Double coset decomposition of symplectic group over a quadratic extension

Question

I'm trying to understand the double coset decomposition of $G(F)\setminus G(E)/K_E$ , where $G = \mathrm{GSp}_{2n}$ is the rank $n$ group of symplectic similitudes, $E/F$ is a quadratic extension of $p$-adic fields and $K_E=G(\mathcal{O}_E)$ is the ring of integers in $E$. My knowledge of buildings is limited at present, but as I understand it, $G(E)$ acts transitively on the vertices of the affine building over $E$, and $K_E$ is the stabilizer of a hyperspecial point, so the quotient $X(E) = G(E)/K_E$ can be identified with the vertices of this building.

If this is correct, then the double coset decomposition becomes a question about orbits of $G(F)$ on $X(E)$.

What is known about the orbits of $G(F)$ on $X(E)$?

Specifically, I would like to find representatives that give some idea as to the geometry of the situation. (I.e., Sage or Matlab might be able to find coset representatives, but I would still have no idea why these are the representatives.)

(I'm reading through Garrett's book on buildings and trying to work through Tits' article in Corvallis, but any other direction for sources would be appreciated as well.)

Answer 1

This question is answered for a large number of reductive groups (including the symplectic) in P. Delorme and V. Sécherre, "An analogue of the Cartan decomposition for p-adic reductive symmetric spaces" (arXiv, MSN, published).