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Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces

Etingof, Pavel and Oblomkov, Alexei and Rains, Eric (2007) Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces. Advances in Mathematics, 212 (2). pp. 749-796. ISSN 0001-8708. doi:10.1016/j.aim.2006.11.008.

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Let D be a simply laced Dynkin diagram of rank r whose affinization has the shape of a star (i.e., D_4, E_6, E_7, E_8). To such a diagram one can attach a group G whose generators correspond to the legs of the affinization, have orders equal to the leg lengths plus 1, and the product of the generators is 1. The group G is then a 2-dimensional crystallographic group: G=Z_ℓ⋉Z^2, where ℓ is 2, 3, 4, and 6, respectively. In this paper, we define a flat deformation H(t, q) of the group algebra C[G] of this group, by replacing the relations saying that the generators have prescribed orders by their deformations, saying that the generators satisfy monic polynomial equations of these orders with arbitrary roots (which are deformation parameters). The algebra H(t, q) for D4 is the Cherednik algebra of type C∨C_1, which was studied by Noumi, Sahi, and Stokman, and controls Askey–Wilson polynomials. We prove that H(t, q) is the universal deformation of the twisted group algebra of G, and that this deformation is compatible with certain filtrations on C[G]. We also show that if q is a root of unity, then for generic t the algebra H(t, q) is an Azumaya algebra, and its center is the function algebra on an affine del Pezzo surface. For generic q, the spherical subalgebra eH(t, q)e provides a quantization of such surfaces. We also discuss connections of H(t, q)with preprojective algebras and Painlevé VI.

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Additional Information:© 2006 Elsevier Inc. Received 27 July 2006, Accepted 22 November 2006, Available online 16 January 2007. The work of P.E. and A.O. was partially supported by the NSF grant DMS-9988796 and the CRDF grant RM1-2545-MO-03. P.E. is very grateful to M. Artin for many useful explanations about noncommutative algebraic geometry. We are also grateful to J. Starr for discussions about del Pezzo surfaces, and to W. Crawley-Boevey, A. Malkin and M. Vybornov for explanations about preprojective algebras of quivers.
Funding AgencyGrant Number
Civilian Research & Development Foundation (CRDF)RM1-2545-MO-03
Subject Keywords:Generalized double affine Hecke algebras of rank 1; Quantized del Pezzo surfaces
Issue or Number:2
Record Number:CaltechAUTHORS:20171018-155901135
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Official Citation:Pavel Etingof, Alexei Oblomkov, Eric Rains, Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces, In Advances in Mathematics, Volume 212, Issue 2, 2007, Pages 749-796, ISSN 0001-8708, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:82471
Deposited By: Tony Diaz
Deposited On:18 Oct 2017 23:37
Last Modified:15 Nov 2021 19:50

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