GL_%282%29%28C%29的一些不可约表示.pdf

Vo1.52,No.3DOI:10.3969/J.ISSN.1000-5137.2023.03.003Some irreducible representations of GL2(C)Journal of Shanghai Normal University(Natural Sciences)Jun.,2023CHEN Xiaoyu,LAI Yuanxu,LI Zhize(Mathematics and Science College,Shanghai Normal University,Shanghai 200234,China)Abstract:Let G=GL2(C),and let B be the standard Borel subgroup of G,and let CG(resp.CB)be the group algebra of G(resp.B)over the field of complex numbers.For any character of B,definethe naive induced module M(0)=CG cB 0.In this paper,we prove that if 3 is antidominant,thenM(0)is irreducible.Thus,we give a class of new infinite-dimensional irreducible representations ofGL2(C).Key words:reductive group;naive induced module;Bruhat decompositionCLC number:O 186.5Document code:AArticle ID:1000-5137(2023)03-0295-08GL2(C)的一些不可约表示陈晓煜，赖元旭，李支泽(上海师范大学数理学院，上海2 0 0 2 34)摘要：设 G=GL2(C),并且 B 是 G 的标准 Borel子群，并且 CG,CB分别是群 G 和群 B的在复数域C上的群代数对于任意B的特征标0,定义G的离散诱导模M(0=CGcB0.证明了当是反支配权时，M()是个不可约表示由此给出了一类GL2(C)全新的、无限维的不可约表示.关键词：简约群；朴素诱导模；Bruhat分解Received date:2023-04-20Foundation item:Shanghai Sailing Program(17YF1413800);The National Natural Science Foundation of China(11701373)Biography:CHEN Xiaoyu(1985),male,associate professor,research area:algebraic groups and quantum groups.E-mail:引用格式：陈晓煜，赖元旭，李支泽GL2(C)的一些不可约表示J.上海师范大学学报（自然科学版),2 0 2 3,52(3)295-302.Citation format:CHEN X Y,LAI Y X,LI Z Z.Some irreducible representations of GL2(C)J.Journal of ShanghaiNormal University(Natural Sciences),2023,52(3):295-302.2961IntroductionJ.Shanghai Normal Univ.(Nat.Sci.)Jun.2023The classification of irreducible representations(up to isomorphism)of a given group is a fundamental problemin the representation theory.Let G be a connected reductive group over a field k,e.g.GLn(k),SLn(k).Therepresentation of G plays a prominent role in various areas of mathematics such as algebraic geometry and numbertheory.In the case k=Fq,the algebraic closure of Fg(Fq is the finite field of q elements)or k=C,the field ofcomplex numbers,the irreducible rational representations of G was classified by CHEVALLEY in 195811.In 1973,BOREL and TITS classified all finite-dimensional irreducible representations of G when k is an infinite field and thusverified a conjecture of Steinberg 2.In the case k=F,the classification of reducible ordinary representations ofG was given in 3 by DELIGNE and LUSZTIG,and the classification of irreducible modular representations wasgiven in 4 by ROUQUIER and BONNAFE.Despite the fruitful results mentioned above,lttle was known about the infinite-dimensional irreducible repre-sentations of G when k is an infinite field.In 2014,XI began to study the infinite-dimensional representations ofG when k=Fq.He constructed such representations(in particular,the infinite-dimensional Steinberg modules)via the union of irreducible representations of the finite groups G(Fqr)5.XI showed that the infinite-dimensionalSteinberg module is irreducible if the base field is of characteristic zero or characteristic of Fq.In 2015,YANGproved in 6 that the infinite-dimensional Steinberg module is irreducible for any base field when k=Fq.LetB be the standard Borel subgroup defined over Fg and k is another field,and let 3 be a character of B,and defineM(0)k*=kG kB 0.Assume that k=Fq.In 2019,CHEN and DONG proved in 7 that M(0)k has finite lengthand determined all composition factors when the characteristic of k is not equal to that of Fq,and M(O)k may haveinfinite length when k=F.In 8 and 9,CHEN and DONG determined the decomposition of M(tr)k for anyfield k when M(0)k has finite length.Recently,CHEN classifed all irreducible kG-modules with B-stable linewhen k=k=F,10.In this paper,we deal with k=Ik=C and G=GL2(C).Let 3 be a character of B,we give a sufficientcondition for the irreducibility of M()=CG cB 0.This paper is organized as follows:in section 2,we recallsome basic facts of group representations,and give the main result.In section 3,we recall the Bruhat decompositionfor GL2(C)which will be frequently used later.In section 4,we give the proof of the main result.2Basic definition and main resultIn