  For the discrete series representations of ${\rm GL}(n)$ over a
non-archimedean local field $F$, we define a notion of functions similar to
"zonal spherical functions" for unramified principal series. We prove the
existence of such functions in the level $0$ case. As for unramified principal
series, they give rise to explicit coefficients. We deduce a local proof of
Matringe's criterion of distinction of discrete series, in the level $0$ case,
for the Galois symmetric space ${\rm GL}(n,F)/{\rm GL}(n,F_0 )$, for any
unramified quadratic extension $F/F_0$. We also exhibit explicit test vectors
when these representations are distinguished.
