Representations of quantum groups at a $p$-th root of unity and of semisimple groups in characteristic $p$ : independence of $p$

Representations of quantum groups at a

p

-th root of unity and of semisimple groups in characteristic

p

: independence of

p

Andersen, H. H. ; Jantzen, J. C. ; Soergel, Wolfgang

Astérisque, no. 220 (1994) , 323 p.

MR Zbl | 7 citations dans Numdam

@book{AST_1994__220__1_0,
     author = {Andersen, H. H. and Jantzen, J. C. and Soergel, Wolfgang},
     title = {Representations of quantum groups at a $p$-th root of unity and of semisimple groups in characteristic $p$ : independence of $p$},
     series = {Ast\'erisque},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {220},
     year = {1994},
     mrnumber = {1272539},
     zbl = {0802.17009},
     language = {en},
     url = {http://archive.numdam.org/item/AST_1994__220__1_0/}
}

TY  - BOOK
AU  - Andersen, H. H.
AU  - Jantzen, J. C.
AU  - Soergel, Wolfgang
TI  - Representations of quantum groups at a $p$-th root of unity and of semisimple groups in characteristic $p$ : independence of $p$
T3  - Astérisque
PY  - 1994
IS  - 220
PB  - Société mathématique de France
UR  - http://archive.numdam.org/item/AST_1994__220__1_0/
LA  - en
ID  - AST_1994__220__1_0
ER  -

%0 Book
%A Andersen, H. H.
%A Jantzen, J. C.
%A Soergel, Wolfgang
%T Representations of quantum groups at a $p$-th root of unity and of semisimple groups in characteristic $p$ : independence of $p$
%S Astérisque
%D 1994
%N 220
%I Société mathématique de France
%U http://archive.numdam.org/item/AST_1994__220__1_0/
%G en
%F AST_1994__220__1_0

Andersen, H. H.; Jantzen, J. C.; Soergel, Wolfgang. Representations of quantum groups at a $p$-th root of unity and of semisimple groups in characteristic $p$ : independence of $p$. Astérisque, no. 220 (1994), 323 p. http://numdam.org/item/AST_1994__220__1_0/

Bibliographie
Cité par

[And] H. H. Andersen, Filtrations of cohomology modules for Chevalley groups, Ann. Scient. Éc. Norm. Sup. (4) 16 (1983), 495-528

[APW1] H. H. Andersen, P. Polo, Wen K., Representations of quantum algebras, Invent. math. 104 (1991), 1-59

[APW2] H. H. Andersen, P. Polo, Wen K., Injective modules for quantum groups, Amer. J. Math. 114 (1992), 571-604

[AM] M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Reading, Mass. 1969 (Addison-Wesley)

[BGS] A. Beilinson, V. Ginzburg, W. Soergel, Koszul duality patterns in representation theory, to appear

[Ben] D. J. Benson, Representations and Cohomology I: Basic Representation Theory of Finite Groups and Associative Algebras (Cambridge Studies in Advanced Mathematics 30), Cambridge 1991 (Cambridge Univ)

[BGG1] I. N. Bernshtein, I. M. Gel'Fand, S. I. Gel'Fand, Category of $𝔤$ modules, Funct. Anal. Appl. 10 (1976), 87-92, (russ. orig.:) Об одной категории g-модулей, Функц. анализ и его прил. 10:2 (1976), 1-8

[BGG2] I. N. Bernstein, I. М. Gel'Fand, S. I. Gel'Fand, Schubert cells and cohomology of the spaces G/P, Russian Math. Surveys 28:3 (1973), 1-26, (russ. orig.:) Клетки Шуберта и когомологии пространств G/P, Успехи мат. наук 28:3, 3-26

[Bou1] N. Bourbaki, Algèbre commutative, Paris 1961/62/64/65 (Hermann)

[Bou2] N. Bourbaki, Groupes et algèbres de Lie, Chap. 4, 5 et 6, Paris 1968 (Hermann)

[Cas] L. Casian, Kazhdan-Lusztig conjecture in the negative level case (Kac-Moody algebras of affine type), preprint

[Cli] E. Cline, Simulating algebraic geometry with algebra, III: The Lusz-tig conjecture as a TG1-problem, pp. 149-161 in: P. Fong (ed.), The Arcata Conference on Representations of Finite Groups (1986) Proc. Symp. Pure Math. 47:2, Providence R. I. 1987 (Amer. Math. Soc.)

