Weyl group. Chapter locally finite and nilpotent elements

Transcription

1 Chapter 4 Weyl group In ths chapter, we defne and study the Weyl group of a Kac-Moody Le algebra and descrbe the smlartes and dfferences wth the classcal stuaton Here because the Le algebra g(a) s not fnte dmensonal, we wll need to be more careful wth sums In partcular, we wll need locally fnte and locally nlpotent elements as well as ntegrable representatons In all the chapter g(a) wll be a Kac-Moody Le algebra 41 locally fnte and nlpotent elements We start wth the Defnton 411 (ı) Let T : V V be an endomorphsm of a complex vector space V It s called locally fnte at v V f there exsts a fnte dmensonal subspace W of V contanng v and stable by T If T W s nlpotent, then T s called locally nlpotent at v The endomorphsm T s called locally fnte (resp locally nlpotent) f t s locally fnte (resp locally nlpotent) at every v V (ıı) For a locally fnte T : V V, we can defne an automorphsm expt : V V by expt = n 0 T n and we have the formula exp(kt) = (expt) k If T s locally fnte at v, we can defne (expt)(v) Let us prove some useful formulas: Lemma 412 (ı) Let A be an assocatve algebra, let D be a dervaton on A, let x, y and z be elements n A and let [x,y] = xy yx, then we have the followng formulas: D k [x,y] = [D x,d k y], x k y = ((ad x) y)x k and (ad x) k y = ( 1) x k yx (ıı) Let g be a Le algebra and x, y and z be elements n g, then we have: (ad x) k [y,z] = [(ad x) y, (ad x) k z] 25

2 26 CHAPTER 4 WEYL GROUP Proof : (ı) We prove the frst formula by nducton (t s smply the Lebntz rule For k = 0 ths s true and for k = 1 t s the formula D[x, y] = [x, Dy] + [Dx, y] Compute ( ( ) ) k D k+1 [x,y] = D [D x,d k y] and the result follows from the prevous formula For the second formula, consder the operators L x and R x of left and rght multplcaton by x These operators commute and ad x = L x R x or L x = adx+r x so that the three operators commute Applyng the bnomal formula to L x = adx+r x yelds the frst formula whle applyng t to adx = L x R x yelds the second (ıı) The frst formula s true n U(g) where ad x s a dervaton Applyng t to the adjont representaton gves the result Corollary 413 Let T and S be two endomorphsms of V and assume that T s locally fnte and that {(ad T) k S ; k N} spans a fnte-dmensonal subspace of End(V ), then we have the formula: (expt)s exp( T) = (ad T) n S = (exp(ad T))(S) n 0 Proof : The hypothess on ad T s smply that ad T s locally fnte at S End(V ) In partcular both parts of the equalty are well defned The proof s now a formal computaton usng the prevous Lemma: (expt)s exp( T) = n 0 T n S ( T) n = n 0 n 0 n ( 1) T n ST!(n )! = n 0 Let us prove the followng Lemmas on fnte and nlpotent elements (adt) n S Lemma 414 (ı) Let s be a Le algebra and let π : s gl(v ) be a representaton Assume that y s s such that ad y s locally fnte (resp locally nlpotent) on x s, then π(ad y) s locally fnte (resp locally nlpotent) on π(x) gl(v ) (ıı) Let T End(V ) be a locally fnte (resp locally nlpotent element) and let f : V W be an somorphsm, then f T f 1 s locally fnte (resp locally nlpotent) Proof : (ı) Let U be a fnte dmensonal subspace of s such that x U and ad y stablzes U (and furthermore (ady) U s nlpotent s the locally nlpotent case) Consder the subspace π(u) V whch contans π(x) It s π(ady) stable (and the restrcton s nlpotent n the nlpotent case) (ıı) For ths part smply take f(u) Lemma 415 (ı) Let s be a Le algebra and let x s Defne s x = {y s / (ad x) n y y = 0 for some n y N} Then s x s a Le subalgebra of s (ıı) Let π : s gl(v ) be a representaton of s and let x s Defne V x = {v V / π(x) nv v = 0 for some n v N} Then V x s a s x -submodule of V (ııı) Let π : s gl(v ) be a representaton of s such that s s generated as a Le algebra by the set F V = {x s / ad x s locally fnte on s and π(x) s locally fnte on V } Then

