A Construction of Elements in the Bernstein Center for Quasi-split Groups

The Bernstein center of a reductive p-adic group is the algebra of conjugation invariant distributions on the group which are essentially compact, i.e., invariant distributions whose convolution against a locally constant compactly supported function is again locally constant compactly supported. The center acts naturally on any smooth representation, and if the representation is irreducible, each element of the center acts as a scalar. For a connected reductive quasi-split group, we show certain linear combinations of orbital integrals belong to the Bernstein center. Furthermore, when these combinations are projected to a Bernstein component, they form an ideal in the Bernstein center which can be explicitly described and is often a principal ideal. The elements constructed here should have applications to various questions in harmonic analysis.