素特征域上半单代数群及其李代数表示中的Verma模
摘要
在本文中，我们主要研究素特征域k上连通、单连通的半单代数群G及其李代数g=Lie G表示中的Verma模．本文主要研究成果有下面几个方面：
1．当Z_0(λ)的最高权λ落在基本室C_0时，在域的特征p比较大的情况下，我们决定了Z_0(λ)的所有极大权向量．它们都是单项式形式．而对于这类“单项式形式的”极大权向量，本文给出了一个充分性的界定．
2．我们研究了非限制的广义baby Verma模的不可约性问题．我们知道当p-特征函数χ为零时，一般的广义baby Verma模不是不可约的．但当p-特征函数χ为正则幂零时，广义baby Verma模均是不可约的．当p-特征函数χ具有标准Levi型且当最高权入落在基本室C_0时，我们给出了A_n型李代数表示中的广义baby Verma模U_χ(g)(?)_(U_0(PJ))L_J(λ)不可约的充分条件．在此情况下，我们部分解决了Friedlander-Parshall所提出的相关问题．
3．我们研究了李代数表示理论中的支柱簇和秩簇理论，当p-特征函数χ是秩1时，我们证明了约化包络代数U_χ(g)与限制包络代数U_0(g)(作为左正则模)是(?)g(χ)-等变同构的，从而获得了非限制baby Verma模Z_χ(λ)和限制baby Verma模Z_0(λ)的秩簇之间的关系式：其中(?)_g(χ)={X∈g|χ([X，g])=0}．
Let G denote a connected, simply connected and semi-simple algebraic group over an algebraically closed field k of characteristic p > 0, and g = LieG be its Lie algebra. In this paper, we mainly study the Verma modules in representations of G and g. In this dissertation, The main results are listed below:
1. When the highest weightλof Z_0(λ) lie in the fundamental alcove C_0, we can determine all the maximal weight vector of Z_0(λ) and they are monomials provided that p is bigger than a certain number. For general description of such maximal weight vectors, we give a sufficient condition to judge if a maximal weight vector of a Verma module in the Dist(G)-module category becomes a maximal vector of a baby Verma module in the U_0(g)-module category.
2. We study irreducible non-restricted generalized baby Verma modules. We know that when p-characterχis zero, a baby Verma module is mostly reducible. But when p-characterχis not zero, the generalized baby Verma module may be irreducible. When the p-characterχhas standard Levi form and the highest weightλis included in the fundamental alcove C_0, we get an sufficient condition on generalized baby Verma module U_χ(g) (?)_(U0(P J)) L_J(λ) is irreducible. We partially answered an question addressed by Friedlander and Parshall in the reference [22, 5.1].
3. We study support varieties and rank varieties for g. When the rank of p-characterχis 1,we prov the reduced enveloping algebra U_χ(g) and restricted enveloping algebra U_0(g) are (?)_g(χ)-equivariant isomorphism as left regular modules, so we get the relation between the rank variety of baby Verma module Z_0(λ) and the rank variety of baby Verma module Z_χ(λ):where (?)g(χ) = {X∈g |χ([X,g]) = 0}.
