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Simple characters and coefficient systems on the building

Let F be a non-archimedean local field and G be the group GL(N,F). Let πbe a smooth complex representation of G lying in the Bernstein block B(π) of some simple type in the sense of Bushnell and Kutzko. Refining the approach of the second author and U. Stuhler, we canonically attach to πa subset X_πof the Bruhat-Tits building X of G, as well as a G-equivariant coefficient system C[π] on X_π. Roughly speaking the coefficient system is obtained by taking isotypic components of πaccording to some representations constructed from the Bushnell and Kutzko type of π. We conjecture that when πhas central character, the augmented chain complex associate to C[π] is a projective resolution of πin the category B(π). Moreover we reduce this conjecture to a technical lemma of representation theoretic nature. We prove this lemma when πis an irreducible discrete series of G. We then attach to any irreducible discrete series πof G an explicit pseudo-coefficient f_πand obtain a Lefschetz type formula for the value of the Harish-Chandra character of πat a regular elliptic element. In contrast to that obtained by U. Stuhler and the second author, this formula allows explicit character value computations.