Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $, we prove that any $\Gamma $-Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $. More generally, we prove that for a Zariski dense $\theta $-transverse subgroup $\Gamma <G$, any $(\Gamma , \psi )$-Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $, provided the $\psi $-Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

Let $p \geq 5$ be a prime number, and let $G = {\mathrm {SL}}_2(\mathbb {Q}_p)$. Let $\Xi = {\mathrm {Spec}}(Z)$ denote the spectrum of the centre Z of the pro-p Iwahori–Hecke algebra of G with coefficients in a field k of characteristic p. Let $\mathcal {R} \subset \Xi \times \Xi $ denote the support of the pro-p Iwahori ${\mathrm {Ext}}$-algebra of G, viewed as a $(Z,Z)$-bimodule. We show that the locally ringed space $\Xi /\mathcal {R}$ is a projective algebraic curve over ${\mathrm {Spec}}(k)$ with two connected components and that each connected component is a chain of projective lines. For each Zariski open subset U of $\Xi /\mathcal {R}$, we construct a stable localising subcategory $\mathcal {L}_U$ of the category of smooth k-linear representations of G.

For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter ${\mathcal {L}}^{ss}(\pi )$ to each irreducible representation $\pi $. Our first result shows that the Genestier-Lafforgue parameter of a tempered $\pi $ can be uniquely refined to a tempered L-parameter ${\mathcal {L}}(\pi )$, thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of ${\mathcal {L}}^{ss}(\pi )$ for unramified G and supercuspidal $\pi $ constructed by induction from an open compact (modulo center) subgroup. If ${\mathcal {L}}^{ss}(\pi )$ is pure in an appropriate sense, we show that ${\mathcal {L}}^{ss}(\pi )$ is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal {L}^{ss}(\pi )$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is ${\mathbb {P}}^1$ and a simple application of Deligne’s Weil II.

We study the growth of the local $L^2$-norms of the unitary Eisenstein series for reductive groups over number fields, in terms of their parameters. We derive a poly-logarithmic bound on an average, for a large class of reductive groups. The method is based on Arthur’s development of the spectral side of the trace formula, and ideas of Finis, Lapid and Müller.

As applications of our method, we prove the optimal lifting property for $\mathrm {SL}_n(\mathbb {Z}/q\mathbb {Z})$ for square-free q, as well as the Sarnak–Xue [52] counting property for the principal congruence subgroup of $\mathrm {SL}_n(\mathbb {Z})$ of square-free level. This makes the recent results of Assing–Blomer [8] unconditional.

We formulate and prove the archimedean period relations for Rankin–Selberg convolutions for ${\mathrm {GL}}(n)\times {\mathrm {GL}}(n-1)$. As a consequence, we prove the period relations for critical values of the Rankin–Selberg L-functions for ${\mathrm {GL}}(n)\times {\mathrm {GL}}(n-1)$ over arbitrary number fields.

The loop space of a string manifold supports an infinite-dimensional Fock space bundle, which is an analog of the spinor bundle on a spin manifold. This spinor bundle on loop space appears in the description of two-dimensional sigma models as the bundle of states over the configuration space of the superstring. We construct a product on this bundle that covers the fusion of loops, i.e. the merging of two loops along a common segment. For this purpose, we exhibit it as a bundle of bimodules over a certain von Neumann algebra bundle, and realize our product fibrewise using the Connes fusion of von Neumann bimodules. Our main technique is to establish novel relations between string structures, loop fusion, and the Connes fusion of Fock spaces. The fusion product on the spinor bundle on loop space was proposed by Stolz and Teichner as part of a programme to explore the relation between generalized cohomology theories, functorial field theories, and index theory. It is related to the pair of pants worldsheet of the superstring, to the extension of the corresponding smooth functorial field theory down to the point, and to a higher-categorical bundle on the underlying string manifold, the stringor bundle.

In this paper, we investigate the twisted GGP conjecture for certain tempered representations using the theta correspondence and establish some special cases, namely when the L-parameter of the unitary group is the sum of conjugate-dual characters of the appropriate sign.

