Kannan, S S

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Available at: http://www.ictp.trieste.it/~pub off IC/2000/44United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSPROJECTIVE NORMALITY OF THE WONDERFUL COMPACTIFICATIONOF SEMISIMPLE ADJOINT GROUPSS. Senthamarai Kannan1Chennai Mathematical Institute, 92, GN Chetty Road, Chennai-17, IndiaandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.AbstractWe prove that if G is a semisimple adjoint group over an algebraically closed eld of arbitrarycharacteristic, X is the wonderful compactication of G andbG is a simply connected coveringof G, then for any ample line bundle L over X, the cone over X given by L is normal.MIRAMARE { TRIESTEOctober 20001E-mail: kannan@ictp.trieste.it; kannan@smi.ernet.in1 IntroductionIn [4], what are known as \wonderful compactications" of symmetric varieties were constructedand studied by C. De Concini and C. Procesi. More precisely, if G is a semisimple adjoint groupover the eld of complex numbers, H is the subgroup of all xed points of an involution  ofG that is induced by an involution b of the simply connected coveringbG of G, then, they haveconstructed a complete embedding G=H of the homogeneous space G=H, with boundary being aunion of normal crossing divisors. In particular, one gets such a compactication G for the groupG (G being considered as (GG)=(G)). In [10], E.Strickland has proved that the wonderfulcompactications for the group exists over algebraically closed elds of positive characteristics.In [5], C. De Concini and T.A. Springer have proved that these compactications for arbitrarysymmetric space G=H exists when the base eld is of characteristic p 6= 2. In [6], G. Faltingsraised the question: For whatbG- linearised line bundles L, the cone over G=H given by L isnormal?The aim of this paper is to provide an armative answer to his question for the case of thewonderful compactication of a semisimple adjoint group G over an algebraically closed eld ofarbitrary characteristic. To be more precise, we prove that if G is a semisimple adjoint groupover an algebraically closed eld of arbitrary characteritic,bG is a simply connected covering ofG, then for anybGbG- linearised very ample line bundle L over the woderful compacticationG, the cone over G given by L is normal. In section 3, we prove this result when the base eld isthe eld of complex numbers. For a more precise statement, see Theorem 3.4. In section 4, weprove the result for algebraically closed elds of positive characteristic by using the propertiesof good ltrations.2 Notations and basic TheoremsThroughout sections 2 and 3, we x the following notations: Let G be a semisimpleadjoint group over the eld of complex numbers. Let  :bG  ! G be a simply connected coveringof G. Let k[bG] denote the co-ordinate ring ofbG. Let T be a maximal torus of G and B be aBorel subgroup of G containing T . Let B be the opposite Borel subgroup of G determined byT and B. LetbT denote the pull back of T inbG. LetbB (resp.dB ) denote the pull back of B(resp. B ) inbG. Let W denote the Weyl group of G with respect to T .Let  denote the set of roots with respect to T as above and +  denote the set ofpositive roots with respect to B as above.2Let   +denote the set of simple roots. We label the elements of  by : := f1; 2; :::lg; l = rank(G):Let  denote the weight lattice of G with respect to T . Let E := NR denote the R span of .Let (:; :) denote the positive denite bilinear form on E induced by the Killing form of the Liealgebra of G and dene h; i =2(;)(;). Now, consider the fundamental weights. Since, theyform a dual basis of the simple coroots, we label them:f$1; :::$lg;where $iis dual to i. Let w0denote the longest element of W with respect to .In this paragraph, we recall some basic properties of the wonderful compactication of thesymmetric spaces. Let G be a semisimple adjoint group, let  be an involution of G that isinduced by an involution b ofbG. Let H denote the subgroup of all xed points of  in G. LetH0denote the xed points of b inbG. Then, there exist a - stable maximal torus T of G and aBorel subgroup B of G containing T with the following property :1. The dimension of ft 2 T : (t) = t 1g is maximal.2. For any positive root  (with respect to B), if () is a positive root, then () = .Let f1; 2;    jg be the set of all simple roots