  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 and uses ideas due to Schneider-Stuhler.
Specifically, we introduce a class of coefficient systems on Bruhat-Tits
buildings of perverse sheaves sheaves on affine algebraic groups over an
algebraic closure of a finite field and to each supercuspidal depth-zero
representation of an unramified $p$-adic group we associate a formal sum of
these coefficient systems, called a model for the representation. Then, using a
character formula due to Schneider-Stuhler and a fixed-point formula in etale
cohomology we show that each model defines a distribution which coincides with
the Harish-Chandra character of the corresponding representation, on the set of
regular elliptic elements. The paper includes a detailed treatment of SL(2),
Sp(4) and GL(n) as examples of the theory.
