Cherednik algebra for the normalizer

Fallet, Henry

Comptes Rendus. Mathématique, Tome 360 (2022) no. G1, pp. 47-52.

Ginzburg, Guay, Opdam and Rouquier established an equivalence of categories between a quotient category of the category $𝒪$ for the rational Cherednik algebra and the category of finite dimension modules of the Hecke algebra of a complex reflection group $W$. We announce a generalization of this result to the extension of the Hecke algebra associated to the normalizer of a reflection subgroup.

Classification : 20C08

[1] Bellamy, Gwyn; Thiel, Ulrich Highest weight theory for finite-dimensional graded algebras with triangular decomposition, Adv. Math., Volume 330 (2018), pp. 361-419 | DOI | MR | Zbl

[2] Bonnafé, Cédric; Rouquier, Raphaël Cherednik algebras and Calogero-Moser cells (2017) (https://arxiv.org/abs/1708.09764)

[3] Broué, Michel; Malle, Gunter; Rouquier, Raphaël Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math., Volume 1998 (1998) no. 500, pp. 127-190 | MR | Zbl

[4] Cannings, Rob C.; Holland, Martin P. Differential operators on varieties with a quotient subvariety, J. Algebra, Volume 170 (1994) no. 3, pp. 735-753 | DOI | MR | Zbl

[5] Etingof, Pavel Proof of the Broué–Malle–Rouquier conjecture in characteristic zero (after I. Losev and I. Marin-G. Pfeiffer), Arnold Math. J., Volume 3 (2017) no. 3, pp. 445-449 | DOI | Zbl

[6] Etingof, Pavel; Ginzburg, Victor Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism, Invent. Math., Volume 147 (2002) no. 2, pp. 243-348 | DOI | MR | Zbl

[7] Fallet, Henry Opérateurs de Dunkl–Opdam, Catégorie $𝒪$, Algèbres de Cherednik (PhD thesis in preparation at Université Picardie Jule-Verne)

[8] Ginzburg, Victor Lectures on D-modules (1998) (Online lecture notes, available at Sabin Cautis’ webpage http://www. math. columbia. edu/~ scautis/dmodules/dmodules/ginzburg.pdf, with collaboration of Baranovsky, V. and Evens S)

[9] Ginzburg, Victor; Guay, Nicolas; Opdam, Eric; Rouquier, Raphaël On the category $𝒪$ for rational Cherednik algebras, Invent. Math., Volume 154 (2003) no. 3, pp. 617-651 | DOI | MR | Zbl

[10] Gobet, Thomas; Henderson, Anthony; Marin, Ivan Braid groups of normalizers of reflection subgroups (2020) (https://arxiv.org/abs/2002.05468, to appear in Ann. Inst. Fourier)

[11] Gobet, Thomas; Marin, Ivan Hecke algebras of normalizers of parabolic subgroups (2020) (https://arxiv.org/abs/2006.09028)

[12] Godbillon, Claude Eléments de topologie algébrique, Editions Hermann, 1971

[13] Holmes, Randall R.; Nakano, Daniel K. Brauer-type reciprocity for a class of graded associative algebras, J. Algebra, Volume 144 (1991) no. 1, pp. 117-126 | DOI | MR | Zbl

[14] Marin, Ivan Artin groups and Yokonuma–Hecke algebras, Int. Math. Res. Not., Volume 2018 (2018) no. 13, pp. 4022-4062 | MR | Zbl

[15] Rotman, Joseph J. An introduction to homological algebra, Springer, 2008