Let $D$ a division quaternion algebra over a number field $F$, and consider $(V,q)$ be a $D$-hermitian space of $D$-dimension $2$, and introduce its group of isometries
\begin{align\*}
\mathrm{GU}(V, q) & = \left\{ g \in GL(V) \ : \forall x, y \in V, q(gx,gy) = q(x,y)
 \right\}
\end{align\*}


These groups are known to be the inner forms of $GSp(4)$ over $F$. Moreover, they have compact automorphic quotient if and only if $D$ is ramified at a certain real place and $q$ is positive-definite or negative-definite (since anisotropy is sufficient at one place to have global anisotropy, by the local-global principle for quadratic forms).


Let take such an inner form with compact automorphic quotient. For every finite place $p$, there is a unique hermitian space giving the unique non-trivial inner form of $\mathrm{GSp(4)}$ ovet $F\_p$, and it is isotropic.



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> What can be said about the representations of this unique non-trivial
>  inner form? Are they all supercuspidal? (or: is this group compact
>  modulo the center?)
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I am interested in the possibility of using matrix coefficients for selecting representations.



Are all these representations supercuspidal
First of all, you will only get a non-trivial inner form at finitely many places (where $D$ ramifies). Even then, the group is not compact mod center, so not all representations will be supercuspidal.


However, the representations have been classified using the theta correspondence:



> 
> Wee Teck Gan and Welly Tantono, The local Langlands conjecture for GSp(4) II: the case of inner forms, American J. of Math 136 (2014), no. 3, 761-805.
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