Given a smooth genus three curve C, the moduli space of rank two stable vector bundles on C with trivial determinant embeds in ${\mathbb {P}}^8$ as a hypersurface whose singular locus is the Kummer threefold of C; this hypersurface is the Coble quartic. Gruson, Sam and Weyman realized that this quartic could be constructed from a general skew-symmetric four-form in eight variables. Using the lines contained in the quartic, we prove that a similar construction allows to recover $\operatorname {\mathrm {SU}}_C(2,L)$, the moduli space of rank two stable vector bundles on C with fixed determinant of odd degree L, as a subvariety of $G(2,8)$. In fact, each point $p\in C$ defines a natural embedding of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ in $G(2,8)$. We show that, for the generic such embedding, there exists a unique quadratic section of the Grassmannian which is singular exactly along the image of $\operatorname {\mathrm {SU}}_C(2,{\mathcal {O}}(p))$ and thus deserves to be coined the Coble quadric of the pointed curve $(C,p)$.

We introduce a new invariant, the conductor exponent, of a generic irreducible Casselman–Wallach representation of $\operatorname {\mathrm {GL}}_n(F)$, where F is an archimedean local field, that quantifies the extent to which this representation may be ramified. We also determine a distinguished vector, the newform, occurring with multiplicity one in this representation, with the complexity of this vector measured in a natural way by the conductor exponent. Finally, we show that the newform is a test vector for $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_n$ and $\operatorname {\mathrm {GL}}_n \times \operatorname {\mathrm {GL}}_{n - 1}$ Rankin–Selberg integrals when the second representation is unramified. This theory parallels an analogous nonarchimedean theory due to Jacquet, Piatetski-Shapiro, and Shalika; combined, this completes a global theory of newforms for automorphic representations of $\operatorname {\mathrm {GL}}_n$ over number fields. By-products of the proofs include new proofs of Stade’s formulæ and a new resolution of the test vector problem for archimedean Godement–Jacquet zeta integrals.

We consider the homology theory of étale groupoids introduced by Crainic and Moerdijk [A homology theory for étale groupoids. J. Reine Angew. Math. 521 (2000), 25–46], with particular interest to groupoids arising from topological dynamical systems. We prove a Künneth formula for products of groupoids and a Poincaré-duality type result for principal groupoids whose orbits are copies of an Euclidean space. We conclude with a few example computations for systems associated to nilpotent groups such as self-similar actions, and we generalize previous homological calculations by Burke and Putnam for systems which are analogues of solenoids arising from algebraic numbers. For the latter systems, we prove the HK conjecture, even when the resulting groupoid is not ample.

We show the thinness of $7$ of the $40$ hypergeometric groups having a maximally unipotent monodromy in $\mathrm{Sp}(6)$.

A suitable notion of weak amenability for dual Banach algebras, which we call weak Connes amenability, is defined and studied. Among other things, it is proved that the measure algebra M(G) of a locally compact group G is always weakly Connes amenable. It can be a complement to Johnson’s theorem that $L^1(G)$ is always weakly amenable [10].

We consider self-propelled rigid bodies interacting through local body-attitude alignment modelled by stochastic differential equations. We derive a hydrodynamic model of this system at large spatio-temporal scales and particle numbers in any dimension $n \geq 3$. This goal was already achieved in dimension $n=3$ or in any dimension $n \geq 3$ for a different system involving jump processes. However, the present work corresponds to huge conceptual and technical gaps compared with earlier ones. The key difficulty is to determine an auxiliary but essential object, the generalised collision invariant. We achieve this aim by using the geometrical structure of the rotation group, namely its maximal torus, Cartan subalgebra and Weyl group as well as other concepts of representation theory and Weyl’s integration formula. The resulting hydrodynamic model appears as a hyperbolic system whose coefficients depend on the generalised collision invariant.

