Comparison of germ expansion on inner forms of \(GL(n)\) (Q1576590)

Let \(k\) be a non-Archimedean local field of characteristic zero, and let \(D\) be a division algebra over \(k\) of index \(m\). Let \(\pi'\) be a discrete series representation of \(GL_n (D)\), and let \(\pi\) be a representation of \(GL_{mn} (k)\) associated to \(\pi'\) by the Deligne-Kazhdan-Vigneras correspondence. In this paper the author proves that the germ expansion of \(\pi'\) and that of \(\pi\) are closely related. In particular, he shows that certain coefficients in the germ expansion of a discrete series representation of \(GL_{mn} (k)\) can be expressed in terms of the dimension of a certain space of Whittaker models of \(GL_n (D)\).