By Robert S. Doran, Paul J., Jr. Sally, Loren Spice

ISBN-10: 0821849859

ISBN-13: 9780821849859

This quantity includes the lawsuits of the AMS designated 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 instructed a robust analogy with genuine teams. From this starting, the topic has built a shocking number of instruments and purposes. to say quite a few, Moy-Prasad's improvement of Bruhat-Tits idea relates research to team 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 an important position in classifying representations; and personality sheaves, initially constructed via Lusztig within the context of finite teams of Lie sort, even have connections to characters of $p$-adic teams. The papers during this quantity current either expository and study articles on those and similar themes, proposing a extensive photo of the present cutting-edge in $p$-adic harmonic research. The suggestions are liberally illustrated with examples, frequently acceptable for an upper-level graduate scholar in illustration conception or quantity thought. The concrete case of the two-by-two precise linear workforce is a continuing touchstone

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Springer, A purity result for ﬁxed point varieties in ﬂag manifolds, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 31 (1984), no. 2, 271–282. Tonny A. Springer and Robert Steinberg, Conjugacy classes, Seminar on Algebraic Groups and Related Finite Groups. (Held at The Institute for Advanced Study, Princeton, NJ, 1968/69) (A. Borel and R. ), Lecture Notes in Mathematics, vol. 131, Springer– Verlag, Berlin, 1970, pp. 167–266. Tonny A. Springer, Linear Algebraic Groups, Second Edition, Modern Birkh¨ auser Classics, Birkh¨ auser, Boston, 2009.

G(Λ ) := q −1/2 Λ(− )r (X 2 ). 4], Waldspurger works with an additive character ψ of depth 0. 4]. 2], where our G(Λ ) is denoted by G (Φ). 2. measdΛ z (Rθ ) = and H(Λ , kθ ) = ⎧ ⎨q r+1 , θ= ⎩ θ= q r+1/2 , ⎧ ⎨(−1)r+1 , θ= ⎩ θ= G(Λ ), 31 13 ; . 3], this formula agrees with the one given in [42, p. 1235]. Proof. Note that H(Λ , kθ ) is just what is called H(Λ , Q) in [46, p. 11], where Q = Normθ is the norm form on kθ . 2 loc. cit. shows that the choice does not matter). For deﬁniteness, we take the lattice to be Rθ .

In the ﬁrst example above, X is nilpotent, so the corresponding coset does not correspond to an unreﬁned minimal K-type. , the ﬁltration associated to the point xC ). If r ∈ Z, then every coset in the quotient space gxC ,−r /gxC ,(−r)+ has the form −r Xβsplit +gxC ,(−r)+ . As before, we may take β to lie in f× . Similarly, if r ∈ Z+ 12 , then every coset in the quotient space gxC ,−r /gxC ,(−r)+ either contains a nilpotent element or has the form (up to conjugation by GxC ) − r Xβθ,η + gxC ,(−r)+ , where θ ∈ { , } and η ∈ {1, }.

### Harmonic Analysis on Reductive, p-adic Groups by Robert S. Doran, Paul J., Jr. Sally, Loren Spice

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