Higher symplectic reflection algebras and non-homogeneous N-Koszul property

We study a class of associative algebras associated to finite groups acting on a vector space. These algebras are non-homogeneous N-Koszul algebra generalizations of symplectic reflection algebras [P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism, Invent. Math. 147 (2002) 243–348]. The Koszul property was generalized to homogeneous algebras of degree N>2 in [R. Berger, Koszulity for nonquadratic algebras, J. Algebra 239 (2001) 705–734]. In the present paper, the extension of the Koszul property to non-homogeneous algebras is realized through a PBW theorem. This PBW theorem is the generalization to the N-case of a quadratic result obtained by Braverman and Gaitsgory [A. Braverman, D. Gaitsgory, Poincaré–Birkhoff–Witt theorem for quadratic algebras of Koszul type, J. Algebra 181 (1996) 315–328] (see also [A. Polishchuk, L. Positselski, Quadratic Algebras, Univ. Lecture Ser., vol. 37, Amer. Math. Soc., Providence, RI, 2005]). We work in the general setting where the ground ring is an arbitrary von Neumann regular ring.