# Permutation group of Satake parameters

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 $$G_{p}$$?

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?"