the followng Theorem 61 Let g be a symmetrzable Kac Moody algebra over a commutatve rng R wth 1 Let α, I, be the smple roots and e, f the Chevalley generators of g Let VR λ be an R form of an ntegrable hghest weght module V λ for g, correspondng to domnant ntegral weght λ and defnng representaton ρ : g End(VR λ ) Then G λ (R) = χ α (s) = exp(ρ(se )), χ α (t) = exp(ρ(tf )) s, t R Aut(V λ R ) s a representaton theoretc Kac Moody group assocated to g A smlar constructon for G λ over arbtrary felds was used n [CG] to construct Kac Moody groups over fnte felds When R = C (or R = Q), we defne the ntegral subgroup G λ (Z) to be the group G λ (Z) = χ α (s), χ α (t) s, t Z, I It s non trval to prove that ths group concdes wth the Chevalley group over Z, namely the subgroup of G λ (C) preservng V λ (Z) Ths s proven n [Ca] The followng was also proven n [Ca] Theorem 62 ([Ca]) As a subgroup of G λ (C) (or G λ (Q)) the group G λ (Z) has the followng generatng sets: (1) χ α (1) and χ α (1), and (2) χ α (1) and w α (1) = χ α (1)χ α ( 1)χ α (1) We now take λ to be a regular weght, and we defne the weght topology on G λ (R) by takng stablzers of elements of VR λ as a sub base of neghborhoods of the dentty The completon of G λ (R) n ths topology wll be referred to as the Carbone Garland completon and denoted by Ĝ λ (A) Snce G λ (R) s a homomorphc mage of G(R), we can thnk of Ĝλ (R) as a completon of G(R) rather than G λ (R) 6

7 7 Unqueness of representaton theoretc Kac Moody groups over Z For G smply laced and hyperbolc, we consder a map from the fnte presentaton of G(R) of [AC] to the representaton theoretc group G λ (R), over a commutatve rng R wth 1 We defne the map ρ λ : G(R) ρ λ G λ (R) from generators of G(R) to generators of G λ (R): for t R, u R, {1,, l} X (t) χ α (t) S w α h (u) h α (u) Theorem 71 When R = Z, ths map extends to a group homomorphsm ρ λ,z : G(Z) G λ (Z) Remark The exstence of an abstract group homomorphsm from Tts group G(Q) G λ (Q) and ts restrcton G(Z) G λ (Z) can be deduced from [T] and [CER] However, we are nterested only n ths homomorphsm from the fnte presentaton of G(Z) n [AC] and n determnng the generators of ts kernel Thus we prove that the relatons n the fnte presentaton of G(Z) are satsfed n G λ (Z) Proof The presentaton n [AC] for G(Z) does not make sgns of structure constants explct, but rather holds formally for any choce of sgn of the structure constants The known relatons n G λ (Z) also hold up to a sgn n the structure constants Hence we verfy that ρ λ,z s a homomorphsm up to a sgn n the structure constants From now on, we wll wrte ρ λ to denote ρ λ,z f the context allows We show that the relatons n G(Z) are satsfed by ther mages n G λ (Z) under ρ λ We refer to the labellng of the relatons n Table 2 (1) Consder the mage of S 4 = 1 under ρ λ For any we have: Snce the w α ρ λ (S 4 ) = w 4 α generate the extended Weyl group W and w 4 α = 1 n W, t follows that ρ λ (S 4 ) = w 4 α = 1 Thus, the mage of the relaton S 4 = 1 s satsfed n Gλ (Z) (2) In G λ (Z), we have w α χ α = χ wα (α ) for any par, For each we have w α (α ) = α Hence Consder now the mage of [S 2, X ] n G λ (Z): ρ λ ([S 2, X ]) = ρ λ (S S X S 1 w α χ α w 1 α = χ α S 1 X 1 7 ) = w α w α χ α w 1 α w 1 α χ 1 α

8 Ths reduces to ρ λ ([S 2, X ]) = w α ( w α χ α ) χ 1 α = w α χ α χ 1 α = χ α χ 1 α = 1 Thus the mage of the relaton [S 2, X ] = 1 s satsfed n G λ (Z) (3) Smlarly, consder the mage of X S X S 1 X Applyng ρ λ, we have Ths last term equals w α, so that ρ λ (X S X S 1 X ) = χ α w α χ α χ α = χ α χ α χ α ρ λ (X S X S 1 X ) = w α = ρ λ (S ) Thus, the mage of the relaton S = X S X S 1 X s satsfed n G λ (Z) (4) Suppose now that are not adacent on the Dynkn dagram Usng the presentaton of W n [KP], we have w α w α = 1 Hence, w α w α = w α w α, so that ρ λ (S S ) = ρ λ (S S ) Thus the mage of the relaton S S = S S for,, not adacent on the Dynkn dagram, s satsfed n G λ (Z) (5) Consder the mage of [S, X ] under ρ λ : ρ λ ([S, X ]) = ρ λ (S X S 1 In ths last term, we evaluate w α (α ): X 1 ) = w α χ α χ 1 α = χ wα (α )χ 1 α w α (α ) = α α (α )α = α (a )α However, for,, not adacent on the Dynkn dagram, we have a = 0, so that If are not adacent, then w α (α ) = α (a )α = α ρ λ ([S, X ]) = χ α χ 1 α = 1 Thus, the mage of [S, X ] = 1 for,, not adacent on the Dynkn dagram s satsfed n G λ (Z) (6) For, and, not adacent, we may compute n the rank 2 root subsystem of type A 1 A 1 We have [χ α, χ α ] = 1, so that ρ λ ([X, X ]) = 1 Thus the mage of the relaton [X, X ] = 1 for not adacent, s satsfed n G λ (Z) (7) We now suppose that are adacent on the Dynkn dagram Usng the presentaton of W n [KP], we have Hence, w α w α w α = w α w α w α, so that w α w α w α w 1 α = 1 ρ λ (S S S ) = w α w α w α = w α w α w α = ρ λ (S S S ) Thus the mage of the relaton S S S = S S S for,, adacent s satsfed n G λ (Z) (8) We have the relaton w α w 2 α w 1 α = w 2 α ( w α ) 2a n G λ (Z) (see [CER]) If are adacent, then a = 1 Hence w α w 2 α w 1 α = w 2 α w 2 α 8

