Many phenomena in geometry and analysis can be explained via the theory of $D$-modules, but this theory explains close to nothing in the non-archimedean case, by the absence of integration by parts. Hence there is a need to look for alternatives. A central example of a notion based on the theory of $D$-modules is the notion of holonomic distributions. We study two recent alternatives of this notion in the context of distributions on non-archimedean local fields, namely $\mathscr{C}^{\text{exp}}$-class distributions from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] and WF-holonomicity from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We answer a question from Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)] by showing that each distribution of the $\mathscr{C}^{\text{exp}}$-class is WF-holonomic and thus provides a framework of WF-holonomic distributions, which is stable under taking Fourier transforms. This is interesting because the $\mathscr{C}^{\text{exp}}$-class contains many natural distributions, in particular, the distributions studied by Aizenbud and Drinfeld [‘The wave front set of the Fourier transform of algebraic measures’, Israel J. Math.207(2) (2015), 527–580 (English)]. We show also another stability result of this class, namely, one can regularize distributions without leaving the $\mathscr{C}^{\text{exp}}$-class. We strengthen a link from Cluckers et al. [‘Distributions and wave front sets in the uniform nonarchimedean setting’, Trans. Lond. Math. Soc.5(1) (2018), 97–131] between zero loci and smooth loci for functions and distributions of the $\mathscr{C}^{\text{exp}}$-class. A key ingredient is a new resolution result for subanalytic functions (by alterations), based on embedded resolution for analytic functions and model theory.

For a locally compact group G, we study the distality of the action of automorphisms T of G on SubG, the compact space of closed subgroups of G endowed with the Chabauty topology. For a certain class of discrete groups G, we show that T acts distally on SubG if and only if Tn is the identity map for some $n\in\mathbb N$. As an application, we get that for a T-invariant lattice Γ in a simply connected nilpotent Lie group G, T acts distally on SubG if and only if it acts distally on SubΓ. This also holds for any closed T-invariant co-compact subgroup Γ in G. For a lattice Γ in a simply connected solvable Lie group, we study conditions under which its automorphisms act distally on SubΓ. We construct an example highlighting the difference between the behaviour of automorphisms on a lattice in a solvable Lie group and that in a nilpotent Lie group. We also characterise automorphisms of a lattice Γ in a connected semisimple Lie group which act distally on SubΓ. For torsion-free compactly generated nilpotent (metrisable) groups G, we obtain the following characterisation: T acts distally on SubG if and only if T is contained in a compact subgroup of Aut(G). Using these results, we characterise the class of such groups G which act distally on SubG. We also show that any compactly generated distal group G is Lie projective.

We show a Siegel–Weil formula in the setting of exceptional theta correspondence. Using this, together with a new Rankin–Selberg integral for the Spin L-function of $\text{PGSp}_{6}$ discovered by Pollack, we prove that a cuspidal representation of $\text{PGSp}_{6}$ is a (weak) functorial lift from the exceptional group $G_{2}$ if its (partial) Spin L-function has a pole at $s=1$.

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak {sl}_2$, where they use singular blocks of category $\mathcal {O}$ for $\mathfrak {sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak {s}\mathfrak {l}_n$ over a field $\mathbf {k}$ of characteristic p with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification constructed by Cautis, Kamnitzer, and Licata using coherent sheaves on cotangent bundles to Grassmanians. In particular, the latter admits an abelian refinement. With respect to this abelian refinement, the stratified Mukai flop induces a perverse equivalence on the derived categories for complementary Grassmanians. This is part of a larger project to give a combinatorial approach to Lusztig’s conjectures for representations of Lie algebras in positive characteristic.

Soient $F$ un corps global, et $G$ un groupe réductif connexe défini sur $F$. On prouve que si deux données endoscopiques de $G$ sont équivalentes en presque toute place de $F$, alors elles sont équivalentes. Le résultat est encore vrai pour l’endoscopie (ordinaire) avec caractère. On donne aussi, pour $F$ global ou local et $G$ quasi-simple simplement connexe, une description des données endoscopiques elliptiques de $G$.

In this paper, we study Finsler warped product metrics with relatively isotropic Landsberg curvature. We obtain the differential equations that characterize such metrics. Then we give some examples.

We construct analogues of Rankin–Selberg integrals for Speh representations of the general linear group over a $p$-adic field. The integrals are in terms of the (extended) Shalika model and are expected to be the local counterparts of (suitably regularized) global integrals involving square-integrable automorphic forms and Eisenstein series on the general linear group over a global field. We relate the local integrals to the classical ones studied by Jacquet, Piatetski-Shapiro and Shalika. We also introduce a unitary structure for Speh representation on the Shalika model, as well as various other models including Zelevinsky’s degenerate Whittaker model.