[CPS] E. Cline, B. Parshall, L. Scott, Infinitesimal Kazhdan-Lusztig theories, pp. 43-73 in: V. Deodhar (ed.), Kazhdan-Lusztig Theory and Related Topics, Proc. Chicago 1989 (Contemporary Mathematics 139), Providence, R. I. 1992 (Amer. Math. Soc.)

[CR] C. W. Curtis, I. Reiner, Methods of Representation Theory: With Applications to Finite Groups and Orders, Vol. I, (Pure and Applied Mathematics), New York etc. 1981 (Wylie)

[DCK1] C. De Concini, V. G. Kac, Representations of quantum groups at roots of 1, pp. 471-506 in: A. Connes et al. (eds.), Operator Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Colloque Dixmier), Proc. Paris 1989 (Progress in Mathematics 92), Boston etc. 1990 (Birkhäuser)

[DCK2] C. De Concini, V. G. Kac, Representations of quantum groups at roots of 1: Reduction to the exceptional case, pp. 141-149 in: A. Tsuchiya, T. Eguchi, M. Jimbo (eds.), Infinite Analysis, Part A, Proc. Kyoto 1991(Advanced Series in Mathematical Physics 16), River Edge, N. J., 1992 (World Scientific)

[Dem] M. Demazure, Invariants symétriques entiers des groupes de Weyl et torsion, Invent. math. 21 (1973), 287-301

[Don] S. Donkin, Rational Representations of Algebraic Groups (Lecture Notes in Mathematics 1140), Berlin etc. 1985 (Springer)

[FP1] E. Friedlander, B. Parshall, Modular representation theory of Lie algebras, Amer. J. Math. 110 (1988), 1055-1093

[FP2] E. Friedlander, B. Parshall, Deformations of Lie algebra representations, Amer. J. Math. 112 (1990), 375-395

[GJ] O. Gabber, A. Joseph, Towards the Kazhdan-Lusztig conjecture, Ann. Scient. Éc. Norm. Sup. (4) 14 (1981), 261-302

[GK] V. Ginzburg, S. Kumar, Cohomology of quantum groups at roots of unity, Duke Math. J. 69 (1993), 179-198

[Hu1] J. E. Humphreys, Algebraic groups and modular Lie algebras, Mem. Amer. Math. Soc. 71 (1967)

[Hu2] J. E. Humphreys, Modular representations of classical Lie algebras and semisimple groups, J. Algebra 19 (1971), 51-79

[Hu3] J. E. Humphreys, Reflection Groups and Coxeter Groups (Cambridge Studies in Advanced Mathematics 29), Cambridge 1990 (Cambridge Univ)

[Ja1] J. C. Jantzen, Über das Dekompositionsverhalten gewisser modularer Darstellungen halbeinfacher Gruppen und ihrer Lie-Algebren, J. Algebra 49 (1977), 441-469

[Ja2] J. C. Jantzen, Moduln mit einem höchsten Gewicht (Lecture Notes in Mathematics 750), Berlin etc. 1979 (Springer)

[Ja3] J. C. Jantzen, Über Darstellungen höherer Frobenius-Kerne halbeinfacher algebraischer Gruppen, Math. Z. 164 (1979), 271-292

[Ja4] J. C. Jantzen, Darstellungen halbeinfacher Gruppen und ihrer Frobenius-Kerne, J. reine angew. Math. 317 (1980), 157-199

[Ja5] J. C. Jantzen, Support varieties of Weyl modules, Bull. London Math. Soc. 19 (1987), 238-244

[Ja6] J. C. Jantzen, Representations of Algebraic Groups (Pure and Applied Mathematics 131), Orlando, Fla. 1987 (Academic Press)

[Kan] M. Kaneda, On the inverse Kazhdan-Lusztig polynomials for affine Weyl groups, J. reine angew. Math. 381 (1987), 116-135

[KT] M. Kashiwara, T. Tanisaki, Characters of the negative level highest weight modules for affine Lie algebras, preprint 1994

[Kat] S. Kato, On the Kazhdan-Lusztig polynomials for affine Weyl groups, Adv. in Math. 55 (1985), 103-130

