Let \(F\) be a local non-Archimedean field with ring of integers denoted by \(\mathfrak o_F\), and let \(N\) be a positive integer. The aim of the article is to obtain explicit formulas for the characters of certain representations of the group \(\mathrm{GL}(N,F)\) belonging to the discrete series.
Let \(V\) be an \(F\)-vector space of dimension \(N\), and let \(G:=\mathrm{Aut}_F(V)\simeq\mathrm{GL}(N,F)\). Every square-integrable irreducible smooth representation \(\pi\) of \(G\) contains by restriction a simple type \((J,\lambda)\) as proved by Bushnell and Kutzko. Such a type is attached to some data, including a principal hereditary \(\mathfrak o_F\)-order \(\mathfrak A\) in \(A:=\mathrm{End}_F(V)\simeq\mathrm{M}_N(F)\), and an element \(\beta\) in \(A\) that generates a field \(E=F[\beta]\) and normalizes the order \(\mathfrak A\). Let \(e:=e(\mathfrak A)\) be the period of \(\mathfrak A\) and let \(\mathfrak K(\mathfrak A)\) denote the normalizer of \(\mathfrak A\) in \(G\). We put \(B:=\mathrm{End}_E(V)\) and \(\mathfrak B:=\mathfrak A\cap B\).
Let \(K\) be a fixed unramified extension of \(E\) of degree \(f:=N/([E:F]e)\) such that \(K^\times\subset \mathfrak K(\mathfrak B)\), and let \(C\) denote the \(K\)-algebra \(\mathrm{End}_K(V)\). Then the order \(\mathfrak C:=\mathfrak B\cap C\) is a minimal hereditary order of \(C\) and \(\mathfrak K(\mathfrak C)\subset\mathfrak K(\mathfrak B)\subset\mathfrak K(\mathfrak A)\). Let \(I\) denote the group of units of \(\mathfrak C\). We set \(H:=\mathrm{GL}_K(V)\simeq C^\times\). Then \(I\) is an Iwahori subgroup of \(H\).
Using the work of Bushnell, Henniart and Kutzko on the relation between types and the Plancherel formula, the author proves (see Theorem 6.3) the following result: An irreducible smooth representation of \(G\) is tempered if and only if the corresponding \(I\)-spherical irreducible smooth representation of \(H\) is tempered.
Let \(\mathcal H(H,I)\) be the Hecke-Iwahori algebra of \(H\), that is, the convolution algebra of compactly supported functions \(f: H\to\mathbb C\) that are bi-invariant under the action of \(I\).
Let \(\mathrm{St}_H\) be the Steinberg representation of \(H\), and let \(f_0\) be the variant due to Laumon of the Kottwitz pseudo-coefficient of \(\mathrm{St}_H\). By definition, for every irreducible tempered smooth representation \((\sigma,\mathcal W)\) of \(H\), with trivial central character, one has \[ \mathrm{Tr}(\sigma(f_0),\mathcal W)=\begin{cases} 1&\text{if \(\sigma\simeq\mathrm{St}_H\),}\\ 0&\text{otherwise.}\end{cases} \] Let \((J',\lambda')\) the modified type that has been attached by Bushnell and Kutzko to \((J,\lambda)\), and let \(\mathcal H(G,\lambda')\) denote the convolution algebra of \(\lambda'\)-spherical functions on \(G\). Then there exists a unitary isomorphism of Hecke algebras (see Corollary 5.3) \[ \mathcal H(H,I)\to\mathcal H(G,\lambda') \] such that \(\pi\) corresponds to \(\mathrm{St}_H\) via the associate equivalence of categories. Following an idea of Henniart, by transferring \(f_0\) through the above isomorphism, the author contructs and computes a pseudo-coefficient of \(\pi\). The article provides the computation of the orbital integrals of this pseudo-coefficient in order to get character formulas in the two following cases:

1.

The extension \(E/F\) is unramified, and the element for which one computes the character is minimal and generates in \(A\) an unramified extension.

2.

The extension \(E/F\) is totally ramified, the regular elliptic element is minimal and generates a totally ramified extension.