Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups, II

Let $M\stackrel {\rho _0}{\curvearrowleft }S$ be a $C^\infty $ locally free action of a connected simply connected solvable Lie group S on a closed manifold M. Roughly speaking, $\rho _0$ is parameter rigid if any $C^\infty $ locally free action of S on M having the same orbits as $\rho _0$ is $C^\infty $ conjugate to $\rho _0$ . In this paper we prove two types of result on parameter rigidity. First let G be a connected semisimple Lie group with finite center of real rank at least $2$ without compact factors nor simple factors locally isomorphic to $\mathop {\mathrm {SO}}\nolimits _0(n,1)(n\,{\geq}\, 2)$ or $\mathop {\mathrm {SU}}\nolimits (n,1)(n\geq 2)$ , and let $\Gamma $ be an irreducible cocompact lattice in G. Let $G=KAN$ be an Iwasawa decomposition. We prove that the action $\Gamma \backslash G\curvearrowleft AN$ by right multiplication is parameter rigid. One of the three main ingredients of the proof is the rigidity theorems of Pansu, and Kleiner and Leeb on the quasi-isometries of Riemannian symmetric spaces of non-compact type. Secondly we show that if $M\stackrel {\rho _0}{\curvearrowleft }S$ is parameter rigid, then the zeroth and first cohomology of the orbit foliation of $\rho _0$ with certain coefficients must vanish. This is a partial converse to the results in the author’s [Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups. Geom. Topol. 21(1) (2017), 157–191], where we saw sufficient conditions for parameter rigidity in terms of vanishing of the first cohomology with various coefficients.

A Poisson transform adapted to the Rumin complex

Let [Formula: see text] be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use [Formula: see text]-invariant differential forms on [Formula: see text] to construct [Formula: see text]-equivariant Poisson transforms mapping differential forms on [Formula: see text] to differential forms on [Formula: see text]. Such invariant forms can be constructed using finite-dimensional representation theory. In this general setting, we first prove that the transforms that always produce harmonic forms are exactly those that descend from the de Rham complex on [Formula: see text] to the associated Bernstein–Gelfand–Gelfand (or BGG) complex in a well defined sense. The main part of this paper is devoted to an explicit construction of such transforms with additional favorable properties in the case that [Formula: see text]. Thus, [Formula: see text] is [Formula: see text] with its natural CR structure and the relevant BGG complex is the Rumin complex, while [Formula: see text] is complex hyperbolic space of complex dimension [Formula: see text]. The construction is carried out both for complex and for real differential forms and the compatibility of the transforms with the natural operators that are available on their sources and targets are analyzed in detail.

Prime II1 factors arising from irreducible lattices in products of rank one simple Lie groups

We prove that if Γ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the {\mathrm{II}_{1}} factor {L(\Gamma)} is prime. In particular, we deduce that the {\mathrm{II}_{1}} factors associated to the arithmetic groups {\mathrm{PSL}_{2}(\mathbb{Z}[\sqrt{d}])} and {\mathrm{PSL}_{2}(\mathbb{Z}[S^{-1}])} are prime for any square-free integer {d\geq 2} with {d\not\equiv 1~{}(\operatorname{mod}\,4)} and any finite non-empty set of primes S. This provides the first examples of prime {\mathrm{II}_{1}} factors arising from lattices in higher rank semisimple Lie groups. More generally, we describe all tensor product decompositions of {L(\Gamma)} for icc countable groups Γ that are measure equivalent to a product of non-elementary hyperbolic groups. In particular, we show that {L(\Gamma)} is prime, unless Γ is a product of infinite groups, in which case we prove a unique prime factorization result for {L(\Gamma)}.

