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Spherical functions on spherical varieties

From: American Journal of Mathematics
Volume 135, Number 5, October 2013
pp. 1291-1381 | 10.1353/ajm.2013.0046



Let ${\bf X}={\bf H}\backslash{\bf G}$ be a homogeneous spherical variety for a split reductive group ${\bf G}$ over the integers $\frak o$ of a $p$-adic field $k$, and $K={\bf G}(\frak o)$ a hyperspecial maximal compact subgroup of $G={\bf G}(k)$. We compute eigenfunctions (``spherical functions'') on $X={\bf X}(k)$ 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 ${\bf X}$ 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 $\pi$ is non-zero on $\pi^K$), (2) the unramified Plancherel formula for $X$, including a formula for the ``Tamagawa measure'' of ${\bf X}(\frak o)$, and (3) a computation of the most continuous part of ${\bf H}$-period integrals of principal Eisenstein series.