A review of conjectures due to Drinfeld, Beilinson, Gaitsgory et al. and of results of Gaitsgory on the quantum Langlands correspondence.
Un survol des conjectures de Drinfeld, Beilinson, Gaitsgory et al. et de résultats de Gaitsgory sur la correspondance de Langlands quantique.
@article{AFST_2014_6_23_1_129_0,
author = {Vadim Schechtman},
title = {Dualit\'e de {Langlands} quantique},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {129--158},
publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
address = {Toulouse},
volume = {6e s{\'e}rie, 23},
number = {1},
year = {2014},
doi = {10.5802/afst.1400},
mrnumber = {3204734},
zbl = {06293506},
language = {fr},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1400/}
}
TY - JOUR AU - Vadim Schechtman TI - Dualité de Langlands quantique JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2014 SP - 129 EP - 158 VL - 23 IS - 1 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1400/ DO - 10.5802/afst.1400 LA - fr ID - AFST_2014_6_23_1_129_0 ER -
%0 Journal Article %A Vadim Schechtman %T Dualité de Langlands quantique %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2014 %P 129-158 %V 23 %N 1 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1400/ %R 10.5802/afst.1400 %G fr %F AFST_2014_6_23_1_129_0
Vadim Schechtman. Dualité de Langlands quantique. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 1, pp. 129-158. doi : 10.5802/afst.1400. https://afst.centre-mersenne.org/articles/10.5802/afst.1400/
[1] Arinkin (D.), Gaitsgory (D.).— Singular support of coherent sheaves, and the geometric Langlands conjecture, arXiv :1201.6343.
[2] Beilinson (A.), Bernstein (J.).— A proof of Jantzen conjectures, I.M.Gelfand Seminar, Adv. Soviet Math. (AMS), 16, Part 1, p. 1-51 (1993). | MR | Zbl
[3] Beilinson (A.), Drinfeld (V.).— Quantization of Hitchin integrable system and Hecke eigensheaves.
[4] Beilinson (A.), Schechtman (V.).— Determinant bundles and Virasoro algebras, Comm. Math. Phys., 118, p. 651-701 (1988). | MR | Zbl
[5] Belavin (A.), Polyakov (A.), Zamolodchikov (A.).— Infinite conformal symmetry in 2D quantum field theory, Nucl. Phys. B241, p. 333-380 (1984). | Zbl
[6] Berezin (F.A.), Gelfand (I.M.).— Quelques remarques sur la théorie des fonctions sphériques sur les variétés symétriques riemanniennes (russe), Trudy MMO (Travaux de la Société Mathématique de Moscou) 5, p. 311-351 (1956). | MR | Zbl
[7] Braverman (A.), Gaitsgory (D.).— Geometric Eisenstein series, Invent. Math. 150, p. 287-384 (2002). | MR | Zbl
[8] Bezrukavnikov (R.), Finkelberg (M.), Schechtman (V.).— Factorizable sheaves and quantum groups, Lect. Notes in Math. 1691, Springer-Verlag, Berlin (1998). | MR | Zbl
[9] Finkelberg (M.), Mirkovich (I.).— Semiinfinite flags. I. Case of global curve ; Feigin (B.), Finkelberg (M.), Kuznetsov (A.), Mirkovich (I.), Semiinfinite flags. II. Local and global intersection cohomology of quasimaps spaces, dans : Differential topology, infinite-dimensional Lie algebras, and applications, p. 81-112 et p. 113-148, AMS Transl. Ser. 2, 194, Providence, RI (1999). | MR | Zbl
[10] Frenkel (E.), Ben-Zvi (D.).— Vertex algebras and algebraic curves, 2nd Edition, Math. Surveys and Monographs 88, AMS, Providence, RI (2004). | MR | Zbl
[11] Frenkel (E.), Gaitsgory (D.), Vilonen (K.).— Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. Math. 153, p. 699-748 (2001). | MR | Zbl
[12] Frenkel (E.), Gaitsgory (D.), Vilonen (K.).— On the geometric Langlands conjecture, JAMS 15, p. 367-417 (2001). | MR | Zbl
[13] Gaitsgory (D.).— Twisted Whittaker models and factorizable sheaves, Selecta Math. (N.S.) 13, p. 617-659 (2008). | MR | Zbl
[14] Gaitsgory (D.).— On a vanishing conjecture appearing in the Geometric Langlands correspondence, Ann. Math. 160, p. 617-682 (2004). | MR | Zbl
[15] Ginzburg (V.).— Perverse sheaves on a loop group and Langlands duality, arXiv :alg-geom/2511007. | MR
[16] Helgason (S.).— Groups and geometric analysis, Academic Press (1984). | MR | Zbl
[17] Kazhdan (D.), Lusztig (G.).— Tensor structures arising from affine Lie algebras I-IV JAMS, 6, p. 905-947, p. 949-1011 (1993) ; 7, p. 335-381, p. 383-453 (1994). | MR | Zbl
[18] Lafforgue (L.).— Chtoukas de Drinfeld et correspondance de Langlands, Inv. Math. 147, p. 1-241 (2002). | MR | Zbl
[19] Laumon (G.).— Transformation de Fourier généralisée, arXiv :alg-geom/9603004.
[20] Maaß(H.).— Über eine neue Art von nichtanalytichen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 21, p. 141-183 (1949). | MR | Zbl
[21] Macdonald (I.G.).— Spherical functions on a -adic Chevalley group, Bull. Amer. Math. Soc. Vol. 74, 3, p. 520-525 (1968). | MR | Zbl
[22] Mirković (I.), Vilonen (K.).— Geometric Langlands duality and representations of algebraic groups over commutative rings, Ann. of Math. (2) 166, p. 95-143 (2007). | MR | Zbl
[23] Satake (I.).— Theory of spherical functions on reductive algebraic groups over -adic fields, Publ. Math. IHES 18, p. 5-69 (1963). | Numdam | MR | Zbl
[24] Stoyanovsky (A.).— On quantization of the geometric Langlands correspondence, arXiv :math/9911108.
[25] Stoyanovsky (A.).— Quantum Langlands duality and Conformal field theory, arXiv :math/0610974.
Cited by Sources: