Heisenberg algebras and rational double affine Hecke algebras

Shan, P.¹ ; Vasserot, E.¹

¹Université Paris 7, UMR CNRS 7586, F-75013 Paris, France

Journal of the American Mathematical Society, Tome 25 (2012) no. 4, pp. 959-1031

Résumé

We relate the filtration by the support on the Grothendieck group $[\mathcal {O}]$ of the category $\mathcal {O}$ of cyclotomic rational double affine Hecke algebras to a representation-theoretic grading on $[\mathcal {O}]$, defined using the action of an affine Lie algebra and of a Heisenberg algebra on the Fock space. This implies a recent conjecture of Etingof. The proof uses a categorification of the Heisenberg action, which is new, and a categorification of the affine Lie algebra action, which was introduced by the first author in an earlier paper.

Détail

DOI : 10.1090/S0894-0347-2012-00738-3

Bibliographie

[1] Andersen, Henning Haahr, Polo, Patrick, Wen, Ke Xin Representations of quantum algebras Invent. Math. 1991 1 59

[2] Bezrukavnikov, Roman, Etingof, Pavel Parabolic induction and restriction functors for rational Cherednik algebras Selecta Math. (N.S.) 2009 397 425

[3] Berest, Yuri, Etingof, Pavel, Ginzburg, Victor Finite-dimensional representations of rational Cherednik algebras Int. Math. Res. Not. 2003 1053 1088

[4] Carter, R. W. Lie algebras of finite and affine type 2005

[5] Cherednik, Ivan Double affine Hecke algebras 2005

[6] Deligne, P. Catégories tannakiennes 1990 111 195

[7] Dipper, Richard, Donkin, Stephen Quantum 𝐺𝐿_{𝑛} Proc. London Math. Soc. (3) 1991 165 211

[8] Dipper, Richard, James, Gordon 𝑞-tensor space and 𝑞-Weyl modules Trans. Amer. Math. Soc. 1991 251 282

[9] Donkin, S. The 𝑞-Schur algebra 1998

[10] Doty, Stephen, Giaquinto, Anthony Presenting Schur algebras Int. Math. Res. Not. 2002 1907 1944

[11] Frenkel, Igor B., Jing, Naihuan, Wang, Weiqiang Vertex representations via finite groups and the McKay correspondence Internat. Math. Res. Notices 2000 195 222

[12] Ginzburg, Victor On primitive ideals Selecta Math. (N.S.) 2003 379 407

[13] Ginzburg, Victor, Guay, Nicolas, Opdam, Eric, Rouquier, Raphaël On the category 𝒪 for rational Cherednik algebras Invent. Math. 2003 617 651

[14] Gordon, I. G. Quiver varieties, category 𝒪 for rational Cherednik algebras, and Hecke algebras Int. Math. Res. Pap. IMRP 2008

[15] Jantzen, Jens Carsten Representations of algebraic groups 2003

[16] Jimbo, Michio A 𝑞-analogue of 𝑈(𝔤𝔩(𝔑+1)), Hecke algebra, and the Yang-Baxter equation Lett. Math. Phys. 1986 247 252

[17] Jimbo, Michio, Misra, Kailash C., Miwa, Tetsuji, Okado, Masato Combinatorics of representations of 𝑈_{𝑞}(̂𝔰𝔩(𝔫)) at 𝔮 Comm. Math. Phys. 1991 543 566

[18] Kac, Victor G. Infinite-dimensional Lie algebras 1990

[19] Kashiwara, Masaki, Schapira, Pierre Sheaves on manifolds 1994

[20] Kleshchev, Alexander Linear and projective representations of symmetric groups 2005

[21] Kirillov, A. N., Reshetikhin, N. 𝑞-Weyl group and a multiplicative formula for universal 𝑅-matrices Comm. Math. Phys. 1990 421 431

[22] Kumar, Shrawan Kac-Moody groups, their flag varieties and representation theory 2002

[23] Lyle, Sinéad, Mathas, Andrew Blocks of cyclotomic Hecke algebras Adv. Math. 2007 854 878

[24] Leclerc, Bernard, Miyachi, Hyohe Some closed formulas for canonical bases of Fock spaces Represent. Theory 2002 290 312

[25] Lusztig, George Introduction to quantum groups 1993

[26] Macdonald, I. G. Symmetric functions and Hall polynomials 1995

[27] Miemietz, Vanessa On representations of affine Hecke algebras of type 𝐵 Algebr. Represent. Theory 2008 369 405

[28] Nakanishi, Tomoki, Tsuchiya, Akihiro Level-rank duality of WZW models in conformal field theory Comm. Math. Phys. 1992 351 372

[29] Pressley, Andrew, Segal, Graeme Loop groups 1986

[30] Parshall, Brian, Wang, Jian Pan Quantum linear groups Mem. Amer. Math. Soc. 1991

[31] Rouquier, Raphaël 𝑞-Schur algebras and complex reflection groups Mosc. Math. J. 2008

[32] Shan, Peng Crystals of Fock spaces and cyclotomic rational double affine Hecke algebras Ann. Sci. Éc. Norm. Supér. (4) 2011 147 182

[33] Suzuki, Takeshi Double affine Hecke algebras, conformal coinvariants and Kostka polynomials C. R. Math. Acad. Sci. Paris 2006 383 386

[34] Uglov, Denis Canonical bases of higher-level 𝑞-deformed Fock spaces and Kazhdan-Lusztig polynomials 2000 249 299

[35] Varagnolo, M., Vasserot, E. Cyclotomic double affine Hecke algebras and affine parabolic category 𝒪 Adv. Math. 2010 1523 1588