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We define new deformations of group algebras of Coxeter groups  W  and of subgroups of even elements in them by deforming the braid relations. We show that these deformations are algebraically flat if and only if they are formally flat, and that this happens if and only if the group  W  has no finite parabolic subgroups of rank 3. We explain the connection of our deformations with the Hecke algebras of orbifolds defined by the first author and with generalized double affine Hecke algebras defined by the authors and A. Oblomkov.

New deformations of group algebras of Coxeter groups