Volume 2019, Issue 750

The preservation principle of the local theta correspondence predicts the existence of a chain of irreducible supercuspidal representations of p -adic classical groups. In this paper, we give an explicit characterization of the chain starting from an irreducible supercuspidal representations of a unitary group of one variable or an orthogonal group of two variables. In particular, we define the Lusztig-like correspondence of generic cuspidal data for p -adic groups and establish its relation with local theta correspondence of supercuspidal representations for p -adic dual pairs.

This is the second in a series of papers ([7] and [6] are the others) that studies the behavior of harmonic maps into the Weil–Petersson completion 𝒯¯{\overline{\mathcal{T}}} of Teichmüller space. The boundary of 𝒯¯{\overline{\mathcal{T}}} is stratified by lower-dimensional Teichmüller spaces and the normal space to each stratum is a product of copies of a singular space 𝐇¯{\overline{\bf H}} called the model space. The significance of 𝐇¯{\overline{\bf H}} is that it captures the singular behavior of the Weil–Petersson geometry of 𝒯¯{\overline{\mathcal{T}}}. The main result of the paper is that certain subsets of 𝐇¯{\overline{\bf H}} are essentially regular in the sense that harmonic maps to those spaces admit uniform approximation by affine functions. This is a modified version of the notion of essential regularity introduced by Gromov–Schoen in [12] for maps into Euclidean buildings and is one of the key ingredients in proving superrigidity. In the process, we introduce new coordinates on 𝐇¯{\overline{\bf H}} and estimate the metric and its derivatives with respect to the new coordinates. These results form the technical core for studying the analytic behavior of harmonic maps into the completion of Teichmüller space and are utilized in our subsequent paper [6], where we prove the holomorphic rigidity of the Teichmüller space and several rigidity results for the mapping class group.

In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel (n,0){(n,0)}-form, just like the Calabi–Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

The understanding of the topology of the spectra of quantum Schubert cell algebras hinges on the description of their prime factors by ideals invariant under the maximal torus of the ambient Kac–Moody group. We give an explicit description of these prime quotients by expressing their Cauchon generators in terms of sequences of normal elements in chains of subalgebras. Based on this, we construct large families of quantum clusters for all of these algebras and the quantum Richardson varieties associated to arbitrary symmetrizable Kac–Moody algebras and all pairs of Weyl group elements. Along the way we develop a quantum version of the Fomin–Zelevinsky twist map for all quantum Richardson varieties. Furthermore, we establish an explicit relationship between the Goodearl–Letzter and Cauchon approaches to the descriptions of the spectra of symmetric CGL extensions.

We answer three related open questions about the model theory of valued differential fields introduced by Scanlon. We show that they eliminate imaginaries in the geometric language introduced by Haskell, Hrushovski and Macpherson and that they have the invariant extension property. These two results follow from an abstract criterion for the density of definable types in enrichments of algebraically closed valued fields. Finally, we show that this theory is metastable.

We study representations of wreath product analogues of categories of finite sets. This includes the category of finite sets and injections (studied by Church, Ellenberg, and Farb) and the opposite of the category of finite sets and surjections (studied by the authors in previous work). We prove noetherian properties for the injective version when the group in question is polycyclic-by-finite and use it to deduce general twisted homological stability results for such wreath products and indicate some applications to representation stability. We introduce a new class of formal languages (quasi-ordered languages) and use them to deduce strong rationality properties of Hilbert series of representations for the surjective version when the group is finite.

We construct sharply 2-transitive groups of characteristic 0 without regular normal subgroups. These groups act sharply 2-transitively by conjugation on their involutions. This answers a long-standing open question.

In this short note we show how to modify the construction of non-split sharply 2-transitive groups of characteristic 0 given in [2] to allow for arbitrary fields of characteristic 0.

We give a complete answer to which homogeneous groups admit homogeneous distances for which the Besicovitch Covering Property (BCP) holds. In particular, we prove that a stratified group admits homogeneous distances for which BCP holds if and only if the group has step 1 or 2. These results are obtained as consequences of a more general study of homogeneous quasi-distances on graded groups. Namely, we prove that a positively graded group admits continuous homogeneous quasi-distances satisfying BCP if and only if any two different layers of the associated positive grading of its Lie algebra commute. The validity of BCP has several consequences. Its connections with the theory of differentiation of measures is one of the main motivations of the present paper. As a consequence of our results, we get for instance that a stratified group can be equipped with some homogeneous distance so that the differentiation theorem holds for each locally finite Borel measure if and only if the group has step 1 or 2. The techniques developed in this paper allow also us to prove that sub-Riemannian distances on stratified groups of step 2 or higher never satisfy BCP. Using blow-up techniques this is shown to imply that on a sub-Riemannian manifold the differentiation theorem does not hold for some locally finite Borel measure.