1．当Z_0(λ)的最高权λ落在基本室C_0时，在域的特征p比较大的情况下，我们决定了Z_0(λ)的所有极大权向量．它们都是单项式形式．而对于这类“单项式形式的”极大权向量，本文给出了一个充分性的界定．
2．我们研究了非限制的广义baby Verma模的不可约性问题．我们知道当p-特征函数χ为零时，一般的广义baby Verma模不是不可约的．但当p-特征函数χ为正则幂零时，广义baby Verma模均是不可约的．当p-特征函数χ具有标准Levi型且当最高权入落在基本室C_0时，我们给出了A_n型李代数表示中的广义baby Verma模U_χ(g)(?)_(U_0(PJ))L_J(λ)不可约的充分条件．在此情况下，我们部分解决了Friedlander-Parshall所提出的相关问题．
3．我们研究了李代数表示理论中的支柱簇和秩簇理论，当p-特征函数χ是秩1时，我们证明了约化包络代数U_χ(g)与限制包络代数U_0(g)(作为左正则模)是(?)g(χ)-等变同构的，从而获得了非限制baby Verma模Z_χ(λ)和限制baby Verma模Z_0(λ)的秩簇之间的关系式：其中(?)_g(χ)={X∈g|χ([X，g])=0}．
Let G denote a connected, simply connected and semi-simple algebraic group over an algebraically closed field k of characteristic p > 0, and g = LieG be its Lie algebra. In this paper, we mainly study the Verma modules in representations of G and g. In this dissertation, The main results are listed below:
1. When the highest weightλof Z_0(λ) lie in the fundamental alcove C_0, we can determine all the maximal weight vector of Z_0(λ) and they are monomials provided that p is bigger than a certain number. For general description of such maximal weight vectors, we give a sufficient condition to judge if a maximal weight vector of a Verma module in the Dist(G)-module category becomes a maximal vector of a baby Verma module in the U_0(g)-module category.
2. We study irreducible non-restricted generalized baby Verma modules. We know that when p-characterχis zero, a baby Verma module is mostly reducible. But when p-characterχis not zero, the generalized baby Verma module may be irreducible. When the p-characterχhas standard Levi form and the highest weightλis included in the fundamental alcove C_0, we get an sufficient condition on generalized baby Verma module U_χ(g) (?)_(U0(P J)) L_J(λ) is irreducible. We partially answered an question addressed by Friedlander and Parshall in the reference [22, 5.1].
3. We study support varieties and rank varieties for g. When the rank of p-characterχis 1,we prov the reduced enveloping algebra U_χ(g) and restricted enveloping algebra U_0(g) are (?)_g(χ)-equivariant isomorphism as left regular modules, so we get the relation between the rank variety of baby Verma module Z_0(λ) and the rank variety of baby Verma module Z_χ(λ):where (?)g(χ) = {X∈g |χ([X,g]) = 0}.
引文
[1] Andersen H. H., On the structure of Weyl modules, Math. Z. 170 (1980), 1-14.
[2] Bala P. and Carter R., Classes of unipotent elements in simple algebraic groups I, Math. Proc. Camb. Phil. Soc. 79 (1976), 401-425.
[3] Bala P. and Carter R., Classes of unipotent elements in simple algebraic groups Ⅱ, Math. Proc. Camb. Phil. Soc. 80 (1976), 1-18.
[4] Beilinson A. , Bernstein I. N., Localization de g-modules, C. R. Acad. Sc.Paris, 202 (1981), 15-18.
[5] Beilinson A. , Bernstein I. N., Localization de g-modules, C. R. Acad. Sc.Paris, 202 (1981), 15-18.
[6] Bendel P., Generalized Reduced Enveloping Algebras for Restricted Lie Algebras, J. Algebra 218 (1999), 373-411.
[7] Bernstein I. N., Gel'fand I. M. and Gel'fand S. I., Structure of Representationgenerated by vectors of highest weight, Funct. Anal. Appl. 5 (1971), 1-9.
[8] Bernstein I. N., Gel'fand I. M. and Gel'fand S. I., Differential operators on the base affine space and a study of g-modules, in "Lie Groups and Their Representations" (I. M. Gel'fand, Ed.), pp.21-64, Hilger, London, 1975.
[9] Bernstein I. N., Gel'fand I. M. and Gel'fand S. I., A csrtain category of O-modules, (Russian) Funct. Anal. Appl. 10 (1976), 1-8.
[10] Brown K. A. and Gordon I., The ramification of centers: Lie algebra in positive characteristic and quantised enveloping algebras, Math. Zeit., 238 (2001), 733-779.
[11] Brylinski J. L. and Kashiwara M., Kazhdan-Lusztig conjecture and Holonomicsystems, Inv. Math., 64 (1981), 387-410.
[12]曹锡华 王建磐，线性代数群表示导论(上册)，科学出版社.1987．
[13] Carter R. and Lusztig G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193-242.
