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For a unimodular totally disconnected locally compact group $G$ we introduce and study an analogue of the Hattori-Stallings rank $\tilde{\rho}(P)\in\mathbf{h}_G$ for a finitely generated projective rational discrete left $\mathbb Q[G]$-module $P$. Here $\mathbf{h}_G$ denotes the $\mathbb Q$-vector space of left invariant Haar measures of $G$. Indeed, an analogue of Kaplansky's theorem holds in this context (cf. Theorem A). As in the discrete case, using this rank function it is possible to define a rational discrete Euler-Poincar\'e characteristic $\tilde{\chi}_G$ whenever $G$ is a unimodular totally disconnected locally compact group of type $\mathrm{FP}_\infty$ of finite rational discrete cohomological dimension. E.g., when $G$ is a discrete group of type $\mathrm{FP}$, then $\tilde{\chi}_G$ coincides with the ''classical'' Euler-Poincar\'e characteristic times the counting measure $\mu_{\{1\}}$. For a profinite group $\mathcal{O}$, $\tilde{\chi}_{\mathcal{O}}$ equals the probability Haar measure $\mu_{\mathcal{O}}$ on $\mathcal{O}$. Many more examples are calculated explicitly (cf. Example 1.7 and Section 5). In the last section, for a totally disconnected locally compact group $G$ satisfying an additional finiteness condition, we introduce and study a formal Dirichlet series $\zeta_{_{G,\mathcal{O}}}(s)$ for any compact open subgroup $\mathcal{O}$. In several cases it happens that $\zeta_{_{G,\mathcal{O}}}(s)$ defines a meromorphic function $\tilde{\zeta}_{_{G,\mathcal{O}}}\colon \mathbb{C} \to\bar{\mathbb C}$ of the complex plane satisfying miraculously the identity $\tilde{\chi}_G=\tilde{\zeta}_{_{G,\mathcal{O}}}(-1)^{-1}\cdot\mu_{\mathcal{O}}$. Here $\mu_{\mathcal{O}}$ denotes the Haar measure of $G$ satisfying $\mu_{\mathcal{O}}(\mathcal{O})=1$., Comment: 42 pages

Let $(W,S)$ be a Coxeter system of finite rank and let $J,K\subset S$. We study the rationality of the Poincar\'e series of the set of representatives of minimal length of $(W_J,W_K)$-double cosets of $W$: we conclude that it depends mostly on the rationality of the Poincar\'e series of the normalizers of finite parabolic subgroups of $W$. For affine Weyl groups, we prove that all these series are rational and we give some explicit examples., Comment: 22 pages

Let $G$ be an inner form of a general linear group over a non-archimedean locally compact field of residue characteristic $p$, let $R$ be an algebraically closed field of characteristic different from $p$ and let $\mathscr{R}_R(G)$ be the category of smooth representations of $G$ over $R$. In this paper, we prove that a block (indecomposable summand) of $\mathscr{R}_R(G)$ is equivalent to a level-$0$ block (a block in which every object has non-zero invariant vectors for the pro-$p$-radical of a maximal compact open subgroup) of $\mathscr{R}_R(G')$, where $G'$ is a direct product of groups of the same type of $G$.

Let $G$ be a direct product of inner forms of general linear groups over non-archimedean locally compact fields of residue characteristic $p$ and let $K^1$ be the pro-$p$-radical of a maximal compact open subgroup of $G$. In this paper we describe the (intertwining) Hecke algebra $\mathscr{H}(G,K^1)$, that is the convolution $\mathbb{Z}$-algebra of functions from $G$ to $\mathbb{Z}$ that are bi-invariant for $K^1$ and whose supports are a finite union of $K^1$-double cosets. We produce a presentation by generators and relations of this algebra. Finally we prove that the level-$0$ subcategory of the category of smooth representations of $G$ over a unitary commutative ring $R$ such that $p\in R^{\times}$ is equivalent to the category of modules over $\mathscr{H}(G,K^1)\otimes_\mathbb{Z} R$., Comment: 17 pages, minor corrections, updated bibliography

Given F a locally compact, non-discrete, non-archimedean field of characteristic different from 2 and R an integral domain such that a non-trivial smooth F-character with values in the multiplicative group of R exists, we construct the (reduced) metaplectic group attached to R. We show that it is in most cases a double cover of the symplectic group over F. Finally we define a faithful infinite dimensional R-representation of the metaplectic group analogue to the Weil representation in the complex case.