This quantity comprises the court cases of the AMS unique consultation on Harmonic research and Representations of Reductive, $p$-adic teams, which used to be hung on January sixteen, 2010, in San Francisco, California. one of many unique guiding philosophies of harmonic research on $p$-adic teams used to be Harish-Chandra's Lefschetz precept, which urged a powerful analogy with genuine teams. From this starting, the topic has constructed a shocking number of instruments and functions. to say quite a few, Moy-Prasad's improvement of Bruhat-Tits thought relates research to staff activities on in the neighborhood finite polysimplicial complexes; the Aubert-Baum-Plymen conjecture relates the neighborhood Langlands conjecture to the Baum-Connes conjecture through a geometrical description of the Bernstein spectrum; the $p$-adic analogues of classical symmetric areas play a necessary position in classifying representations; and personality sheaves, initially built via Lusztig within the context of finite teams of Lie style, even have connections to characters of $p$-adic teams. The papers during this quantity current either expository and learn articles on those and similar themes, featuring a large photo of the present cutting-edge in $p$-adic harmonic research. The strategies are liberally illustrated with examples, often applicable for an upper-level graduate pupil in illustration thought or quantity concept. The concrete case of the two-by-two certain linear staff is a continuing touchstone.

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