[14] Carter R. and Payne J., On homomorphism between Weyl modules and Specht modules, Math. Proc. Camb. Phil. Soc. 87 (1980), 419-425.
[15] Cline E., Parshall B. and Scott, L., Abstract Kazhdan-Lusztig theoreies, Tohoku Math. J. (2) 45 (1993), 511-533.
[16] Collingwood D. H., McGovern W. M., Nilpotent Orbits in Semisimple Lie Algebras, New York 1993 (Van Nostrand).
[17] Franklin J., Homomorphisms between Verma modules in characteristic p, J. Algebra 112 (1988), 58-85.
[18] Franklin J., Homomorphisms between Verma modules and Weyl modules in characteristic p, Ph.D. Thesis, Warwick 1981.
[19] Friedlander E. M. and Parshall B., Rational actions associated to the adjoint representations, Ann. scient. Ec. Norm. Sup. 20:4 (1987), 215-226.
[20] Friedlander E. M. and Parshall B., Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), 1055-1093.
[21] Friedlander E. M. and Parshall B., Geometry of p-unipotent Lie algebras, J. Algebra 109 (1987), 25-45
[22] Friedlander E. M. and Parshall B., Support varieties for restricted Lie algebra, Invent, math. 86 (1986),553-562
[23] Friedlander E. M. and Parshall B., Deformations of Lie algebra representations,Amer. J. Math. 112(1990), 375-395.
[24] Gerstenhaber M., Dominance over the classical groups, Ann. of Math. 74 (1961), 532-569.
[25] Humphreys J., Introduction to Lie algebras and their representations,
[26] Gordon I. and Premet A., Block representation type of reduced envelopingalgebras, Trans. Amer. Math. Soc. 354 (2001), 549-1581.
[27] Humphreys J., Modular representations of classical Lie algebras and semisimple groups, J. Algebra 19 (1971), 51-79.
[28] Humphreys J., Ordinary and Modular Representations of Chevalley Groups, Lecture notes in Mathematics 528. 1976.
[29] Irving R. S., The structure of certain highest weight modules for SL_3, J. Algebra 99 (1986), 438-457.
[30] Irving R. S., Projective modules in the category O_s: self-duality, Trans. Amer. Math. soc. 291 (1985),701-732.
[31] Jantzen J. C., Kohomologie Von p-Lie-Algebra und nilpotente Elemente,Abh. Math. Sem. Univ. Hamburg 56 (1986), 191-219.
[32] Jantzen J. C, Uber Darstellungen hoherer Frobenius-Kerne halbein-facher algebraischer Gruppen, Math. Z., 164(1979), 271-292.
[33] Jantzen J. C, Representations of Lie algebras in prime characteristic, in ; A. Broer (Ed. ), Representation Theories and Algebraic Geometry,??Proceedings, Montreal, 1997, in: NATO ASI Series, Vol. C514, Kluwer, Dordrecht, 1998, pp. 185-235.
[34] Jantzen J. C, Subregular nilpotent representations of sl_n and so_(2n+1), Math. Prop. Cambridge Philos Soc. 126 (1999) 223-257.
[35] Jantzen J. C, Representations of so_5 in prime characteristic, preprint July 1997 (Aarhus series 1997:13)
[36] Jantzen J. C, Modular representations of reductive Lie algebras, Journal of Pure and Applied Algebra 152 (2000), 133-185.
[37] Jantzen J. C, Reresentations of Alagebraic Groups, 2nd edn, American Mathematical Society, Providence, RI, 2003.
[38] Jantzen J. C, Nilpotent Orbits in representation Theory, in Lie Theory Lie Algebras and Representations (2003).
[39] Kaneda M., Extensions of modules for infinitesimal algebraic groups, J. Algebra, 122 (1989), 188-210.
[40] Kazhdan D. Lusztig G., Representations of Coxeter groups and Hecke algebras, Inv. Math., 53 (1979), 165-184.
[41] Koppinen M. Homomorphism between neighbouring Weyl modules, J. Algebra 103 (1986), 302-319.