3 41 LOCALLY FINITE AND NILPOTENT ELEMENTS 27 the Le algebra s s spanned by F V as a C vector space In partcular, f s s generated as a Le algebra by the set F of ts ad -locally fnte vectors, then F spans s as a vector space If dms <, then any v V les n a fnte-dmensonal s-submodule of V Proof : The ponts (ı) and (ıı) follow drectly from the formulas 2 and 4 of Lemma 412 (ııı) Let x and y n F V and t C Because ady s locally fnte, we may consder exp(ady) and even exp(t ady) Ths s an endomorphsm of s so we may apply t to x and get (exp(t ady))(x) s We want to prove that ths element s n F V For ths we want to apply Corollary 413 to π((exp(t ady))(x)) But because π s a Le algebra morphsm, we have π((ady) n x) = (adπ(y)) n π(x) and because of Lemma 414 we have π(exp(ad y))(x)) = (exp(ad π(y)))(π(x)) But π(y) s locally fnte and ad π(y) s locally fnte on π(x) so we may apply Corollary 413 to get π(exp(ady))(x)) = (exp(π(y)))π(x)(exp( π(y))) Because π(x) s locally fnte, ths proves thanks to Lemma 414 (ıı) that π(exp(ad y))(x)) s locally fnte The same proof shows that ad (exp(ad y))(x))s locally fnte and that exp(ad y))(x)) F V Now we have the formula (exp(tady))x x lm = [y,x] t 0 t Provng that the lnear span of elements n F V s a Le subalgebra of s The last result follows from the prevous one and Poncaré-Brkhoff-Wtt Theorem Let us now prove the followng characterzatons of locally nlpotent elements Lemma 416 (ı) Let y 1,y 2, be a system of generators of a Le algebra g and let x g such that (ad x) N y = 0 for some postve ntegers N Then ad x s locally nlpotent on g (ıı) Let v 1,v 2, be a system of generators of a g-module V and let x g be such that ad x s locally nlpotent on g and x N (v ) = 0 for some postve nteger N Then x s nlpotent on V Proof : (ı) If an element s locally nlpotent on a vector space bass, then t s locally nlpotent on the space Ths together wth formula (ıı) of the Lemma 412 concludes the proof (ıı) To prove ths result, we need to prove that a power of x wll kll an element of the form y 1 y s (v) where y g and v V We apply the second formula of Lemma 412 (ı) to x k y 1 y k n U(g) and get the result Corollary 417 The elements ad e and ad f and locally nlpotent on g(a) Proof : We gve two proofs We have ade (f j ) = 0 and (ade ) 1 a,j (e j ) = 0 for j Furthermore, ade (e ) = 0, (ade ) 2 (h) = 0 for any h h Indeed, we have (ad e ) 2 (h) = α,h ad e (e ) = 0 Fnally, we have (ad e ) 3 (f ) = 0 and the result follows from Lemma 416 (ı) The same proof works for adf Second proof Let x = e and consder g(a) x = {y g(a) / (ad x) ny y = 0 for some n y N} as n Lemma 415 We know that g(a) x s a Le subalgebra and because of the defnng relatons of g(a) and of Proposton 326, we know that h and all the e j and the f j are n g(a) x so that g(a) x = g(a)

4 28 CHAPTER 4 WEYL GROUP 42 Integrable representatons and Weyl group 421 Integrable representatons Defnton 421 (ı) A g(a)-module V s h-dagonalsable (sometmes also called a weght module) f there s a decomposton V = λ h V α where V λ = {v V / h(v) = λ,h v h h} The subspace V λ s called a weght space, λ s called a weght f V λ 0 and dmv λ s the multplcty of λ denoted by mult V λ (ıı) An h-dagonalsable module V over g(a) s called ntegrable f e and f are locally nlpotent on V for all n [1,n] Let us gve a Proposton explanng the termnology of ntegrable representatons Proposton 422 Let V be an ntegral representaton of g(a) and let g () be the sl 2 somorphc Le subalgebra of g(a) defned by e, f and α Then V decomposes as a drect sum of fnte dmensonal rreducble h-nvarant modules and n partcular the acton of g () can be ntegrated to an acton of SL 2 (C) Proof : We only need to apply Lemma 415 Because g () s fnte dmensonal, we know that any element v V sts n a g () -stable fnte dmensonal subspace of V These fnte dmensonal subspaces are ntegrable Proposton 423 The adjont representaton of g(a) s ntegrable Proof : We already know that the adjont representaton s h-dagonalsable and the rest follows from Corollary Defnton of the Weyl group and acton on ntegrable representatons Defnton 424 (ı) For any [1, n], defne the reflecton s Auth by s (λ) = λ λ,α α, for λ h It s the reflecton wth respect to the hyperplane {λ h / λ,α = 0} In partcular we have s 2 = 1 (ıı) Let W the subgroup of Auth generated by the reflectons s for [1,n] Ths group s the Weyl group of the Kac-Moody Le algebra g(a) and the reflectons s are called smple reflectons The fathful representaton of W n h s called the standard representaton of W (ııı) Dualzng the representaton, we get the congradent representaton W Auth, whch s explctly gven for any 1 n by s (h) = h α,h α, for h h (ıv) The length of an element w W s the smallest k such that we can wrte w = s 1 s k wth the s j smple reflectons Defnton 425 Let π : g(a) gl(v ) be an ntegrable representaton of g(a) We may defne the followng element s (π) End(V ) by s (π) = (expπ(f ))(exp( π(e )))(exp(π(f ))) Proposton 426 Let π : gl(a) V be an ntegrable representaton Let λ h and s W a smple reflecton (ı) We have the equalty s (π)(v λ ) = V s (λ), n partcular mult V λ = mult V wλ for all w W (ıı) For any v V and x g(a), we have s (π)(xv) = (s (ad )x)(s (π)v) In partcular s (ad ) s a Le algebra automorphsm of g(a) (preserves the Le bracket)

