Depth - Zero Character Sheaves

In this paper we provide a geometric framework for the study of characters of depth-zero representations of unramified groups over local fields with finite residue fields which is built directly on Lusztig's theory of character sheaves for groups over finite fields. We introduce depth-zero character sheaves, which are coefficient systems of ℓ-adic sheaves on Bruhat-Tits buildings for p-adic groups (ℓ = p). We associate a formal sum of depth-zero character sheaves, called a model, to each supercuspidal depth-zero representation of an unramified p-adic group. Using a character formula due to Schneider-Stuhler and a fixed-point formula inétale cohomology we show that each model defines a distribution which coincides with the Harish-Chandra character of the representation on the set of regular elliptic elements. The paper includes a detailed treatment of SL(2) and Sp(4) as examples of the theory.