The Bernstein Center of the Category of Smooth W(k)[GL_n(F)]-Modules

The Bernstein center plays a role in the representation theory of locally profinite groups analogous to that played by the center of the group ring in the representation theory of finite groups. When F is a finite extension of Q_p, we discuss the Bernstein center of the category of smooth representations of GL_n(F) over the Witt vectors of an algebraically closed field of characteristic l not equal to p. We will prove results on the basic structure of the Bernstein center, and describe a conjecture that has implications for the local Langlands correspondence in algebraic families.