Motivic nature of character values of depth-zero representations
  It is shown that the values of Harish-Chandra distribution characters on definable compact subsets of the set of topologically unipotent elements of symplectic or special orthogonal p-adic groups can be expressed as the trace of Frobenius action on virtual Chow motives. The result is restricted to a class of depth-zero representations that can be obtained by inflation from Deligne-Lusztig representations. The proof relies on arithmetic motivic integration. 
Julia Gordon