Soit $G$ le groupe $\mathrm{GL}(N,F)$, où $F$ est un corps localement compact et non archimédien. En utilisant la théorie des types simples de Bushnell et Kutzko, ainsi qu'une idée originale d'Henniart, nous construisons des pseudo-coefficients explicites pour les représentations de la série discrète de $G$. Comme application, nous en déduisons des formules inédites pour la valeur du charactère d'Harish-Chandra de certaines telles représentations en certains éléments elliptiques réguliers.

When $F$ is a $p$-adic field, and $G={\mathbb G}(F)$ is the group of $F$-rational points of a connected algebraic $F$-group, the complex vector space ${\mathcal H}(G)$ of compactly supported locally constant distributions on $G$ has a natural convolution product that makes it into a ${\mathbb C}$-algebra (without an identity) called the Hecke algebra. The Hecke algebra is a partial analogue for $p$-adic groups of the enveloping algebra of a Lie group. However, $\mathcal{H}(G)$ has drawbacks such as the lack of an identity element, and the process $G \mapsto \mathcal{H}(G)$ is not a functor. Bernstein introduced an enlargement $\mathcal{H}\,\hat{\,}(G)$ of $\mathcal{H}(G)$. The algebra $\mathcal{H}\,\hat{\,} (G)$ consists of the distributions that are left essentially compact. We show that the process $G \mapsto \mathcal{H}\,\hat{\,} (G)$ is a functor. If $\tau \colon G \rightarrow H$ is a morphism of $p$-adic groups, let $F(\tau) \colon \mathcal{H}\,\hat{\,} (G) \rightarrow \mathcal{H}\,\hat{\,} (H)$ be the morphism of $\mathbb{C}$-algebras. We identify the kernel of $F(\tau)$ in terms of $\textrm{Ker}(\tau)$. In the setting of $p$-adic Lie algebras, with $\mathfrak{g}$ a reductive Lie algebra, $\mathfrak{m}$ a Levi, and $\tau \colon \mathfrak{g} \to \mathfrak{m}$ the natural projection, we show that $F(\tau)$ maps $G$-invariant distributions on $\mathcal{G}$ to $N_G (\mathfrak{m})$-invariant distributions on $\mathfrak{m}$. Finally, we exhibit a natural family of $G$-invariant essentially compact distributions on $\mathfrak{g}$ associated with a $G$-invariant non-degenerate symmetric bilinear form on ${\mathfrak g}$ and in the case of $SL(2)$ show how certain members of the family can be moved to the group.