The Casselman-Shalika formula for a distinguished model

Banks, William D.

doi:10.1090/S0002-9939-1995-1219718-4

The Casselman-Shalika formula for a distinguished model
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by William D. Banks

Proc. Amer. Math. Soc. 123 (1995), 681-692

DOI: https://doi.org/10.1090/S0002-9939-1995-1219718-4

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Abstract:

Unramified Whittaker functions and their analogues occur naturally in number theory as terms in the Fourier expansions of automorphic forms. Precise information about these functions is useful in many aspects of study, such as in the construction of L-functions. In this paper, the method of Casselman and Shalika is used to derive explicit values for the analogue of the unramified Whittaker function in a distinguished model that arises in connection with quadratic base change.

References

Tetsuya Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin’s method, Math. Ann. 226 (1977), no. 1, 81–94. MR 429751, DOI 10.1007/BF01391220
W. Casselman, The unramified principal series of ${\mathfrak {p}}$-adic groups. I. The spherical function, Compositio Math. 40 (1980), no. 3, 387–406. MR 571057
W. Casselman and J. Shalika, The unramified principal series of $p$-adic groups. II. The Whittaker function, Compositio Math. 41 (1980), no. 2, 207–231. MR 581582
Jeff Hakim, Distinguished $p$-adic representations, Duke Math. J. 62 (1991), no. 1, 1–22. MR 1104321, DOI 10.1215/S0012-7094-91-06201-0
G. Harder, R. P. Langlands, and M. Rapoport, Algebraische Zyklen auf Hilbert-Blumenthal-Flächen, J. Reine Angew. Math. 366 (1986), 53–120 (German). MR 833013
Yangbo Ye, Kloosterman integrals and base change for $\textrm {GL}(2)$, J. Reine Angew. Math. 400 (1989), 57–121. MR 1013725, DOI 10.1515/crll.1989.400.57