球面簇上的球面函数
Spherical functions on spherical varieties
摘要：
设X=H \G是p进域k的整数o上可裂约化群G的齐次球面簇，K=G（o）是G=G（k）的超特殊极大紧子群.我们在G的非分歧（或球形）Hecke代数的作用下计算X=X（k）上的本征函数（“球形函数”），推广了“Casselman-Shalika”类型的许多经典结果。在X上的一些附加假设下，我们还证明了公式的一个变体，它涉及L值的某个商，并且我们提出了几个应用，例如：（1）关于无重数情况下“好测试向量”的说明（即，不可约非分歧表示π上的H不变泛函在π^K上非零），（2）X的非分歧Plancherel公式，包括X（o）的“Tamagawa测度”公式;（3）主Eisenstein级数H-周期积分最连续部分的计算。
英文摘要：
Let X=H\G be a homogeneous spherical variety for a split reductive group G over the integers o of a p-adic field k, and K=G(o) a hyperspecial maximal compact subgroup of G=G(k). We compute eigenfunctions ("spherical functions") on X=X(k) under the action of the unramified (or spherical) Hecke algebra of G, generalizing many classical results of "Casselman-Shalika" type. Under some additional assumptions on X we also prove a variant of the formula which involves a certain quotient of L-values, and we present several applications such as: (1) a statement on "good test vectors" in the multiplicity-free case (namely, that an H-invariant functional on an irreducible unramified representation πis non-zero on π^K), (2) the unramified Plancherel formula for X, including a formula for the "Tamagawa measure" of X(o), and (3) a computation of the most continuous part of H-period integrals of principal Eisenstein series.
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