[KL1] D. Kazhdan, G. Lusztig, Affine Lie algebras and quantum groups, Internat. Math. Res. Notices 1991, no. 2, 21-29, in: Duke Math. J. 62 (1991)

[KL2] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras I, J. Amer. Math. Soc. 6 (1993), 905-947

[KL3] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras II, J. Amer. Math. Soc. 6 (1993), 949-1011

[KL4] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras III, J. Amer. Math. Soc. (to appear)

[KL5] D. Kazhdan, G. Lusztig, Tensor structures arising from affine Lie algebras IV, preprint

[KR] A. N. Kirillov, N. Reshetikhin, $q$ -Weyl group and a multiplicative formula for universal R-matrices, Commun. Math. Phys. 134 (1990), 421-431

[KK] B. Kostant, S. Kumar, The nil Hecke-ring and cohomology of $G / P$ for a Kac-Moody group G, Adv. in Math. 62 (1986), 187-237.

[LS] S. Z. Levendorskiĭ, Ya. S. Soibelman, Some applications of the quantum Weyl groups, J. Geom. Phys. 7 (1990), 241-254

[Lu1] G. Lusztig, Some problems in the representation theory of finite Chevalley groups, pp. 313-317 in: B. Cooperstein, G. Mason (eds.), The Santa Cruz Conference on Finite Groups (1979), Proc. Symp. Pure Math. 37, Providence, R. I. 1980 (Amer. Math. Soc.)

[Lu2] G. Lusztig, Hecke algebras and Jantzen's generic decomposition patterns, Adv. in Math. 37 (1980), 121-164

[Lu3] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), 237-249

[Lu4] G. Lusztig, Modular representations and quantum groups, pp. 59-77 in: A. J. Hahn, D. G. James, Z.-X. Wan (eds.), Classical Groups and Related Topics, Proc. Beijing 1987 (Contemporary MathMathematics 82) Providence, R. I. 1989 (Amer. Math. Soc.)

[Lu5] G. Lusztig, On quantum groups, J. Algebra 131 (1990), 466-475

[Lu6] G. Lusztig, Finite dimensional Hopf algebras arising from quantized universal enveloping algebras, J. Amer. Math. Soc. 3 (1990), 257-296

[Lu7] G. Lusztig, Quantum groups at roots of 1, Geom. Dedicata 35 (1990), 89-114

[Lu8] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), 447-498

[Lu9] G. Lusztig, Canonical bases arising from quantized enveloping algebras II, Progr. Theor. Phys. Suppl. 102 (1990), 175-201

[Lu10] G. Lusztig, Introduction to Quantum Groups (Progress in Mathematics 110), Boston etc. 1993 (Birkhäuser)

[Lu11] G. Lusztig, Monodromic systems on affine flag manifolds, preprint 1993

[PW1] B. Parshall, J.-P. Wang, Quantum linear groups, Mem. Amer. Math. Soc. 439 (1991)

[PW2] B. Parshall, J.-P Wang, Cohomology of infinitesimal quantum groups I, Tôhoku Math. J. (2) 44 (1992), 395-423

[So1] W. Soergel, $𝔫$ -Cohomology of simple highest weight modules on walls and purity, Invent. math. 98 (1989), 565-580

[So2] W. Soergel, Kategorie $𝒪$ , Perverse Garben und Moduln über den Koinvarianten zur Weylgruppe, J. Amer. Math. Soc. 3 (1990), 421-445

[So3] W. Soergel, The combinatorics of Harish-Chandra bimodules, J. reine angew. Math. 429 (1992), 49-74

[VK] B. Yu. Veisfeiler, V. G. Kats, Irreducible representations of Lie $p$ -algebras, Funct. Anal. Appl. 5 (1971), 111-117, (russ. orig.:) О неприводимых представлениях р-алгебр Ли, Функц. анализ и его прил. 5:2 (1976), 28-36

[Ver] D.-N. Verma, The rôle of affine Weyl groups in the representation theory of algebraic Chevalley groups, pp. 653-705 in: I. M. Gel'fand (ed.), Lie Groups and Their Representations, Proc. Budapest 1971, London 1975 (Hilger)

[Xi1] Xi Nanhua, Representations of finite dimensional Hopf algebras arising from quantum groups, preprint (1989)

[Xi2] Xi Nanhua, Irreducible modules of quantized enveloping algebras at roots of 1, preprint (1994)