[42] Lepowsky J., Generalized Verma modules, the Cartan Helgason theorem and the Harish Chandra homomorphism, J. Algebra 49 (1977), 470-495.
[43]Li Yiyang(李宜阳)and Shu Bin(舒斌)，Filtrations in Modular Representationsof Reductive Lie algebras, to appear in Algebra Colloquium.
[44] Lusztig G.,Some problems in the representation theory of finite Chevalley groups, Proc.Symp.Pure Math.37, Amer.Math.Soc. 1980, 313-317.
[45] Lusztig G.,Periodic W-graphs, Represent.Th.1(1997),207-279 (electronic).
[46] Mil'ner A., Maximal degree of irreducible Lie algebra representations over a field of positive characteristic, Funct. Anal. Appl. 14:2 (1980), 67-68.
[47] Mirkovic I. and Rumynin D., centers of reduced enveloping algebras, Math. Zeit. 231 (1999), 123-132.
[48] Premet A., Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture, Invent. Math. 121 (1995), 79-117.
[49] Premet A., An analogue of the Jacobson-Morozov theorem for Lie algebra of reductive groups, Trans. Amer. Math. Soc. 347 (1995), 2961-2988.
[50] Premet A., Support varities of non-restricted modules over Lie algebras of reductive groups, J. London Math. Soc. (2) 55 (1997), 236-250
[51] Premet A., Complexity of Lie algebras and nilpotent elements of the stabilizers of linear forms, Math. Zeit. 228 (1998), 255-282.
[52] Shapovalov N. N., On a bilinear form on the universal enveloping algebra of complex semisimple Lie algebra, Funct. Anal. Appl. 6 (1972), 307-312.
[53] Springer T. and Steinberg R., Conjugacy Classes, in Seminar on Algebraic Groups and Related Finite Groups, Lecture Note in Math. 131, Springer-Verlag, NewYouk, 1970, 167-266.
[54] Veisfeiler B. Yu. and Kats V. G., Irreducible representations of Lie palgebras,Funct. Anal. Appl. 5 (1971), 111-117.
[55] Verma D.-N., Structure of Cerntain Induced Representationof Complex Semisimple Lie Algebras, Bull. Amer. Math. Soc. 74 (1968), 160-168.
[56]Wang Jianpan(王建磐)，Sheaf cohomology of G/B and tensor products of Weyl Modules, Journal of Algebra 77 (1982), 162-185.
[57]Xi Nanhua(席南华)，Irreducible Modules of Quantized Enveloping Algebras at Roots of 1, Publications of the Research Institute for MathematicsSciences 32 (1996), 276-352.
[58]Xi Nanhua(席南华)，Maximal and Primitive Elements in Weyl Modules for Type A_2, Journal of Algebra 215 (1999), 735-756.
[59]Xi Nanhua(席南华)，Some Maximal Elements in A Baby Verma Module,The 4th International Coference on Representations Theory.
[60]Ye Jiachen(叶家琛)，Filtrations of principal indecomposable modules of Frobenius kernels of reductive groups, Math. Zeit. 189 (1985), 515-527.
[2] Bala P. and Carter R., Classes of unipotent elements in simple algebraic groups I, Math. Proc. Camb. Phil. Soc. 79 (1976), 401-425.
[3] Bala P. and Carter R., Classes of unipotent elements in simple algebraic groups Ⅱ, Math. Proc. Camb. Phil. Soc. 80 (1976), 1-18.
[4] Beilinson A. , Bernstein I. N., Localization de g-modules, C. R. Acad. Sc.Paris, 202 (1981), 15-18.
[5] Beilinson A. , Bernstein I. N., Localization de g-modules, C. R. Acad. Sc.Paris, 202 (1981), 15-18.
[6] Bendel P., Generalized Reduced Enveloping Algebras for Restricted Lie Algebras, J. Algebra 218 (1999), 373-411.
[7] Bernstein I. N., Gel'fand I. M. and Gel'fand S. I., Structure of Representationgenerated by vectors of highest weight, Funct. Anal. Appl. 5 (1971), 1-9.
