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Let $L(s)=\prod_{p}L_{p}(s)$ the Euler product of an L-function in the relevant right half-plane. As $ L_{p}(s)=\prod_{j=1}^{d}(1-\alpha_{j}(p)p^{-s} )^{-1}$, the permutation group $ G_{p}$ of the Satake parameters $\alpha_{j}(p) $ preserves $ L_{p} $. Which results are known about the general structure of $ G_{p}$?

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This is a silly question. The Satake paramaters $(\alpha_1(p),\ldots,\alpha_d(p))$ can be more precisely thought of as a conjugacy class in $\mathrm{GL}_n(\mathbb{C})$. In particular, the Satake parameters are, by definition, only defined up to permutations: really, you should think of $(\alpha_1(p),\ldots,\alpha_d(p))$ being the same Satake parameters as $(\alpha_{\sigma(1)}(p),\ldots,\alpha_{\sigma(d)}(p))$ (this makes a lot more sense at the level of representations of $\mathrm{GL}_n$ over local fields and the Satake isomorphism, but that would take us too far afield).

So you are just asking "what is $S_n$, the group of permutations on $n$-letters?"

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