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We elaborate the theory of the stable Bernstein center of a $p$-adic group$G$, and apply it to state a general conjecture on test functions for Shimuravarieties due to the author and R. Kottwitz. This conjecture provides aframework by which one might pursue the Langlands-Kottwitz method in a verygeneral situation: not necessarily PEL Shimura varieties with arbitrary levelstructure at $p$. We give a concrete reinterpretation of the test functionconjecture in the context of parahoric level structure. We also use the stableBernstein center to formulate some of the transfer conjectures (the"fundamental lemmas") that would be needed if one attempts to use the testfunction conjecture to express the local Hasse-Weil zeta function of a Shimuravariety in terms of automorphic $L$-functions.

The stable Bernstein center and test functions for Shimura varieties