Are all these representations supercuspidal

Question

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.

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?)

I am interested in the possibility of using matrix coefficients for selecting representations.

Answer 1

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.