Abstract
Let H be a split reductive group over a local non-Archimedean field, and let \(\check {H}\) denote its Langlands dual group. We present an explicit formula for the generating function of an unramified L-function associated to a highest weight representation of the dual group, considered as a series of elements in the Hecke algebra of H. This offers an alternative approach to a solution of the same problem by Wen-Wei Li. Moreover, we generalize the notion of “Satake transform” and perform the analogous calculation for a large class of spherical varieties.
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Notes
- 1.
A “rational family” can be defined as an element of \(\operatorname {Hom} (\mathcal M(X)^K, \mathbb {C}[\tilde A_X]) \otimes _{\mathbb {C}[\tilde A_X]} \mathbb {C}(\tilde A_X)\); equivalently, it is a \(\mathbb {C}(\tilde A_X)\)-valued function on X∕K, with only a finite number of poles.
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Sakellaridis, Y. (2018). Inverse Satake Transforms. In: Müller, W., Shin, S., Templier, N. (eds) Geometric Aspects of the Trace Formula. SSTF 2016. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-319-94833-1_11
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DOI: https://doi.org/10.1007/978-3-319-94833-1_11
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