A Simplified Optimal-Switching-Sequence MPC with Finite-Control-Set Moving Horizon Optimization for Grid-Connected Inverter

Optimal-switching-sequence model-predictive-control (OSS-MPC) strategy, a popular kind of continuous-control-set MPC (CCS-MPC), has been used to address the variable frequency of finite-control-set MPC (FCS-MPC). Since the digital signal processor (DSP) does not have a dedicated divider, it requires a large amount of computation for the division operations in OSS searching. Here, a simplified OSS-MPC absorbing the merits of FCS-MPC is proposed to reduce the computational burden of conventional OSS-MPC. The proposed method uses a novel cost function whose candidates are the finite center vectors instead of switching sequences, which can avoid duty cycle computation and convert the moving horizon optimization of CCS-MPC into that of FCS-MPC. Besides, the derivatives of the active and reactive powers are divided into the constant terms and variable terms. The constant terms are extracted from the cost function. Experimental and simulation results show that the computational amount of the proposed algorithm is only 36.34% of that of the conventional OSS-MPC. Meanwhile, the simplified OSS-MPC still maintains the excellent dynamic and steady state performance of conventional OSS-MPC.

Evidence of pair-density wave in spin-valley locked systems

Cooper pairs with a finite center-of-mass momentum form a remarkable state in which the superconducting order parameter is modulated periodically in space. Although intense interest in such a “pair-density wave” (PDW) state has emerged due to recent discoveries in highTcsuperconductors, there is little theoretical understanding of the mechanism driving this exotic state. The challenge is that many competing states lie close in energy in seemingly simple models, such as the Hubbard model, in the strongly correlated regime. Here, we show that inversion symmetry breaking and the resulting spin-valley locking can promote PDWs over more commonly found spin stripes through frustration against magnetic order. Specifically, we find the first robust evidence for a PDW within density matrix renormalization group simulation of a simple fermionic model. Our results point to a tantalizing possibility in hole-doped group VI transition metal dichalcogenides, with spin-valley locked band structure and moderate correlations.

A compact imbedding of Riemannian Symmetric Spaces

Let G be a connected real semisimple Lie group with finite center and θ be a Cartan involution of G. Suppose that K is the maximal compact subgroup of G corresponding to the Cartan involution θ. The coset space X = G/K is then a Riemannian symmetric space. In this paper, by choosing the reduced root system Σ0 = {α ∈ Σ | 2α /∈ Σ; α 2 ∈/ Σ} insteads of the restricted root system Σ and using the action of the Weyl group, firstly we construct a compact real analytic manifold Xb 0 in which the Riemannian symmetric space G/K is realized as an open subset and that G acts analytically on it, then we consider the real analytic structure of Xb 0 induced from the real analytic srtucture of AbIR, the compactification of the corresponding vectorial part.

Ergodic boundary representations

We prove a von Neumann-type ergodic theorem for averages of unitary operators arising from the Furstenberg–Poisson boundary representation (the quasi-regular representation) of any lattice in a non-compact connected semisimple Lie group with finite center.

A remark on fullness of some group measure space von Neumann algebras

Recently Houdayer and Isono have proved, among other things, that every biexact group $\unicode[STIX]{x1D6E4}$ has the property that for any non-singular strongly ergodic essentially free action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ on a standard measure space, the group measure space von Neumann algebra $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(X)$ is full. In this paper, we prove the same property for a wider class of groups, notably including $\text{SL}(3,\mathbb{Z})$. We also prove that for any connected simple Lie group $G$ with finite center, any lattice $\unicode[STIX]{x1D6E4}\leqslant G$, and any closed non-amenable subgroup $H\leqslant G$, the non-singular action $\unicode[STIX]{x1D6E4}\curvearrowright G/H$ is strongly ergodic and the von Neumann factor $\unicode[STIX]{x1D6E4}\ltimes L^{\infty }(G/H)$ is full.

Amount of failure of upper-semicontinuity of entropy in non-compact rank-one situations, and Hausdorff dimension

Recently, Einsiedler and the authors provided a bound in terms of escape of mass for the amount by which upper-semicontinuity for metric entropy fails for diagonal flows on homogeneous spaces $\unicode[STIX]{x1D6E4}\setminus G$, where $G$ is any connected semisimple Lie group of real rank one with finite center, and $\unicode[STIX]{x1D6E4}$ is any non-uniform lattice in $G$. We show that this bound is sharp, and apply the methods used to establish bounds for the Hausdorff dimension of the set of points that diverge on average.