5 42 INTEGRABLE REPRESENTATIONS AND WEYL GROUP 29 (ııı) For v V λ, we have s (π) 2 (v) = ( 1) λ,α v and f v 0, then λ,α Z (ıv)let m,j be the order of s s j n W, then f m,j <, we have s (π)s j (π)s (π) }{{} m,j factors = s j (π)s (π)s j (π) }{{} m,j factors Proof : (ı) Let us prove the ncluson s (π)(v λ ) V s (λ) A very smlar proof gves the ncluson s (π) 1 (V λ ) V s (λ) and the result follows from the fact that s s an nvoluton (the element s (π) s however not an nvoluton as proved n (ııı)) Let v V λ and consder the element h(s (π)(v)) n V If α, h = 0, then h and e commute n U(g(A)) and also h and f commute n U(g(A)) In partcular, we get h(s (π)(v)) = s (π)(h(v)) = λ,h s (π)(v) But n that case we have s (λ) = λ λ, α α and s (λ),h = λ, h We thus have the result for the hyperplane of weghts orthogonal to α To prove the result we only need to prove t on one element not n that orthogonal (by lnearty) For example, t s enough to prove t for h = α that s to say t s enough to prove the relaton α (s (π)v) = λ,α s (π)v for v V λ or s (π) 1 (α (s (π)v)) = λ,α v for v V λ or the equalty s (π) 1 π(α )s (π) = π(α ) of elements n End(V ) Usng Corollary 413, we get the equalty s (π) 1 π(α )s (π) = π((exp( adf ))(expade )(exp( adf ))(α )) But because of the defnng relatons of g(a), we get the equaltes (exp( ad f ))(α ) = α 2f, (exp( adf ))(f ) = f, (expade )(α ) = α 2e and (exp ade )(f ) = f + α e We thus get the formula (exp( adf ))(expade )(exp( adf ))(α ) = α (ıı) We compute s (π)(xv) = s (π)π(x)s (π) 1 s (π)(v) but the composton s (π)π(x)s (π) 1 s equal to (expπ(f ))(expπ( e ))(expπ(f ))x(expπ( f ))(expπ(e ))(expπ( f )) and applyng Corollary 413 we obtan the equalty s (π)π(x)s (π) 1 = π((expadf )(exp( ade ))(expadf )x) = π(s (ad)(x)) (ııı) To prove ths result, we may assume that g(a) = g () sl 2 and by Lemma 415 we may assume that V s fnte dmensonal By Proposton 325, we obtan that f v 0 then λ,α Z Furthermore, we have the commutatve dagram sl 2 π End(V ) Exp exp SL 2 Aut(V ) and t s enough to check the relaton (Exp(f)Exp( e)exp(f)) 2 = 1 = Exp(πh) n SL 2 Ths s an easy calculaton Now we compute Exp(πh)(v) = exp(π λ,h )v = ( 1) λ,h v (ıv) To prove ths part, we may assume that A s a 2 2 matrx of the from A = ( 2 a,j a j, 2 Let us frst prove that we may assume that a = a,j a j, s an nteger n the nterval [0, 3] For ths we prove that f a 4, then the order m,j of s s j s nfnte We know that both a,j and a j, are non )

