Comparison of germ expansion on inner forms of GL(n)

We prove that the germ expansion of a discrete series representation π' on GLn(D) where D is a division algebra over k of index m and the germ expansion of the representation π of GLmn(k) associated to π' by the Deligne–Kazhdan–Vigneras correspondence are closely related, and therefore certain coefficients in the germ expansion of a discrete series representation of GLmn(k) can be interpreted (and therefore sometimes calculated) in terms of the dimension of a certain space of (degenerate) Whittaker models on GLn(D)