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It is a convention to say Kloosterman sums and Bessel functions are similar.

There are papers talking about Bessel functions on $p$-adic group (associated with a representation) such as Baruch's: http://www2.math.technion.ac.il/~embaruch/GLn/func.pdf

But I did not see why it is related to classical Kloosterman sums even in the case of $\mathrm{GL}(2,\mathbb Q_p)$.

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    $\begingroup$ It is explained e. g. by Sarnak in Kloosterman, quadratic forms and modular forms (see the text around the display (16) there) $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 9, 2017 at 6:54
  • $\begingroup$ The one you mentioned by Sarnak is not very satisfying. I am still seeking angle from representation theory, such as Baruch's paper. $\endgroup$
    – 7-adic
    Commented Mar 9, 2017 at 7:15
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    $\begingroup$ Were you able to calculate it and did not see how your calculation was related to Kloosterman sums, or were you unable to calculate it? $\endgroup$
    – Will Sawin
    Commented Mar 9, 2017 at 7:35
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    $\begingroup$ I don't know which corresponds to which. I am confused because the Bessel function on GL(2,Q_p) is associated with a representation whereas the Kloosterman sum only depends on three integer numbers (not Satake parameters). $\endgroup$
    – 7-adic
    Commented Mar 10, 2017 at 6:17
  • $\begingroup$ This is kinda late, but for a generic principal series representation of $\mathrm{GL}_2\left( \mathbb{F}_q \right)$, induced from the characters $\alpha_1,\alpha_2$, we get $J_{\pi, \psi} \pmatrix{ & 1 \\ a& } = q^{-1} \sum_{ \substack{x \in \mathbb{F}_q \\ xy=-a^{-1}} } \alpha_1^{-1}(x) \alpha_2^{-1}(y) \psi(x+y)$, which is a Kloosterman sum. These sort of formulas are given in Piatetski-Shapiro's book "Complex Representations of $GL(2,K)$ for Finite Fields $K$" (see page 61). $\endgroup$
    – darkl
    Commented Nov 29, 2021 at 23:41

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