that are not xed by . Then,  induces apermutation e on the set f1; 2;    jg.Also, there is only one closed G-orbit in the wonderful compactication G=H, say G=P andthe restriction map Pic(G=H)  ! Pic(G=P ) is injective. For a dominant weight , let Vdenote the irreducible representation ofbG with highest weight . Let  be a dominant weightsuch that VH0is non zero. Then, [by Theorem 8.3, pp 30-31 [4]], the global sections of the linebundle Lon G=H is given by:2.1 H0(G=H;L) =L2 V, where ,denote the set of all dominant weights  that are ofthe form  =   (Pie(i)mi(i  (i))), with mi2 Z0.Now, we apply these to our special involution  : GG  ! GG sending (x; y) 7! (y; x).Here, the symmetric space is the group G = (G  G)=(G). Let X denote the wonderfulcompactication of G.Here, we should take T  T as a - stable maximal torus with the above maximal property.We should take B  B (T  B) as a Borel subgroup of G  G containing T  T with theabove property. Here, the unique closed G  G- orbit is G=B  G=B , and the image ofPic(X)  ! Pic(G=BG=B ) consists of line bundles associated to weights of the form (; ).For any dominant weight , let Ldenote the line bundle over X associated to  and let3H0(X;L) denote the global sections of Lover X. Let  denote the dominant ordering on .That is    if and only if    =Pi=1miifor some mi2 Z0.Now, applying the above statement 2.1 to our special involution of GG, we have2.2 H0(X;L) =LEnd(V), where the sum is taken over all dominant weights  such that  .3 Projective normalityWe now prove that for any line bundle Lon X, there is a unique (up to multiplication by aconstant) injective homomorphism: H0(X;L)  ! k[bG] (3:0:1)ofbGbG modules.For a proof of this fact: Since G = G  G=(G) is a G  G stable open subset of X, andthe natural map  :bG =bG bG=(bG)  ! G = G  G=(G) is a surjection, the naturalmap  :bG =bG bG=(bG)  ! X is a dominant map and hence for any line bundle L overX, H0(X;L) is abG bG- sub module of thebG bG module H0(bG;L). Since k[bG] is a uniquefactorisation domain, the pull back (L) onbG is trivial and hence we haveH0(bG;L) = k[bG]:Moreover, this identication is unique up to constants since the units of the k- algebra k[bG] areonly nonzero constants. Thus, (3.0.1) holds.We now prove that for any two line bundles Land L, the following diagram is commutative:H0(X;L)NH0(X;L)  ! H0(X;LNL)??y??yk[bG]Nk[bG]  ! k[bG](3:0:2)Here the horizontal arrows are the natural maps, the vertical map on the left is andthe vertical map on the right is +.For a proof of this observation:By the Peter-Weyl Theorem, thebG bG- module k[bG] =LEnd(V), where the sum istaken over all dominant weights . Using this and the fact that H0(X;L) =LEnd(V),it is easy to see that (3.0.2) holds.Now, we prove the following Lemma.4Lemma 3.1. Let  and  be two dominant weights. Let 1be a dominant weight such that1  and 1be a dominant weight such that 1 . Let  be a dominant weight such that Vis abG- submodule of the tensor product V1NV1. Then, the image of End(V1)NEnd(V1)under the multiplication map H0(X;L)NH0(X;L)  ! H0(X;L+) contains End(V).Proof: SincebG is linearly reductive and Vis a quotient of V1NV1, we can assume thatVis a submodule of V1NV1.Now, consider thebB action onEnd(V1) on the right and consider thedB action on End(V1)on the right. By the observation 2.2 and the hypothesis 1  and 1 , we have End(V1) H0(X;L) and End(V1)  H0(X;L). So, by our identication (3.0.1), we have End(V1) H0(X;L)  k[bG] and End(V1)  H0(X;L)  k[bG]. Therefore, by the Borel-Weil Theorem [cf[1]], we have V1= ff 2 End(V1) : f(xb) = 1(b):f(x) for x 2bG; b 2bBg = ff 2 k[bG] : f(xb) =1(b):f(x) for x 2bG; b 2bBg and V1= ff 2 End(V1) : f(xb ) = w0(1)(b ):f(x) for x 2bG; b 2dB g = ff 2 k[bG] : f(xb ) = w0(1)(b ):f(x) for x 2bG; b 2dB g.Fix our choice of V1and V1as above and let f =Pifi gi2 V1NV1, fi2 V1,gi2 V1be such that (Pifi gi)(x; x) is zero, for every x 2bG . Then, we have f((xb; xb )) =(1(b))(w0(1)(b ))f(x; x) = 0 for every x 2bG, b 2 B and b 2 B . Since the elements(xb; xb ) form a dense open set inbGbG, f must vanish identically onbGbG. Thus, V1NV1ismapped injectively via the multiplication map End(V1)NEnd(V1)  k[bG]Nk[bG]  ! k[bG].By the hypothesis 1 , 1  and by Theorem(2.1), we have End(V1)  H0(X;L)and End(V1)  H0(X;L). Therefore, by the observation (3.0.2), thebG module V1NV1is mapped injectively into H0(X;L+) via