We give a simple description of the coordinate ring of the universal centralizer associated to a simply connected semisimple group. To this end, we prove a general result on Weil restriction of affine schemes X over the Cartan subalgebra 𝔱 equipped with a compatible action of the Weyl group W. Specifically, we show that the coordinate ring of the scheme Res^W(X) of W-fixed points of Weil restriction of X to the GIT quotient 𝔱//W can be obtained from the coordinate ring of X by applying Demazure operators if and only if the scheme Res^W(X) is integral.

We investigate the effect of a hyperbolic (or, more generally, isolated as an invariant set) closed Reeb orbit on the dynamics of a Reeb flow on the (2n-1)-dimensional standard contact sphere, extending two results previously known for Hamiltonian diffeomorphisms to the Reeb setting. In particular, we show that under very mild dynamical convexity type assumptions, the presence of one hyperbolic closed orbit implies the existence of infinitely many simple closed Reeb orbits. The second main result of the paper is a higher-dimensional Reeb analogue of the Le Calvez-Yoccoz theorem, asserting that no closed orbit of a non-degenerate dynamically convex Reeb pseudo-rotation is locally maximal, i.e., isolated as an invariant set. The key new ingredient of the proofs is a Reeb variant of the crossing energy theorem.

In this paper we continue investigating connections between Floer theory and dynamics of Hamiltonian systems, focusing on the barcode entropy of Reeb flows. Barcode entropy is the exponential growth rate of the number of not-too-short bars in the Floer or symplectic homology persistence module. The key novel result is that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set. This, combined with the fact that the topological entropy bounds the barcode entropy from above, established by Fender, Lee and Sohn, implies that in dimension three the two types of entropy agree. The main new ingredient of the proof is a variant of the Crossing Energy Theorem for Reeb flows.

Given a hyperspherical $G$-variety $\mathscr X$ we consider the zero moment level $\Lambda_{\mathscr X}\subset{\mathscr X}$ of the action of a Borel subgroup $B\subset G$. We conjecture that $\Lambda_{\mathscr X}$ is Lagrangian. For the dual $G^\vee$-variety ${\mathscr X}^\vee$, we conjecture that that there is a bijection between the sets of irreducible components $\mathrm{Irr}\Lambda_{\mathscr X}$ and $\mathrm{Irr}\Lambda_{{\mathscr X}^\vee}$. We check this conjecture for all the hyperspherical equivariant slices, and for all the basic classical Lie superalgebras.

It has been known for a long time that Ext's between IC-sheaves may often be expressed in terms of Hom's between cohomology groups. We prove a more general result under weaker assumptions. The result is used to describe the action of the derived Satake equivalence on !-pure objects and show that the equivalence enjoys a new kind of functoriality with respect to morphisms of reductive groups. We find and prove normality of the symplectic dual X^! for many smooth affine Hamiltonian G-varieties X, including X=T^*(G/H) for all connected reductive subgroups H of G. We also describe the symplectic duals M^! in the case of Coulomb branches and prove that M^! has symplectic singularities.

. This paper is a follow-up to the authors' recent work on barcode entropy. We study the growth of the barcode of the Floer complex for the iterates of a compactly supported Hamiltonian diffeomorphism. In particular, we introduce sequential barcode entropy which has properties similar to barcode entropy, bounds it from above and is more sensitive to the barcode growth. In the same vein, we explore another variant of barcode entropy based on the total persistence growth and revisit the relation between the growth of periodic orbits and topological entropy. We also study the behavior of the spectral norm, aka the y-norm, under iterations. We show that the y-norm of the iterates is separated from zero when the map has sufficiently many hyperbolic periodic points and, as a consequence, it is separated from zero C infinity-generically in dimension two. We also touch upon properties of the barcode entropy of pseudo-rotations and, more generally, y-almost periodic maps.