9 Multplyng by w α on both sdes yelds w α w α 2 = w α 2 w α 3 Snce w α 4 = 1 for all n G λ (Z), we can rewrte the rght hand sde as w α 2 We have w α 2 = w α 2 for all, so w α w α 2 = w α 2 Fnally, multplyng by w α 2 on both sdes gves Thus, the mage of the relaton S 2 S S 2 (9) Now consder the words w α χ α w 1 α ρ λ (S 2 S S 2 ) = w α 2 w α w α 2 = = ρ λ (S 1 ) = S 1 and w 1 α χ α w α for, adacent s satsfed n G λ (Z) w α χ α w 1 α = χ wα (α ) n G λ (Z) As before, we have for any, Usng the relaton w 4 α = 1 for all, we may rewrte the second word above as w 1 α χ α w α = w 3 α χ α w 3 α For, adacent on the Dynkn dagram, a = 1, so Thus Smlarly, We may smplfy the frst word: and the second word: Ths yelds We have w α (α ) = α α (α )α = α (a )α = α + α w α χ α w 1 α = χ wα (α ) = χ α +α w α (α ) = α α (α )α = α (a )α = α + α = α + α w α χ α w 1 α = χ α +α χ α w α = w α 3 χ α w α 3 = w α 2 ( w α χ α w α ) w α 2 = w α 2 χ α +α w α 2 χ α w α = w α 2 χ α +α w α 2 = w α ( w α χ α +α ) = w α χ wα (α +α ) w α (α + α ) = w α (α ) + w α (α ) Snce w α (α ) = α + α and w α (α ) = a, we have Hence whch yelds Hence w α (α + α ) = a χ α w α = w α χ wα (α +α ) = w α χ α w 1 α χ α w α = w α χ α w 1 α = χ wα (α ) = χ α +α w α χ α w 1 α = χ α +α = w 1 α χ α w α Multplyng by w α and then w α on both sdes gves the equalty w α w α χ α = χ α w α w α, whch s the mage of X S S = S S X under ρ λ Thus, the mage of the relaton X S S = S S X s satsfed n G λ (Z) (10) To determne the mage of S 2 X S 2 under ρ λ, note that S 2 = h ( 1), so that ρ λ (S 2 X S 2 ) = ρ λ ( h ( 1)X h ( 1) 1 ) = h ( 1)χ α (1)h ( 1) 1 9

10 We have the relaton h (u)χ α (v)h (u) 1 = χ α (vu a ) n G λ (Z) for any, ([CER]) Hence, for, adacent, we see that ρ λ (S 2 X S 2 ) = h ( 1)χ α (1)h ( 1) 1 = χ α ( 1) = χ 1 α = ρ λ (X ) 1 Thus, the mage of the relaton S 2 X S 2 = X 1 for, adacent s satsfed n G λ (Z) (11) For any prenlpotent par of real roots α and β, by [T] and [CG] there s a Chevalley relaton n G λ of the form [χ α (u), χ β (v)] = mα + nβ re + m, n > 0 χ mα+nβ (C mnαβ u m v n ) between root groups U α and U β Here u, v R, C mnαβ are ntegers (structure constants) We clam that adacent smple roots α and α form a prenlpotent par By Lemma 6 of [AC], n the smply laced hyperbolc case, a par of real roots s prenlpotent f (α, β) 1 But for adacent smple roots α and α, (α, α ) = 1 Thus the par α, α s prenlpotent and we have C mnαβ {±1} We have (mα + nα mα + nα ) = m 2 (α α ) + 2mn(α α ) + n 2 (α α ) = 2m 2 2mn + 2n 2 But ( ) = 2 for all elements of re To determne when mα + nα re for m, n > 0, we seek non negatve solutons of 2m 2 2mn + 2n 2 = 2(m 2 mn + n 2 ) = 2 or m 2 mn + n 2 = 1 It s easy to see that the only soluton s m = n = 1 Thus the rght hand sde of the Chevalley commutaton relaton s χ α +α = χ wα (α ) = w α χ α w 1 α and the mage of the relaton [X, X ] = S X S 1 (12) Fnally, consder ρ λ ([X, S X S 1 ]) for, adacent We have ρ λ ([X, S X S 1 for, adacent s satsfed n G λ (Z) ]) = [ρ λ (X ), ρ λ (S X S 1 )] = [χ α, w α χ α ] We have w α χ α = χ wα (α ) As and are adacent, a = 1, thus t follows that w α (α ) = α + α, so that ρ λ ([X, S X S 1 ] = [χ α, w α χ α ] = [χ α, χ α +α ] We note that [χ α, χ α +α ] = 1 when 2α + α s not a root We have (2α + α 2α + α ) = 4(α α ) + 2(α α ) + 2(α α ) + (α α ) = = 6 As ( ) 0 for all elements of m and ( ) = 2 for all elements of re, t follows that 2α + α m and 2α + α re, thus 2α + α s not a root Thus ρ λ ([X, S X S 1 ]) = [χ α, χ α +α ] = 1 and the mage of the relaton [X, S X S 1 ] = 1 for, not adacent, s satsfed n G λ (Z) 10

