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 $s=1$.

Motivated by the Bruhat and Cartan decompositions of general linear groups over local fields, we enumerate double cosets of the group of label-preserving automorphisms of a label-regular tree over the fixator of an end of the tree and over maximal compact open subgroups. This enumeration is used to show that every continuous homomorphism from the automorphism group of a label-regular tree has closed range.

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 introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple proof that effectiveness of an étale groupoid is implied by a Cuntz–Krieger uniqueness theorem for a universal groupoid C*-algebra.

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].

Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $\text{Sym}(n)$ (in the sofic case) or the finite-dimensional unitary groups $\text{U}(n)$ (in the hyperlinear case)? In the case of $\text{U}(n)$, the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $\text{U}(n)$ with respect to the Frobenius norm $\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$. Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability, that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $p$-adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.

To study when a paratopological group becomes a topological group, Arhangel’skii et al. [‘Topological games and topologies on groups’, Math. Maced. 8 (2010), 1–19] introduced the class of $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable spaces. We show that every $\unicode[STIX]{x1D707}$-complete (or normal) $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable semitopological group is a topological group. We prove that the product of a $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable space and a strongly Fréchet $(\unicode[STIX]{x1D6FC},G_{\unicode[STIX]{x1D6F1}})$-favourable space is $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourable. We also show that continuous closed irreducible mappings preserve the $(\,\unicode[STIX]{x1D6FD},G_{\unicode[STIX]{x1D6F1}})$-unfavourableness in both directions.

We study the locally compact abelian groups in the class ${\mathfrak E_{ \lt \infty }}$, that is, having only continuous endomorphisms of finite topological entropy, and in its subclass $\mathfrak E_0$, that is, having all continuous endomorphisms with vanishing topological entropy. We discuss the reduction of the problem to the case of periodic locally compact abelian groups, and then to locally compact abelian p-groups. We show that locally compact abelian p-groups of finite rank belong to ${\mathfrak E_{ \lt \infty }}$, and that those of them that belong to $\mathfrak E_0$ are precisely the ones with discrete maximal divisible subgroup. Furthermore, the topological entropy of endomorphisms of locally compact abelian p-groups of finite rank coincides with the logarithm of their scale. The backbone of the paper is the Addition Theorem for continuous endomorphisms of locally compact abelian groups. Various versions of the Addition Theorem are established in the paper and used in the proofs of the main results, but its validity in the general case remains an open problem.

Furstenberg has associated to every topological group $G$ a universal boundary $\unicode[STIX]{x2202}(G)$ . If we consider in addition a subgroup $H<G$ , the relative notion of $(G,H)$ -boundaries admits again a maximal object $\unicode[STIX]{x2202}(G,H)$ . In the case of discrete groups, an equivalent notion was introduced by Bearden and Kalantar (Topological boundaries of unitary representations. Preprint, 2019, arXiv:1901.10937v1) as a very special instance of their constructions. However, the analogous universality does not always hold, even for discrete groups. On the other hand, it does hold in the affine reformulation in terms of convex compact sets, which admits a universal simplex $\unicode[STIX]{x1D6E5}(G,H)$ , namely the simplex of measures on $\unicode[STIX]{x2202}(G,H)$ . We determine the boundary $\unicode[STIX]{x2202}(G,H)$ in a number of cases, highlighting properties that might appear unexpected.

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 study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.

We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

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.