this section,we recall some basics for group representations and the structure of reductive groups.Let V beVol.52,No.3a vector space over the field C of complex numbers,and let GL(V)be the group of invertible linear maps of V,andlet G be a group,with identity element 1.Definition 1(linear representation).A linear representation of G is a group homomorphismCHEN X Y,LAI Y X,LI Z Z.Some irreducible representations of GL2(C)297p:G GL(V),g p(g).For all v e V,p(g)u is abbreviated to gu.When p is given,we say that V is a representation space of G(or evensimply,by abuse of language,a representation of G).Definition 2(sub-representation).Let W be a subspace of V,then W is called a sub-representation of V ifgw E W for all g E G and w E W.Definition 3(irreducible representation).Let V be a representation of G,V is said to be irreducible if theonly sub-representations of V are 0 and V.We denote by CG the group algebra of G over C.It is well known that a representation of G is identified witha CG-module,and we will indiscriminately use the terminology“linear representation or“module.Definition 4 (induced representation).Let H be a subgroup of G,and let W be a left CH-module.Then thetensor product of the right CH-module CG and the left CH-module WCGO.WCHis a left CG-module which is denoted by Ind W,and is called the induced representation from CH-module W.Following the notation above,it is well-known that a basis of the induced representation is given by the follow-ing proposition(cf.11).Proposition 1 Suppose that w;li e I)is a basis of W and(g,H I j E J)is the set of left cosets of G withrespect to H.Then the set(giwilieI,jEJ)forms a basis of IndW.One of the most commonly used methods to get a new representation of G is to study the induced representationsfrom some subgroups of G.Let G=GL2(C)and let B be the standard Borel subgroup of G(the set of upper-triangular matrixes in G).298Then B=U T,whereJ.Shanghai Normal Univ.(Nat.Sci.)I a E clTJun.2023c,deSince U is a normal subgroup of B,we have the natural homomorphism :B-T and ker =U.Letbe a homomorphism from group T into group C.Then the pullback of 3 by T,denoted by 3 for convenience(=o),is a group homomorphism from B into C i.e.a character of B.Let Ce be the 1-dimensional space affording the character a.We are interested in the induced module CG cB Cewhich is denoted by M().A nature question is whether M(0)is an irreducible module.Definition 5(antidominant).A character of B is called antidominant if there is an n E Zo,such thatnfor all t E Cx.The main result of this paper is the following theorem.Theorem 1 If is antidominant,then M(0)is irreducible.3Bruhat decomposition of GL2(C)For any reductive group of G,one has the Bruhat decomposition(cf.12,Chapter 8).In this section we recallthe Bruhat decomposition of GL2(C).In order to get a basis of M(0),We need to understand G/B.Let B)G/Bbe the set of double cosets of G with respect to B.It is well known thatG=BUBsB,wheres:Proposition 2 There is a bijection :U (s)B BsB.Proof Let bi sb2 be an arbitrary element of BsB.Since B=U T,we have b1=ut where u E U,t E T,then bisb2=utsb2=us(stsb2),where us(stsb2)e UsB.We define(u,s,b)=usb which is a bijection.Combining G=B U BsB and proposition 2.We see that the set of all represent elements of cosets in G/Bcan be written as I2 U us|u E U).For any E C,we set e(a)which is called Bruhat decomposition of GL2(C).EU.01Vol.52,No.3Proposition 3For any E Cx we haveCHEN X Y,LAI Y X,LI Z Z.Some irreducible representations of GL2(C)se(a)s=e(a299(1)ProofFirst,the left side of(1)equals toE G,se(a)and the right side of(1)equals to0)(6-)and thus the result is proved.4Proof of the main resultThroughout this section,we assume that O is antidominant.In order to show that M(0)is an irreducible CG-module,it is enough to show M(0)=CG.c for any O+E M(0).Since the set of all representatives of cosets inG/B can be written as I2 U us I u E U)and proposition 1,we see that us 1 u E U)U1 1 forms abasis of M(0).We abbreviate usl for us 1,1 for 1 1,then we have0 (1-e(a).a=ci e(ai)-e(ai+a)s1 E CG.a.i=1T=l1+cie(ai)sl where l,ci E C,r E Z0.