Motivic Nature of Character Values of Depth-zero Representations

In the present paper, it is shown that the values of HarishChandra distribution characters on definable compact subsets of the set of topologically unipotent elements of some reductive p-adic groups can be expressed as the trace of Frobenius action on certain geometric objects, namely, Chow motives. The result is restricted to a class of depth-zero representations of symplectic or special orthogonal groups that can be obtained by inflation from Deligne-Lusztig representations. The proof works both in positive and zero characteristic, and relies on arithmetic motivic integration.