Ext-distinction for $p$-adic symmetric spaces

Let $G/H$ be a $p$-adic symmetric space. We compute explicitly the higher relative extension groups for all discrete series representations of $G$ in two examples: the symplectic case and the linear case. The results have immediate applications to the computation of the Euler-Poincar\'e pairing, the alternating sum of the dimensions of the Ext-groups. In the linear case we confirm a conjecture of Wan which asserts the equality of the Euler-Poincar\'e pairing with the geometric multiplicity for any irreducible representation of $G$. In the symplectic case we reduce the verification of Wan's conjecture to the case of discrete series representations. We also determine all relatively supercuspidal representations in both cases.