We study topological entropy of compactly supported Hamiltonian diffeomorphisms from a perspective of persistent homology and Floer theory. We introduce barcode entropy, a Floer-theoretic invariant of a Hamiltonian diffeomorphism, measuring exponential growth under iterations of the number of not-too-short bars in the barcode of the Floer complex. We prove that the barcode entropy is bounded from above by the topological entropy and, conversely, that the barcode entropy is bounded from below by the topological entropy of any hyperbolic invariant set, e.g., a hyperbolic horseshoe. As a consequence, we conclude that for Hamiltonian diffeomorphisms of surfaces the barcode entropy is equal to the topological entropy.

We introduce and study the barcode entropy for geodesic flows of closed Riemannian manifolds, which measures the exponential growth rate of the number of not-too-short bars in the Morse-theoretic barcode of the energy functional. We prove that the barcode entropy bounds from below the topological entropy of the geodesic flow and, conversely, bounds from above the topological entropy of any hyperbolic compact invariant set. As a consequence, for Riemannian metrics on surfaces, the barcode entropy is equal to the topological entropy. A key to the proofs and of independent interest is a crossing energy theorem for gradient flow lines of the energy functional.

The main result of the paper is that for any closed symplectic manifold the spectral norm of the iterates of a Hamiltonian diffeomorphism is locally uniformly bounded away from zero $C^\infty$-generically.

The group scheme of universal centralizers of a complex reductive group G has a quantization called the spherical nil-DAHA. The category of modules over this ring is equivalent, as a symmetric monoidal category, to the category of bi-Whittaker D-modules on G. We construct a braided monoidal equivalence, called the Knop-Ngô functor, of this category with a full monoidal subcategory of the abelian category of Ad(G)-equivariant D-modules, establishing a D-module abelian counterpart of an equivalence established by Bezrukavnikov and Deshpande, in a different way. As an application of our methods, we prove conjectures of Ben-Zvi and Gunningham by relating this equivalence to parabolic induction and prove a conjecture of Braverman and Kazhdan in the D-module setting.

We study lower semi-continuity properties of the volume, i.e., the surface area, of a closed Lagrangian manifold with respect to the Hofer- and γ-distance on a class of monotone Lagrangian submanifolds Hamiltonian isotopic to each other. We prove that volume is γ-lower semi-continuous in two cases. In the first one the volume form comes from a Kähler metric with a large group of Hamiltonian isometries, but there are no additional constraints on the Lagrangian submanifold. The second one is when the volume is taken with respect to any compatible metric, but the Lagrangian submanifold must be a torus. As a consequence, in both cases, the volume is Hofer lower semi-continuous.

We study the multiplicity problem for prime closed orbits of dynamically convex Reeb flows on the boundary of a star-shaped domain in ℝ^2n. The first of our two main results asserts that such a flow has at least n prime closed Reeb orbits, improving the previously known lower bound by a factor of two and settling a long-standing open question. The second main theorem is that when, in addition, the domain is centrally symmetric and the Reeb flow is non-degenerate, the flow has either exactly n or infinitely many prime closed orbits. This is a higher-dimensional contact variant of Franks' celebrated 2-or-infinity theorem and, viewed from the symplectic dynamics perspective, settles a particular case of the contact Hofer-Zehnder conjecture. The proofs are based on several auxiliary results of independent interest on the structure of the filtered symplectic homology and the properties of closed orbits.

The post-compression technique based on self-phase modulation of high-energy pulses leads to an increase in achievable peak power and intensity. Typically, the pulses considered in experiments have been less than 100 fs in duration. Here, the method is applied to the ELFIE laser system at the LULI facility, for a pulse of 7 J energy and an initial measured duration of 350 fs. A 5-mm-thick fused silica window and a 2 mm cyclic-olefin polymer were used as optical nonlinear materials. The 9 cm diameter beam was spectrally broadened to a bandwidth corresponding to 124 fs Fourier-limited pulse duration, and then it was partly post-compressed to 200 fs. After measuring the spatial spectra of the beam fluence, a uniform gain factor of 4 increase in the fluctuations over the studied range of frequencies is observed, due to small-scale self-focusing.

