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Whittaker functions appears in Langlands program. Recently, it is shown that some Whittaker functions can be obtained by integrating a function related to decoration over a geometric crystal in http://arxiv.org/abs/1302.0902 and http://arxiv.org/abs/1308.5451.

My questions are

(1) Why Whittaker function are useful?

(2) It seems that using geometric crystals to obtain Whittaker functions are quite explicit. What are the usual method to compute Whittaker function in number theory?

(3) If we have a Whittaker model, how could we obtain an L-function from it?

Thank you very much.

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  • $\begingroup$ Very late but in case anyone stumbles on this question: 1) Whittaker functions are useful because the Whittaker model is isomorphic to the representation but the action is easy to compute and you can do concrete computations 3) You might want to look into Rankin Selberg integrals and using Whittaker functions as "test vectors". $\endgroup$
    – Μάρκος Καραμέρης
    Commented Oct 16 at 19:28

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