6 30 CHAPTER 4 WEYL GROUP postve so a s non negatve Consder the 2-dmensonal subspace U of h generated by α and α j The matrx of s (resp s j ) n the bass {α,α j } s gven by ( ) ( ( )) 1 a,j 1 0 resp The composton s s j s gven by the matrx ( 1 + a a,j 1 a j, The egenvalues of ths matrx are roots of the polynomal X 2 + (2 a)x + 1 and thus gven by a 2 ± a(a 4) If a > 5, then one egenvalue s bgger than 1 and the composton s s j has to be 2 of nfnte order If a = 4, then the egenvalues are equal to 1 But as the composton s not dentty, t has to be of nfnte order Now we may assume that a = 0,1,2 or 3 Ths mples that g(a) s of type A 1 A 1, A 2, B 2 or G 2 (because Serre relatons are satsfed and because there s no non trval deal meetng h trvally n such Le algebras) By Lemma 415 we may assume that V s fnte dmensonal We have the commutatve dagram g(a) π End(V ) Exp a j, ) exp G(A) Aut(V ) where G(A) s the smply-connected group assocated to g(a) We need to prove the followng relaton S S j S = S j S S j wth m,j terms and where S = Exp(f )Exp( e )Exp(f ) (smlar defnton for S j ) Ths can be proved case-by-case (see for example [Sp98, Proposton 932]) Corollary 427 (ı) For any smple reflecton s, we have s (ad ) h = s as an automorphsm of h (ıı) Assume that for some and j n [1,n] and for w W we have α j = w(α ), then w(α ) = α j Proof : (ı) Take any ntegrable representaton (V, π) of g(a) and let v V of weght λ and x h We have by the prevous Proposton: λ(x)s (π)(v) = s (π)(xv) = (s (ad)(x))(s (π)(v)) = ((s λ)(s (ad)(x)))(s (π)(v)) hence λ(x) = (s λ)(s (ad )(x)) Replacng λ by s λ we get that (s λ)(x) = λ(s (x)) = λ(s (ad )(x)) Ths s true for any weght λ so that s (x) = s (ad)(x) and we have the result (we shall see later that the lnear span of the weghts of ntegrable representatons s h ) We may gve a drect proof of ths result by smple computaton, ndeed, we want to compute s (ad )(h) = exp(ad (f )) exp( ad (e ))exp(ad (f ))(h) for h h But we have the formulas: exp(ad(f ))(h) = h + h,α f, exp( ad (e ))(h) = h + h,α e, exp( ad(e ))(f ) = f + α e and exp(ad(f ))(f ) = f Ths gves the formula s (ad)(h) = h h,α α (ıı) We know from (ı) that there exsts a Le algebra automorphsm ŵ : g(a) g(a) such that ŵ h = w In partcular, we have [ŵ(e ),ŵ(f )] = ŵ[e,f ] = w(α ) But because w(α ) = α j and by the prevous Proposton, we have that ŵ(e ) g αj and ŵ(f ) g αj thus [ŵ(e ), ŵ(f )] = cαj for some c C We thus have w(α ) = cα j Apply α j to ths equalty to get α j (w(α )) = (w 1 (α j )(α ( )) = α (α ) = 2 and α j(cαj ) = 2c thus c = 1 and we are done Defnton 428 Let α be a root such that α = w(α ) for some w n the Weyl group and some ndex [1,n] Such roots wll be called real roots We may defne the coroot of α, denoted α by w(α ) The prevous Corollary mples that ths s well defned

7 Bblography [Bo54] Bourbak, N Groupes et algèbres de Le 4,5,6, Hermann 1954 [Hu72] [Ku02] [PK83] Humphreys, JE Introducton to Le algebras and representaton theory GTM 9, Berln- Hedelberg-New York, Sprnger, 1972 Kumar, S Kac-Moody groups, ther flag varetes and representaton theory Progress n Mathematcs, 204 Brkhäuser Boston, Inc, Boston, MA, 2002 Peterson, DH and Kac, VG Infnte flag varetes and conjugacy theorems, Proc Natl Acad Sc USA, 80 (1983), pp [Sp98] Sprnger, TA Lnear algebrac groups Second edton Progress n Mathematcs, 9 Brkhäuser Boston, Inc, Boston, MA, 1998 Ncolas Perrn, HCM, Unverstät Bonn emal: ncolasperrn@hcmun-bonnde 31