[8] Bernstein I. N., Gel'fand I. M. and Gel'fand S. I., Differential operators on the base affine space and a study of g-modules, in "Lie Groups and Their Representations" (I. M. Gel'fand, Ed.), pp.21-64, Hilger, London, 1975.
[9] Bernstein I. N., Gel'fand I. M. and Gel'fand S. I., A csrtain category of O-modules, (Russian) Funct. Anal. Appl. 10 (1976), 1-8.
[10] Brown K. A. and Gordon I., The ramification of centers: Lie algebra in positive characteristic and quantised enveloping algebras, Math. Zeit., 238 (2001), 733-779.
[11] Brylinski J. L. and Kashiwara M., Kazhdan-Lusztig conjecture and Holonomicsystems, Inv. Math., 64 (1981), 387-410.
[12]曹锡华 王建磐，线性代数群表示导论(上册)，科学出版社.1987．
[13] Carter R. and Lusztig G., On the modular representations of the general linear and symmetric groups, Math. Z. 136 (1974), 193-242.
[14] Carter R. and Payne J., On homomorphism between Weyl modules and Specht modules, Math. Proc. Camb. Phil. Soc. 87 (1980), 419-425.
[15] Cline E., Parshall B. and Scott, L., Abstract Kazhdan-Lusztig theoreies, Tohoku Math. J. (2) 45 (1993), 511-533.
[16] Collingwood D. H., McGovern W. M., Nilpotent Orbits in Semisimple Lie Algebras, New York 1993 (Van Nostrand).
[17] Franklin J., Homomorphisms between Verma modules in characteristic p, J. Algebra 112 (1988), 58-85.
[18] Franklin J., Homomorphisms between Verma modules and Weyl modules in characteristic p, Ph.D. Thesis, Warwick 1981.
[19] Friedlander E. M. and Parshall B., Rational actions associated to the adjoint representations, Ann. scient. Ec. Norm. Sup. 20:4 (1987), 215-226.
[20] Friedlander E. M. and Parshall B., Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), 1055-1093.
[21] Friedlander E. M. and Parshall B., Geometry of p-unipotent Lie algebras, J. Algebra 109 (1987), 25-45
[22] Friedlander E. M. and Parshall B., Support varieties for restricted Lie algebra, Invent, math. 86 (1986),553-562
[23] Friedlander E. M. and Parshall B., Deformations of Lie algebra representations,Amer. J. Math. 112(1990), 375-395.
[24] Gerstenhaber M., Dominance over the classical groups, Ann. of Math. 74 (1961), 532-569.
[25] Humphreys J., Introduction to Lie algebras and their representations,
[26] Gordon I. and Premet A., Block representation type of reduced envelopingalgebras, Trans. Amer. Math. Soc. 354 (2001), 549-1581.
[27] Humphreys J., Modular representations of classical Lie algebras and semisimple groups, J. Algebra 19 (1971), 51-79.
[28] Humphreys J., Ordinary and Modular Representations of Chevalley Groups, Lecture notes in Mathematics 528. 1976.
[29] Irving R. S., The structure of certain highest weight modules for SL_3, J. Algebra 99 (1986), 438-457.
[30] Irving R. S., Projective modules in the category O_s: self-duality, Trans. Amer. Math. soc. 291 (1985),701-732.
[31] Jantzen J. C., Kohomologie Von p-Lie-Algebra und nilpotente Elemente,Abh. Math. Sem. Univ. Hamburg 56 (1986), 191-219.
[32] Jantzen J. C, Uber Darstellungen hoherer Frobenius-Kerne halbein-facher algebraischer Gruppen, Math. Z., 164(1979), 271-292.
[33] Jantzen J. C, Representations of Lie algebras in prime characteristic, in ; A. Broer (Ed. ), Representation Theories and Algebraic Geometry,??Proceedings, Montreal, 1997, in: NATO ASI Series, Vol. C514, Kluwer, Dordrecht, 1998, pp. 185-235.
[34] Jantzen J. C, Subregular nilpotent representations of sl_n and so_(2n+1), Math. Prop. Cambridge Philos Soc. 126 (1999) 223-257.