In this paper we consider uncertainty principles for solutions of certain partial differential equations on $H$-type groups. We first prove that, on $H$-type groups, the heat kernel is an average of Gaussians in the central variable, so that it does not satisfy a certain reformulation of Hardy’s uncertainty principle. We then prove the analogue of Hardy’s uncertainty principle for solutions of the Schrödinger equation with potential on $H$-type groups. This extends the free case considered by Ben Saïd et al. [‘Uniqueness of solutions to Schrödinger equations on H-type groups’, J. Aust. Math. Soc. (3) 95 (2013), 297–314] and by Ludwig and Müller [‘Uniqueness of solutions to Schrödinger equations on 2-step nilpotent Lie groups’, Proc. Amer. Math. Soc. 142 (2014), 2101–2118].

We study lattice embeddings for the class of countable groups $\unicode[STIX]{x1D6E4}$ defined by the property that the largest amenable uniformly recurrent subgroup ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is continuous. When ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ comes from an extremely proximal action and the envelope of ${\mathcal{A}}_{\unicode[STIX]{x1D6E4}}$ is coamenable in  $\unicode[STIX]{x1D6E4}$ , we obtain restrictions on the locally compact groups $G$ that contain a copy of $\unicode[STIX]{x1D6E4}$ as a lattice, notably regarding normal subgroups of  $G$ , product decompositions of  $G$ , and more generally dense mappings from $G$ to a product of locally compact groups.

We prove that the HRT (Heil, Ramanathan, and Topiwala) Conjecture is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent.

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

Let $F$ be a non-Archimedean local field, $G$ a connected reductive group defined and split over $F$, and $T$ a maximal $F$-split torus in $G$. Let $\unicode[STIX]{x1D712}_{0}$ be a depth-zero character of the maximal compact subgroup $T$ of $T(F)$. This gives by inflation a character $\unicode[STIX]{x1D70C}$ of an Iwahori subgroup $\unicode[STIX]{x2110}\subset T$ of $G(F)$. From Roche [Types and Hecke algebras for principal series representations of split reductive$p$-adic groups, Ann. Sci. Éc. Norm. Supér. (4) 31 (1998), 361–413], $\unicode[STIX]{x1D712}_{0}$ defines a reductive $F$-split group $\widetilde{G}^{\prime }$ whose connected component $G^{\prime }$ is an endoscopic group of $G$, and there is an isomorphism of $\mathbb{C}$-algebras $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})\rightarrow \unicode[STIX]{x210B}(\widetilde{G}^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ where $\unicode[STIX]{x210B}(G(F),\unicode[STIX]{x1D70C})$ is the Hecke algebra of compactly supported $\unicode[STIX]{x1D70C}^{-1}$-spherical functions on $G(F)$ and $\unicode[STIX]{x2110}^{\prime }$ is an Iwahori subgroup of $G^{\prime }(F)$. This isomorphism gives by restriction an injective morphism $\unicode[STIX]{x1D701}:Z(G(F),\unicode[STIX]{x1D70C})\rightarrow Z(G^{\prime }(F),1_{\unicode[STIX]{x2110}^{\prime }})$ between the centers of the Hecke algebras. We prove here that a certain linear combination of morphisms analogous to $\unicode[STIX]{x1D701}$ realizes the transfer (matching of strongly $G$-regular semi-simple orbital integrals). If $\operatorname{char}(F)=p>0$, our result is unconditional only if $p$ is large enough.

Suppose that $\mathbf{G}$ is a connected reductive group over a finite extension $F/\mathbb{Q}_{p}$ and that $C$ is a field of characteristic $p$. We prove that the group $\mathbf{G}(F)$ admits an irreducible admissible supercuspidal, or equivalently supersingular, representation over $C$.

We prove a super-rigidity result for algebraic representations over complete fields of irreducible lattices in products of groups and lattices with dense commensurator groups. We derive criteria for the non-linearity of such groups.

Let $F$ be a non-archimedean local field of residual characteristic $p$, $\ell \neq p$ be a prime number, and $\text{W}_{F}$ the Weil group of $F$. We classify equivalence classes of $\text{W}_{F}$-semisimple Deligne $\ell$-modular representations of $\text{W}_{F}$ in terms of irreducible $\ell$-modular representations of $\text{W}_{F}$, and extend constructions of Artin–Deligne local constants to this setting. Finally, we define a variant of the $\ell$-modular local Langlands correspondence which satisfies a preservation of local constants statement for pairs of generic representations.