In the context of symplectic dynamics, pseudo-rotations are Hamiltonian diffeomorphisms with finite and minimal possible number of periodic orbits. These maps are of interest in both dynamics and symplectic topology. We show that a closed, monotone symplectic manifold, which admits a non-degenerate pseudo-rotation, must have a deformed quantum Steenrod square of the top degree element, and hence non-trivial holomorphic spheres. This result (partially) generalizes a recent work by Shelukhin and complements the results by the authors on non-vanishing Gromov-Witten invariants of manifolds admitting pseudo-rotations.

Let G be a complex semisimple group and U its maximal unipotent subgroup. We study the algebra D(G/U) of algebraic differential operators on G/U and also its quasi-classical counterpart: the algebra of regular functions on the cotangent bundle. A long time ago, Gelfand and Graev have constructed an action of the Weyl group on D(G/U) by algebra automorphisms. The Gelfand-Graev construction was not algebraic, it involved analytic methods in an essential way. We give a new algebraic construction of the Gelfand-Graev action, as well as its quasi-classical counterpart. Our approach is based on Hamiltonian reduction and involves the ring of Whittaker differential operators on G/U, a twisted analogue of D(G/U). Our main result has an interpretation, via geometric Satake, in terms of spherical perverse sheaves on the affine Grassmanian for the Langlands dual group.

Continuing the study of Hamiltonian pseudo-rotations of projective spaces, we focus on the conjecture that the fixed-point data set (the actions and the linearized flows at one-periodic orbits) of a pseudo-rotation exactly matches that data for a suitable unique true rotation even though the two maps can have very different dynamics. We prove this conjecture in several instances, e.g., for strongly non-degenerate pseudo-rotations of ${{\mathbb C}{\mathbb P}}^2$ with some notable exceptions, which we call ghost pseudo-rotations. The existence of ghost pseudo-rotations is a completely open question. The conjecture is closely related to the properties of the action and index spectra of pseudo-rotations, and ghost pseudo-rotations, if they exist, satisfy all known restrictions on the fixed-point data for pseudo-rotations but these data are distinctly different from the data for any rotation. The main new ingredient of the proofs is purely combinatorial and of independent interest. This is the index divisibility theorem connecting the divisibility properties of the Conley--Zehnder index sequence for the iterates of a map with the properties of its spectrum.

We give a different and simpler proof of a slightly modified (and weaker) variant of a recent theorem of Shelukhin extending Franks' two-or-infinitely-many theorem to Hamiltonian diffeomorphisms in higher dimensions and establishing a sufficiently general case of the Hofer-Zehnder conjecture. A few ingredients of our proof are common with Shelukhin's original argument, the key of which is Seidel's equivariant pair-of-pants product, but the new proof highlights a different aspect of the periodic orbit dynamics of Hamiltonian diffeomorphisms.

Let G be a reductive group and L a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between Ad-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) D-modules on G and L, respectively. Bezrukavnikov and Yom Din proved, generalizing a classic result of Lusztig, that these functors are exact. In this paper, we consider a special case where L = T is a maximal torus. We give explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra D-module on G x T. We show that this module is flat over D(T), which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of D-modules.

The main theme of this paper is the dynamics of Reeb flows with symmetries on the standard contact sphere. We introduce the notion of strong dynamical convexity for contact forms invariant under a group action, supporting the standard contact structure, and prove that in dimension $2n+1$ any such contact form satisfying a condition slightly weaker than strong dynamical convexity has at least $n+1$ simple closed Reeb orbits. For contact forms with antipodal symmetry, we prove that strong dynamical convexity is a consequence of ordinary convexity. In dimension five or greater, we construct examples of antipodally symmetric dynamically convex contact forms which are not strongly dynamically convex, and thus not contactomorphic to convex ones via a contactomorphism commuting with the antipodal map. Finally, we relax this condition on the contactomorphism furnishing a condition that has non-empty $C^1$-interior.