Let $E/F$ be a quadratic unramified extension of non-archimedean local fields and $\mathbb H$ a simply connected semisimple algebraic group defined and split over F. We establish general results (multiplicities, test vectors) on ${\mathbb H} (F)$-distinguished Iwahori-spherical representations of ${\mathbb H} (E)$. For discrete series Iwahori-spherical representations of ${\mathbb H} (E)$, we prove a numerical criterion of ${\mathbb H} (F)$-distinction. As an application, we classify the ${\mathbb H} (F)$-distinguished discrete series representations of ${\mathbb H} (E)$ corresponding to degree $1$ characters of the Iwahori-Hecke algebra.

We apply Takesaki’s and Connes’s ideas on structure analysis for type III factors to the study of links (a short term of Markov kernels) appearing in asymptotic representation theory.

Clausen a prédit que le groupe des classes d’idèles de Chevalley d’un corps de nombres F apparaît comme le premier K-groupe de la catégorie des F-espaces vectoriels localement compacts. Cela s’est avéré vrai, et se généralise même aux groupes K supérieurs dans un sens approprié. Nous remplaçons F par une $\mathbb {Q}$-algèbre semi-simple, et obtenons le groupe des classes d’idèles noncommutatif de Fröhlich de manière analogue, modulo les éléments de norme réduite une. Même dans le cas du corps de nombres, notre preuve est plus simple que celle existante, et repose sur le théorème de localisation pour des sous-catégories percolées. Enfin, en utilisant la théorie des corps de classes, nous interprétons la loi de réciprocité d’Hilbert (ainsi qu’une variante noncommutative) en termes de nos résultats.

Clausen predicted that Chevalley’s idèle class group of a number field F appears as the first K-group of the category of locally compact F-vector spaces. This has turned out to be true and even generalizes to the higher K-groups in a suitable sense. We replace F by a semisimple $\mathbb {Q}$-algebra and obtain Fröhlich’s noncommutative idèle class group in an analogous fashion, modulo the reduced norm one elements. Even in the number field case, our proof is simpler than the existing one and based on the localization theorem for percolating subcategories. Finally, using class field theory as input, we interpret Hilbert’s reciprocity law (as well as a noncommutative variant) in terms of our results.

We show that for every countable group, any sequence of approximate homomorphisms with values in permutations can be realized as the restriction of a sofic approximation of an orbit equivalence relation. Moreover, this orbit equivalence relation is uniquely determined by the invariant random subgroup of the approximate homomorphisms. We record applications of this result to recover various known stability and conjugacy characterizations for almost homomorphisms of amenable groups.

We extend the Becker–Kechris topological realization and change-of-topology theorems for Polish group actions in several directions. For Polish group actions, we prove a single result that implies the original Becker–Kechris theorems, as well as Sami’s and Hjorth’s sharpenings adapted levelwise to the Borel hierarchy; automatic continuity of Borel actions via homeomorphisms and the equivalence of ‘potentially open’ versus ‘orbitwise open’ Borel sets. We also characterize ‘potentially open’ n-ary relations, thus yielding a topological realization theorem for invariant Borel first-order structures. We then generalize to groupoid actions and prove a result subsuming Lupini’s Becker–Kechris-type theorems for open Polish groupoids, newly adapted to the Borel hierarchy, as well as topological realizations of actions on fiberwise topological bundles and bundles of first-order structures.

Our proof method is new even in the classical case of Polish groups and is based entirely on formal algebraic properties of category quantifiers; in particular, we make no use of either metrizability or the strong Choquet game. Consequently, our proofs work equally well in the non-Hausdorff context, for open quasi-Polish groupoids and more generally in the point-free context, for open localic groupoids.

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

In this paper we take up the classical sup-norm problem for automorphic forms and view it from a new angle. Given a twist minimal automorphic representation $\pi$ we consider a special small $\mathrm{GL}_2(\mathbb{Z}_p)$-type V in $\pi$ and prove global sup-norm bounds for an average over an orthonormal basis of V. We achieve a non-trivial saving when the dimension of V grows.