  We investigate deformations of a skew group algebra that arise from a finite
group acting on a polynomial ring. When the characteristic of the underlying
field divides the order of the group, a new type of deformation emerges that
does not occur in characteristic zero. This analogue of Lusztig's graded affine
Hecke algebra for positive characteristic can not be forged from the template
of symplectic reflection and related algebras as originally crafted by
Drinfeld. By contrast, we show that in characteristic zero, for arbitrary
finite groups, a Lusztig-type deformation is always isomorphic to a
Drinfeld-type deformation. We fit all these deformations into a general theory,
connecting Poincar\'e-Birkhoff-Witt deformations and Hochschild cohomology when
working over fields of arbitrary characteristic. We make this connection by way
of a double complex adapted from Guccione, Guccione, and Valqui, formed from
the Koszul resolution of a polynomial ring and the bar resolution of a group
algebra.