11 8 The kernel of ρ λ When R = K s a feld, Rémy and Ronan took a completon Ĝ(K) of Tts Kac Moody group G(K) n the automorphsm group of the buldng of the BN par for G ([RR] and [Mar], Sec 61) The followng relatonshp between Ĝλ (K) and Ĝ(K) was establshed by [BR] and [Mar], (see also [CER]) Theorem 81 For any regular weght λ, there exsts a (canoncal) contnuous surectve homomorphsm ρ λ : Ĝλ (K) Ĝ(K) The kernel of ρ λ s equal to g B λ g 1 and s contaned n Center(Ĝλ (K)) Ths result was extended further by [Rou] (see also Prop 646 of [Mar]) g Ĝ Theorem 82 ([Rou]) If Char(K) = 0 then ε λ : Ĝλ (K) Ĝ(K) s an somorphsm of topologcal groups From now on, we wll take R = C Lemma 83 The kernel of the map ρ λ : G(C) G λ (C) les n Center(G(C)) Proof Consder the sequence of homomorphsms G(C) ρ λ G λ (C) ι Ĝλ (C) ε λ Ĝ(C) where ι s the ncluson map The composton of these three maps s the natural map from G to Ĝ By Theorem 81, Ker(ε λ ) Center(Ĝλ (C)) By ([RR], 1B), Ker(ε λ ι ρ λ ) = Center(G(C)) H(C) where H(C) = h (a) := s (a) s ( 1) a C, h (a) h (b) = h (ab) for s (a) := X (a)s X (a 1 )S 1 X (a) But Ker(ρ λ ) Ker(ε λ ι ρ λ ) The result follows Snce h α (1) = 1, we defne H(Z) = h α ( 1) = s ( 1) 2 h α ( 1) 2 = 1, {1,, l} Then H(Z) = (Z/2Z) rank(g) and the natural map H(Z) H(C) s nectve 11

12 Inectvty of the natural map G λ (Z) G λ (C) follows from the fact that G λ (Z) s the subgroup of G λ (C) preservng V Z, as proven n [Ca] Inectvty of the natural map ϕ : G(Z) G(C) s not currently known and depends on functoral propertes of Tts group ([T]) Lemma 84 Consder the group homomorphsm ϕ : G(Z) G(C) Then Proof Consder ϕ : G(Z) G(C) Then ϕ(center(g(z))) Center(ϕ(G(Z))) Center(G(C)) ϕ(center(g(z))) Center(ϕ(G(Z))) and K = ϕ(g(z)) = G(Z)/Ker(ϕ) s a subgroup of G(C), so ϕ takes Center(G(Z)) to the center of a subgroup K G(C) But Center(G(C)) s abelan so Center(K) = Center(ϕ(G(Z))) Center(G(C)) The followng dagram commutes: G(Z) ' G(C),Z G (Z) G (C) That s, so (ι ρ λ,z )(G(Z)) = (ρ λ ϕ)(g(z)) Ker(ι ρ λ,z ) = Ker(ρ λ ϕ) We have Ker(ρ λ,z ) Ker(ι ρ λ,z ) = Ker(ρ λ ϕ) Thus f ϕ s nectve, then we have Ker(ι ρ λ,z ) = Ker(ρ λ ι) or Ker(ρ λ,z ) Ker(ι ρ λ,z ) = Ker(ρ λ ι) Center(G(C)) H(C), usng Lemma 83 Furthermore, f ϕ s nectve, then by Lemma 84 we have and We have proven the followng Center(G(Z)) Center(G(C)) Ker(ρ λ,z ) Center(G(Z)) Center(G(C)) H(C) Center(G(Z)) = G(Z) Center(G(C)) G(Z) H(C) = H(Z) Theorem 85 The kernel K λ of the map ρ λ,z : G(Z) G λ (Z) les n H(C) and f the group homomorphsm ϕ : G(Z) G(C) s nectve, then K λ H(Z) = (Z/2Z) rank(g) 12

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