the multiplication map H0(X;L)NH0(X;L) ! H0(X;L+). Since Vis abG- submodule of V1NV1, Vmust be isomorphic to abGsubmodule of the image of End(V1)NEnd(V1). Since the image of End(V1)NEnd(V1) isabGbG submodule of k[bG], it must contain End(V).Therefore, using the diagram (3.0.2), it is easy to see that End(V) is abGbG-submoduleof the image of H0(X;L)NH0(X;L)  ! H0(X;L+) under the multiplication map.This completes the proof of the Lemma.We now prove the following Lemma.Lemma 3.2. Let  and  be two dominant weights. Let  be a dominant weight satisfying   + . Then, there is a dominant weight 1  and there is a dominant weight 1 such that VisbG- submodule of the tensor product V1NV1.Proof: Let  and  be two dominant weights. Let  be a dominant weight such that  + . We wish to prove that there are dominant weights 1, 1such that 1 , 1 5and an element w of the Weyl groupW such that  = 1+w(1), and then using PRV conjecture[cf [9]], we prove that Vis abG- submodule of the tensor product V1NV1.By [sections (13.4) and (21.3)] of [8]],  is a weight of V+. Therefore, we have  = 1+ 2,where 1is a weight of Vand 2is a weight of V. We have    1= (Plimii) for somemi2 Z0.We rst prove that there is a dominant weight 1such that 1  and there is a weight 02of Vsuch that  = 1+ 02by induction on the sumPlimi.IfPlimi= 0, there is nothing to prove.IfPlimi= 1, then, we have 1=   i0for some i02 f1; 2;    lg. Now, if 1is dominant,then there is nothing to prove. Otherwise, h1; i0i is negative. Since  = 1+ 2is dominant,h2; i0i is positive. But, we know that 1+ki0is a weight of Vfor any k 2 f0; 1;    h1; i0igand 2  ki0is a weight of Vfor any k 2 f0; 1;    h2; i0ig [cf [sections (13.4) and (21.3) of[8]]]. In particular, 1+ i0=  is a weight of V(which is ofcourse trivial) and 2  i0is aweight of Vand we also have  =  + 2  i0. Therefore, in this case, we take 1=  and02= 2  i0.Now, let N be a positive integer strictly bigger than 1. Let us assume that the statement istrue for all  such that  = 1+ 2, where 1=   Pli=1miibeing a weight of Vwith thesumPli=1mi N   1 and 2being a weight of V.Let  be a dominant weight such that    and  = 1+ 2where 1is a weight of V, 2is a weight of Vwith 1=   Pimiiand the sumPli=1mi= N . If 1is dominant, then,there is nothing to prove. Otherwise, there is a simple root i0such that h1; i0i < 0.By a proof similar to the case whenPlimi= 1, it is easy to see that 1+ i0is a weightof Vand 2  i0is a weight of V. Also, we have  = (1+ i0) + (2  i0). Therefore, byinduction, there exist a dominant weight 1  and a weight 2of Vsuch that = 1+ 02:We x our choice of 1and 02as above. Since 02is a weight of V, w(02) is also a weight ofVfor every w 2 W . Thus, the unique dominant weight (say 1= w 1(02) ) in the W orbit of02must also be a weight of V.Hence, we have  = 1+ w(1) where 1is a dominant weight satisfying 1 , 1is adominant weight satisfying 1  and w is an element of the Weyl group W .Now, the fact that Vis abG- submodule of the tensor product V1NV1follows from PRVConjecture [cf [9]].6This completes the proof of this Lemma.We haveCorollary 3.3. Let  and  be two dominant weights. Then, the multiplication mapH0(X;L)OH0(X;L)  ! H0(X;L+) is onto:Proof: Proof of this is an immediate consequence of Lemmas 3.1 and 3.2.Theorem 3.4. Let G be a semisimple adjoint group over the eld of complex numbers. Let :bG  ! G be a simply connected covering of G. Let X denote the wonderful compacticationof G. Let L be an ample line bundle over X. Consider the embedding of X in P(H0(X;L)).LetbX denote the cone over X with respect to this embedding of X. Then,bX is normal.Proof: By [4], we have L = Lfor some regular dominant weight . By [II (5.14)(d)] of [7],it suces to prove that the multiplication mapH0(X;L)N ! H0(X;LN) (3:4:1)is surjective for every positive integer N . The proof of this is an immediate consequence ofCorollary 3.3.We haveCorollary 3.5. Let w be an element of the Weyl group W . Let Xwdenote the scheme theoreticclosure of the cell BwB in X. Let L be abGbG linearised very ample line bundle over X andletdXwdenote the cone over the embedding given by L. Then,dXwis normal.Proof: By [Corollary 3, [3]], the restriction mapH0(X;Ln)  ! H0(Xw; Ln) is surjective for every positive integer n    (3:5:1).Now, consider the following commutative diagram of maps:H0(X;L)n ! H0(X;Ln)??y??yH0(Xw; L)n ! H0(Xw; Ln)(3:5:2)Here