  Let $G$ be a split connected reductive group over a non-archimedan local
field $F$. The depth zero stable Bernstein conjecture asserts that there is an
algebra isomorphism between the depth zero stable Bernstein center of $G(F)$
and the ring of functions on the moduli of tame Langlands parameters. An
approach to the depth zero stable Bernstein conjecture was proposed in the work
of Bezrukavnikov-Kazhdan-Varshavsky \cite{BKV}. In this paper we generalize
results and techniques in \cite{BKV} and apply them to give a geometric
construction of elements in the depth zero Bernstein center. We conjecture that
our construction produces all elements in the depth zero Bernstein center. An
an illustration of the method, we give a construction of an algebra embedding
from the (limit of) stable Bernstein centers for finite reductive groups to the
depth zero Bernstein center and a family of elements in the depth zero
Bernstein center coming from Deligne's epsilon factors. The paper is the first
step toward the depth zero stable Bernstein center.