[35] Jantzen J. C, Representations of so_5 in prime characteristic, preprint July 1997 (Aarhus series 1997:13)
[36] Jantzen J. C, Modular representations of reductive Lie algebras, Journal of Pure and Applied Algebra 152 (2000), 133-185.
[37] Jantzen J. C, Reresentations of Alagebraic Groups, 2nd edn, American Mathematical Society, Providence, RI, 2003.
[38] Jantzen J. C, Nilpotent Orbits in representation Theory, in Lie Theory Lie Algebras and Representations (2003).
[39] Kaneda M., Extensions of modules for infinitesimal algebraic groups, J. Algebra, 122 (1989), 188-210.
[40] Kazhdan D. Lusztig G., Representations of Coxeter groups and Hecke algebras, Inv. Math., 53 (1979), 165-184.
[41] Koppinen M. Homomorphism between neighbouring Weyl modules, J. Algebra 103 (1986), 302-319.
[42] Lepowsky J., Generalized Verma modules, the Cartan Helgason theorem and the Harish Chandra homomorphism, J. Algebra 49 (1977), 470-495.
[43]Li Yiyang(李宜阳)and Shu Bin(舒斌)，Filtrations in Modular Representationsof Reductive Lie algebras, to appear in Algebra Colloquium.
[44] Lusztig G.,Some problems in the representation theory of finite Chevalley groups, Proc.Symp.Pure Math.37, Amer.Math.Soc. 1980, 313-317.
[45] Lusztig G.,Periodic W-graphs, Represent.Th.1(1997),207-279 (electronic).
[46] Mil'ner A., Maximal degree of irreducible Lie algebra representations over a field of positive characteristic, Funct. Anal. Appl. 14:2 (1980), 67-68.
[47] Mirkovic I. and Rumynin D., centers of reduced enveloping algebras, Math. Zeit. 231 (1999), 123-132.
[48] Premet A., Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture, Invent. Math. 121 (1995), 79-117.
[49] Premet A., An analogue of the Jacobson-Morozov theorem for Lie algebra of reductive groups, Trans. Amer. Math. Soc. 347 (1995), 2961-2988.
[50] Premet A., Support varities of non-restricted modules over Lie algebras of reductive groups, J. London Math. Soc. (2) 55 (1997), 236-250
[51] Premet A., Complexity of Lie algebras and nilpotent elements of the stabilizers of linear forms, Math. Zeit. 228 (1998), 255-282.
[52] Shapovalov N. N., On a bilinear form on the universal enveloping algebra of complex semisimple Lie algebra, Funct. Anal. Appl. 6 (1972), 307-312.
[53] Springer T. and Steinberg R., Conjugacy Classes, in Seminar on Algebraic Groups and Related Finite Groups, Lecture Note in Math. 131, Springer-Verlag, NewYouk, 1970, 167-266.
[54] Veisfeiler B. Yu. and Kats V. G., Irreducible representations of Lie palgebras,Funct. Anal. Appl. 5 (1971), 111-117.
[55] Verma D.-N., Structure of Cerntain Induced Representationof Complex Semisimple Lie Algebras, Bull. Amer. Math. Soc. 74 (1968), 160-168.
[56]Wang Jianpan(王建磐)，Sheaf cohomology of G/B and tensor products of Weyl Modules, Journal of Algebra 77 (1982), 162-185.
[57]Xi Nanhua(席南华)，Irreducible Modules of Quantized Enveloping Algebras at Roots of 1, Publications of the Research Institute for MathematicsSciences 32 (1996), 276-352.
[58]Xi Nanhua(席南华)，Maximal and Primitive Elements in Weyl Modules for Type A_2, Journal of Algebra 215 (1999), 735-756.
[59]Xi Nanhua(席南华)，Some Maximal Elements in A Baby Verma Module,The 4th International Coference on Representations Theory.
[60]Ye Jiachen(叶家琛)，Filtrations of principal indecomposable modules of Frobenius kernels of reductive groups, Math. Zeit. 189 (1985), 515-527.