the horizontal arrows are the natural multiplication maps and the vertical arrows arerestriction maps.The assertion of the corollary follows from the observations 3.5.1, 3.5.2 and Corollary 3.3.74 Appendix:In this section, we prove the analogue of Theorem 3.4 when the base eld is of positive charac-teristic.Let k be an algebraically closed eld of positive characteristic. Let G be a semisimpleadjoint group over k. LetbG be a simply connected covering of G. Let X denote the wonderfulcompactication of G. Let T , B, B ,bT ,bB,dB be as in section 2.With the notations as in section 2, we introduce some more notations as follows: Let denote the weight lattice of G with respect to T . So,   is the weight lattice of GG withrespect to T  T .For any weight  2 , set V:= H0(G=B;Lw0( )).Denition 4.1. LetbG be as above. Let V be abG module. A ltration ofbG- submodules(0) = V0 V1 V2    Vn= V is a good ltration if each Vi+1=Viis a direct sum of V (i(j))'sfor some dominant weights i(j)'s.As in section 2, let  = f1;   lg denote the set of simple roots with respect to T and B.Let  and  be two dominant weights in .Let L(resp. L) denote the line bundle over X associated to (; ) (resp. to (; ). Let be a dominant weight such that   +). Let  : H0(X;L)NH0(X;L)  ! H0(X;L+)denote the multiplication map.Then, we haveLemma 4.2. ThebGbG- module Vw0( )NVis abGbG-subquotient of the image(H0(X;L)NH0(X;L)).Proof: By Donkin's conjecture [cf Theorem 4.4.3, [11]], a tensor product of two modules withgood ltrations has a good ltration. By Theorem 5.10 of [5], H0(X;L1) (resp. H0(X;L1)) hasa good ltration ofbGbG-submodules such that the successive quotients look like Vw0( )NV,  1(resp.   1). Therefore, the tensor product H0(X;L1)NH0(X;L1) has a goodltration ofbGbG-submodules.Now, let H be a semisimple Chevalley algebraic group over the eld of complex numbers.Let Hpbe the corresponding algebraic group in characteristic p. Now, if Vpis a Hp-modulehaving a good ltration in characteristic p such that it comes from a H-module V modulo p,8then the character of V is same as the chracter of Vpand so the Hp- subquotients of Vpareprecisely the H- direct summands of V .Now, let M(resp. M0) be the largestbGbG submodule of the tensor productH0(X;L1)NH0(X;L1) all of whose BB- weights are  ( w0(); ) (resp. < ( w0(); )).Also, let Nand N0be the corresponding submodules of H0(X;L1+1). Therefore, fromLemmas 3.1 and 3.2, and by the observation above,  induces a non zerobG bG- modulehomomorphism from the quotient (V w0()NV)m= M=M0 ! N=N0= V w0()NVforsome positive integer m. Therefore, by the Frobenius reciprocity [cf Proposition 2.1.6 [11]], theassertion follows.The Theorem (3.4) for positive characteristic follows from Lemma(4.2).Acknowledgements: We thank Professor C.De Concini for suggesting the problem. We thankProfessors M.S.Narasimhan and C.Procesi for their encouragement and useful discussions. Wethank ICTP for the hospitality during the stay. We thank the referee for the useful commentsand for suggesting the problem for positive characteristics.References[1] A.Borel, Linear representations of semi-simple algebraic groups, AMS, Proceedingsof Symposia in Pure Mathematics, volume 29, 1974.[2] N.Bourbaki, Groupes et Algebres de Lie, Ch 4, 5 et 6, Paris: Hermann, 1968.[3] M.Brion and P.Polo, Large Schubert Varieties, Representation Theory 4 (2000),97-126.[4] C.De Concini and C.Procesi, Complete symmetric varieties, pp 1-43, Invariant The-ory, 996, Lecture Notes in Mathematics.[5] C.De Concini and T.A.Springer, Compactication of symmetric varieties, Transfor-mation Groups, Vol.4, No.2-3, 1999, pp. 273-300.[6] G.Faltings, Explicit resolution of local singularities of moduli spaces, J. reine angew.Math. 483 (1997) pp. 183-196.[7] R.Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.[8] J.E.Humphreys, Introduction to Lie algebras and representation theory, Springer -Verlag, 1972.9[9] S.Kumar, Proof of the Parthasarathy-Ranga Rao-Varadarajan Conjecture, Invent.Math. 93, 117-130 (1988).[10] E. Strickland, A Vanishing result for group compactications, Math. Ann. 277(1987), 165-171.[11] Wilberd van der Kallen, On Frobenius Splittings and B-Modules, TIFR Lecturenotes in Maths, Springer-Verlag, 1993.10

Projective normality of the wonderful compactification of semisimple adjoint groups

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