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For the discrete series representations of ${\rm GL}(n)$ over anon-archimedean local field $F$, we define a notion of functions similar to"zonal spherical functions" for unramified principal series. We prove theexistence of such functions in the level $0$ case. As for unramified principalseries, they give rise to explicit coefficients. We deduce a local proof ofMatringe'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 anyunramified quadratic extension $F/F_0$. We also exhibit explicit test vectorswhen these representations are distinguished.

Explicit matrix coefficients and test vectors for discrete series representations