Let $p$ and $\ell$ be two distinct primes. The aim of this paper is to show how, under a certain congruence hypothesis, the mod $\ell$ cohomology complex of the Lubin-Tate tower, together with a natural Lefschetz operator, provides a geometric interpretation of Vignéras’s local Langlands correspondence modulo $\ell$ for unipotent representations.

Soient $\mathrm{F}$ un corps commutatif localement compact non archimédien, de caractéristique résiduelle notée $p$, et $\mathrm{D}$ une $\mathrm{F}$-algèbre à division centrale de dimension finie. Soit $\ell$ un nombre premier différent de $p$. Dans cet article, généralisant les résultats de [GJ], on associe à chaque représentation $\ell$-modulaire lisse irréductible $\pi$ de ${\mathrm{GL}}_m\left(\mathrm{D}\right)$, deux invariants $L(T,\pi )$, $\epsilon (T,\pi ,\psi )$ où $T$ est une variable et $\psi$ est un caractère non trivial de $\mathrm{F}$.

Let $\mathrm{F}$ be a non-Archimedean locally compact field, of residual characteristic $p$, and let $\mathrm{D}$ be a finite-dimensional central division $\mathrm{F}$-algebra. Let $\ell$ be a prime number different from $p$. In this article, generalizing the results of [GJ], we associate, to each $\ell$-modular smooth irreducible representation $\pi$ of ${\mathrm{GL}}_m\left(\mathrm{D}\right)$, two invariants $L(T,\pi )$, $\epsilon (T,\pi ,\psi )$, where $T$ is an indeterminate and $\psi$ is a nontrivial character of $\mathrm{F}$.

We develop the theory of the doubling zeta integral of Piatetski-Shapiro and Rallis for metaplectic groups ${Mp}_{2n}$, and we use it to give precise definitions of the local $\gamma$-factors, $L$-factors, and $ϵ$-factors for irreducible representations of ${Mp}_{2n}\times {GL}_1$, following the footsteps of Lapid and Rallis.

Automorphic induction for ${\mathrm{GL}}_n$ is a case of endoscopic transfer, and its character identity was established by Henniart and Herb, up to a constant of proportionality. We determine this constant in terms of the Kottwitz-Shelstad normalization of transfer factors, which involves certain $\epsilon$-factors.

In this paper, we introduce two conjectures on characterizations of endoscopy structures of irreducible generic cuspidal automorphic representations of odd special orthogonal groups in terms of nonvanishing of certain period of automorphic forms. We discuss a relation between the two conjectures and prove that a special case of Conjecture 1 (and hence Conjecture 2) is true.

We calculate extensions between certain irreducible admissible representations of $p$-adic groups.

We investigate the radial part of the matrix coefficients with minimal $K$-types of the large discrete series representations of $Sp(2;\mathbf{R})$. They satisfy certain difference-differential equations derived from Schmid operators. This system is reduced to a holonomic system of rank 4, which is finally found to be equivalent to higher-order hypergeometric series in the sense of Appell and Kampé de Fériet.

New simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the contribution of “central” unipotent elements in the dimension formula of Siegel cusp forms of any degree. These results lead us to a conjecture on explicit values of dimensions of Siegel cusp forms of any torsion-free principal congruence subgroups of the symplectic groups of general degree.