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Let $F$ be a non-archimedean local number field, $\mathfrak{o}_F$ its ring of integers and $\mathfrak{p}_F$ its maximal ideal. Let $E$ be a quadratic unramified extension over $F$.

Let $G$ be the quasi-split unitary group in 3 variables. Consider the following sequence of open compact subgroups of $G$:

$$K_n = \left( \begin{array}{ccc} \mathfrak{o}_E & \mathfrak{o}_E & \mathfrak{p}_E^{-n} \\ \mathfrak{p}_E^n & 1 + \mathfrak{p}_E^n & \mathfrak{o}_E \\ \mathfrak{p}_E^n & \mathfrak{p}_E^n & \mathfrak{o}_E \end{array} \right) \cap G$$

I do not suceed in computing its index in $K_0$, so if anyone has a clue...

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  • $\begingroup$ Have you looked at Miyauchi's papers on U(3)? He works with these groups, and tells you something things about them, I guess enough to compute these indices. $\endgroup$
    – Kimball
    Commented Aug 9, 2017 at 22:58
  • $\begingroup$ Cross-posted at math.stackexchange.com/questions/2595405/… $\endgroup$
    – Watson
    Commented Jan 7, 2018 at 12:26

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