Abstract: We give a description of the depth-r Bernstein center for non-negative integers r for a connected reductive group G over a non-archimedean local field, as a limit of depth-r standard parahoric Hecke algebras. Using this description, we produce elements of the depth-r center by constructing a map to the depth- r center from a certain subalgebra ( which we call stable functions) of the algebra of invariant functions on the r-th Moy Prasad filtration quotient of hyperspecial parahorics. We use these elements to attach an invariant to each depth-r irreducible representation, and show that this is equal to the semi-simple part of minimal K-types contained in the irreducible representation. This talk is based on a joint work with Tsao-Hsien Chen.