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Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus $\tilde{T}$. Let, $$K=\tilde{G}(O)\subset G=\tilde{G}(F)$$ be a Gelfand pair of compact and locally compact groups.

We know that for all co-character $\lambda\in Hom(\mathbb{G}_m,\tilde{T})$ (assume $\lambda$ is in positive Weyl chamber), we have a decomposition: $$K\lambda(\pi)K=\coprod Kx_i.$$

My question is: Is there any way to find out what $x_i$'s are explicitly? For instance If $F=Q_p$ and $\tilde{G}=PGL_2$ we know that $x_i\in\{\begin{pmatrix}p & \\ & 1\end{pmatrix},\begin{pmatrix}1 & b\\ & p\end{pmatrix}\}$. I was wondering if one can modify the Satake transform somehow so that we can recover the $x_i$'s.

Any help or reference are highly appreciated.

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  • $\begingroup$ Perhaps implicity claiming that $K,G$ is a Gelfand pair is somewhat irrelevant and misdirecting... The rest of the question has a good sense without that implied assertion. $\endgroup$
    – paul garrett
    Commented Mar 13, 2014 at 23:16
  • $\begingroup$ I used an assumption of 'Gelfand pair' so that the Hecke algebra becomes commutative. I thought that the assumption might somehow help the question. $\endgroup$
    – Subhajit Jana
    Commented Mar 14, 2014 at 0:04
  • $\begingroup$ Ah, ok, that the associated Hecke algebra is commutative is certainly one definition... but perhaps not the only. Perhaps other readers understood better than I. $\endgroup$
    – paul garrett
    Commented Mar 14, 2014 at 0:05
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    $\begingroup$ Thought so! Here is an older paper that might have some insight. projecteuclid.org/euclid.jmsj/1230396674 $\endgroup$
    – fretty
    Commented Mar 14, 2014 at 9:43
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    $\begingroup$ And this one... dpollack.web.wesleyan.edu/papers/heckealgebras.pdf $\endgroup$
    – fretty
    Commented Mar 14, 2014 at 9:58

2 Answers 2

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$K\pi^\lambda K$ has a transitive right action of $K$. The stabilizer of $K\pi^\lambda$ for this action is $K\cap \pi^{-\lambda}K\pi^\lambda$. Thus, $K\pi^\lambda K = \coprod_x K\pi^\lambda x$ as $x$ runs over a set of representatives for $(K\cap \pi^{-\lambda}K\pi^\lambda)\backslash K$.

The subgroup $K\cap \pi^{-\lambda}K\pi^\lambda$ has a nice intersections with root subgroups, and I feel it should not be too hard to figure out its index in $K$. For $GL_n$ one can write down an explicit expression in terms of $\lambda$.

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  • $\begingroup$ Thank you Prof. Prasad. I have been trying exactly in this method, but it seems to be too brutal attack and also unable to have any result. For $GL_n$ I have been successful in this way, but not for orthogonal or symplectic. That's why I was thinking if one somehow can use Satake transform effectively. $\endgroup$
    – Subhajit Jana
    Commented Mar 14, 2014 at 8:41
  • $\begingroup$ Ralf Schmidt uses this technique to find coset representatives for GSp4 in his paper "A decomposition of the spaces $S_k(\Gamma_0(N))$ in degree 2 and the construction of hypercuspidal modular forms." found at www2.math.ou.edu/~rschmidt $\endgroup$
    – fretty
    Commented Mar 14, 2014 at 9:44
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W. Casselman's 1980 Compositio article about spherical functions, including discussions of Iwahori-fixed vectors and such, might be what you want, although part of the point there is that explicit formulas become difficult for any but spherical vectors.

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