Representations of \(GL_n (D)\) near the identity (Q6659518)

Let \(p\) be a prime number, and let \(F\) stand for either a finite extension of \(\mathbb{Q}_p\) or \(\mathbb{F}_p((t))\). We denote the cardinality of the residue field of \(F\) by \(q\).\N\NLet \(G\) denote the group \(GL_n(D)\), where \(D\) stands for a central division algebra over \(F\), with finite degree \(d^2\) over \(F\). Let \(\pi\) stand for a finite length smooth representation of \(G\) on an \(R\)-vector space, where characteristic of \(R\) is different than \(p\). For a partition \(\lambda\) of \(n\), let \(\pi_{\lambda}\) stand for a representation of \(G\) nonnormalized parabolically induced from the trivial representation of the upper block triangular subgroup \(P_{\lambda}\) of \(G.\)\N\NFor a pro-\(p\) subgroup \(K\) of \(G\), denote by \([\pi]_K\) the image of the restriction in the Grothendieck group of admissible \(R\)-representations of \(K\). In the paper under the review, the authors show that there is a unique function \(c_{\pi}\) from partitions of \(n\) to \(\mathbb{Z}\) and an open pro-\(p\) subgroup \(K\) of \(G\) such that \([\pi]_K = \sum_{\lambda} c_{\pi}(\lambda) [\pi_{\lambda}]_K\). When \(P_{\lambda}\) is minimal such that \(c_{\pi}(\lambda) \neq 0\), than \(c_{\pi}(\lambda)\) is positive and equals the dimension of a generalized Whittaker model of \(\pi\).\N\NThis result is used to determine the polynomial \(P_{\pi, K}\) with integral coefficients and degree \(d(\pi)\) independent of \(K\) such that, for large enough integers \(j\), \(P_{\pi, K}(q^{dj})\) equals the dimension of fixed points in \(\pi\) under the \(j\)th congruence subgroup \(K_j\) of \(K\).\N\NThe authors also discuss the behavior of \(c_{\pi}\) under the Jacquet-Langlands correspondence.