Homological properties of representations of p-adic groups related to   geometry of the group at infinity
  Geometry of buildings is used to prove some homological properties of the category of smooth representations of a reductive p-adic group (Kazhdan's "pairing conjecture", Bernstein's description of homological duality in terms of Deligne-Lusztig duality). A different proof had been obtained a little earlier by Schneider and Stuhler. 
Roman Bezrukavnikov