We study basic geometric properties of Kottwitz–Viehmann varieties, which are certain generalizations of affine Springer fibers that encode orbital integrals of spherical Hecke functions. Based on the previous work of A. Bouthier and the author, we show that these varieties are equidimensional and give a precise formula for their dimension. Also we give a conjectural description of their number of irreducible components in terms of certain weight multiplicities of the Langlands dual group and we prove the conjecture in the case of unramified conjugacy class.

J.-C. Yoccoz proposed a natural extension of Selberg’s eigenvalue conjecture to moduli spaces of abelian differentials. We prove an approximation to this conjecture. This gives a qualitative generalization of Selberg’s $\frac{3}{16}$ theorem to moduli spaces of abelian differentials on surfaces of genus ${\geqslant}2$.

We study lattices in a product $G=G_{1}\times \cdots \times G_{n}$ of non-discrete, compactly generated, totally disconnected locally compact (tdlc) groups. We assume that each factor is quasi just-non-compact, meaning that $G_{i}$ is non-compact and every closed normal subgroup of $G_{i}$ is discrete or cocompact (e.g. $G_{i}$ is topologically simple). We show that the set of discrete subgroups of $G$ containing a fixed cocompact lattice $\unicode[STIX]{x1D6E4}$ with dense projections is finite. The same result holds if $\unicode[STIX]{x1D6E4}$ is non-uniform, provided $G$ has Kazhdan’s property (T). We show that for any compact subset $K\subset G$, the collection of discrete subgroups $\unicode[STIX]{x1D6E4}\leqslant G$ with $G=\unicode[STIX]{x1D6E4}K$ and dense projections is uniformly discrete and hence of covolume bounded away from $0$. When the ambient group $G$ is compactly presented, we show in addition that the collection of those lattices falls into finitely many $\operatorname{Aut}(G)$-orbits. As an application, we establish finiteness results for discrete groups acting on products of locally finite graphs with semiprimitive local action on each factor. We also present several intermediate results of independent interest. Notably it is shown that if a non-discrete, compactly generated quasi just-non-compact tdlc group $G$ is a Chabauty limit of discrete subgroups, then some compact open subgroup of $G$ is an infinitely generated pro-$p$ group for some prime $p$. It is also shown that in any Kazhdan group with discrete amenable radical, the lattices form an open subset of the Chabauty space of closed subgroups.

Let $\unicode[STIX]{x1D6E4}\leqslant \text{Aut}(T_{d_{1}})\times \text{Aut}(T_{d_{2}})$ be a group acting freely and transitively on the product of two regular trees of degree $d_{1}$ and $d_{2}$. We develop an algorithm that computes the closure of the projection of $\unicode[STIX]{x1D6E4}$ on $\text{Aut}(T_{d_{t}})$ under the hypothesis that $d_{t}\geqslant 6$ is even and that the local action of $\unicode[STIX]{x1D6E4}$ on $T_{d_{t}}$ contains $\text{Alt}(d_{t})$. We show that if $\unicode[STIX]{x1D6E4}$ is torsion-free and $d_{1}=d_{2}=6$, exactly seven closed subgroups of $\text{Aut}(T_{6})$ arise in this way. We also construct two new infinite families of virtually simple lattices in $\text{Aut}(T_{6})\times \text{Aut}(T_{4n})$ and in $\text{Aut}(T_{2n})\times \text{Aut}(T_{2n+1})$, respectively, for all $n\geqslant 2$. In particular, we provide an explicit presentation of a torsion-free infinite simple group on 5 generators and 10 relations, that splits as an amalgamated free product of two copies of $F_{3}$ over $F_{11}$. We include information arising from computer-assisted exhaustive searches of lattices in products of trees of small degrees. In an appendix by Pierre-Emmanuel Caprace, some of our results are used to show that abstract and relative commensurator groups of free groups are almost simple, providing partial answers to questions of Lubotzky and Lubotzky–Mozes–Zimmer.

We introduce a refined version of group cohomology and relate it to the space of polynomials on the group in question. We show that the polynomial cohomology with trivial coefficients admits a description in terms of ordinary cohomology with polynomial coefficients, and that the degree one polynomial cohomology with trivial coefficients admits a description directly in terms of polynomials. Lastly, we give a complete description of the polynomials on a connected, simply connected nilpotent Lie group by showing that these are exactly the maps that pull back to classical polynomials via the exponential map.