=1Since a O,l and ci are not all equal to 0.If all of the Ci are equal to O,then =l1 with l O and we getCG.c=CG.1=M(0),the proof is finished.Now we suppose that c;are not all equal to 0.Since =o T and ker T=U,we have e(a).1=1.Since aiis a finite set,we can take a E C such thatWithout loss of generality,we can assume that c=suitable choice of a).The following idea is motivated by 13,proposition 5.4.Lemma 1If =)Proof Let mi=kis1,then we havehe(a)oherwise we replacea by(1-e(a)a fra=1kie(a)s1,then we have)kis1 E CG.1=1=e(a1)m1+e(a2)m2+.+e(ar)mr E CG.300Using the formulaand noting thatJ.Shanghai Normal Univ.(Nat.Sci.)Jun.2023001011kai01kwe havekai000k(aiS.T(e(a1)m1+e(a2)hm2+.+e(ar)hmr)E CG.c.0Thus,we havee(a1)*m1+e(a2)m2+.+e(ar)mr E CG.for k=1,2,.,r.Let zi=e(a)e CU,i=1,2,and 0h=Z1iiir ini i e CU,k=1,r.Using the identitywe haveThen we obtain the alignsK(zk-z1)(zh-22).(zh-zr)=0,k=1,2,.,r,2-012k+022k2h-2+.+(-1)gr=0,k=1,2,.,r.(zi-0121-1+0221-2+.+(1)r)m1=0,(2-012-1+022-2+(-1)or)m2=0,(2)(zf-01z-1+02zr-2+.+(-1)or)mr=0.Adding the equations above,we obtain(-1)gr（=1=(-1)gr-1C2imi+(-1)-1or-2Z2zmi+.+01Zz-1m-2i=1=1mi2mi:i=1=1(3)Vol.52,No.3Sinceis invertible,combining(2)and(3),we obtainand the result is proved.By lemma 1 we haveNow we assume thathave se(c)1.a=)By proposition 3,and since is antidominant,we have Combining these,we havese(c)1.a=k se(ai+c)s=0=1Now,if there exists c E C such that proof is finished.To see this,it is enough to show the following lemma.Lemma 2 Let fi(ac)=(ai+a)-n i=1,.-,r)be a finite set of complex value functions,then(f;(c)are linearly independent.Proof The Wronsky determinant of(fi()isW(c)=1.fr-1(a)f2-1(a)(a1+)-n(ai+a)-(n+1)C(a1+)-(n+r-1)CHEN X Y,LAI Y X,LIZ Z.Some irreducible representations of GL2(C)Or=e(a1)e(a2).e(ar)=e(a1+a2+.+ar)m1+m2+.+mr E CG.c,kis1 ECG.a.Ifki/O,then s1 E CG.a,thus CG.a=M(0).1=1ki=O,it is clear that there is a c E C such that ai+c O for all ai.For such c,we=1kise(ai+c)s1.1fi(c)fi(c)3010a100ki(ai+c)-+0,then we can replace ki with ki(ai+c)-,and thei=1f2(c)f2(ac)k(a+c)e(a:+c)-1)s1.i-1f()f()fr-1()(a2+)-n(a2+)-(n+1)(a2+)-(n+r-1)(ar+a)-n(ar+a)-(n+1)(ar+a)-(n+r-1)wherer-1 i-1II II(-1)(n+k).Ci=1 k=0302Let =O,we havewhereSince ai a;(i+j),b;+bj,we have W(O)0,thus fi(a),f2(a),.,fr(a)are linearly independent.Thus,by lemma 2 there exists c E C,such that J.Shanghai Normal Univ.(Nat.Sci.)nai(n+1)a1W(0)=C.a1(n+r-1);=a;1,D=II b IIII(-1)(n+k).Jun.2023na2(n+1)a2a2-(n+r-1)j=1i=1 k=0at k;(ai+c)-+0.then the theorem is proved.1nan+1)arr-1 i-11b1=D16-11b21br.：brReferences:1 CHEVALLEY S C.Classification des Groupes de Lie Algebriques M.Paris:Secretariat Mathematique,1958.2 BOREL A,TITS J.Homomorphismes“abstraits de groupes algebriques simples J.Annals of Mathematics,1973,97(3):499-571.3DELIGNE P,LUSZTIG G.Representations of reductive groups over finite fields JJ.Annals of Mathematics,1976,103(1):103-161.4BONNAFE C,ROUQUIER R.Categories derivees et varietes de Deligne-Lusztig J.Publications Mathmatiques deIInstitut des Hautes tudes Scientifiques,2003,97(1):1-59.5XI N H.Some infinite dimensional representations of reductive groups with Frobenius maps J.Science China Mathemat-ics,2014,57(6):1109-1120.6YANG R T.Irreducibility of infinite dimensional Steinberg modules of reductive groups with Frobenius maps JJ.Journalof Algebra,2019,533:17-24.7CHEN X Y,DONG J B.Abstract-induced modules for reductive algebraic groups with Frobenius maps J.InternationalMathematics Research Notices,2022,2022(5):3308-3348.8CHEN X Y,DONG J B.The permutation module on flag varieties in cross characteristic J.Mathematische Zeitschrift,2019,293(1):475-484.9CHEN X Y,DONG J B.The decomposition of permutation module for infinite Chevalley groups J.Science ChinaMathematics,2021,64(5):921-930.10 CHEN X Y.Irreducible modules of reductive groups with B-stable line J.arXiv:2011.04115,2020.11 SERRE J P.Linear Representations of Finite Groups M.New York:Springer,1977.12 CARTER R W.Simple Groups of Lie Type M.London:John Wiley&Son Ltd,1972.13 PUTMAN A,SNOWDEN A.The Steinberg representation is irreducible J.Duke Mathematical Journal,2023,172(4):775-808.(责任编辑：冯珍珍）