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We study the transfer between small special unipotent representations for all equal rank real forms of type E6 and E7. As a consequence, one can verify these modules are unitarity using the results of Wallach and Zhu. Moreover, the K-spectra of these modules can be obtained explicitly. [ABSTRACT FROM AUTHOR]

In this article, by studying the Bernstein degrees and Goldie rank polynomials, we establish a comparison between the irreducible representations of G = GLn(ℂ) possessing the minimal Gelfand–Kirillov dimension and those induced from finite-dimensional representations of the maximal parabolic subgroup of G of type (n − 1,1). We give the transition matrix between the two bases for the corresponding coherent families. [ABSTRACT FROM AUTHOR]

Let G be a special linear group over the real, the complex or the quaternion, or a special unitary group. In this note, we determine all special unipotent representations of G in the sense of Arthur and Barbasch–Vogan, and show in particular that all of them are unitarizable. [ABSTRACT FROM AUTHOR]

Let W be a real symplectic space and (G, G′) an irreducible dual pair in Sp(W), in the sense of Howe, with G compact. Let G ˜ be the preimage of G in the metaplectic group Sp ˜ (W) . Given an irreducible unitary representation Π of G ˜ that occurs in the restriction of the Weil representation to G ˜ , let ΘΠ denote its character. We prove that, for a suitable embedding T of Sp ˜ (W) in the space of tempered distributions on W, the distribution T(Θ̌Π) admits an asymptotic limit, and the limit is a nilpotent orbital integral. As an application, we compute the wave front set of Π′, the representation of G ′ ˜ dual to Π, by elementary means. [ABSTRACT FROM AUTHOR]

In a previous paper, the author and his collaborator studied the problem of lifting Hamiltonian group actions on symplectic varieties and Lagrangian subvarieties to their graded deformation quantizations and apply the general results to coadjoint orbit method for semisimple Lie groups. Only even quantizations were considered there. In this paper, these results are generalized to the case of general quantizations with arbitrary periods. The key step is to introduce an enhanced version of the (truncated) period map defined by Bezrukavnikov and Kaledin for quantizations of any smooth symplectic variety X, with values in the space of Picard Lie algebroid over X. As an application, we study quantizations of nilpotent orbits of real semisimple groups satisfying certain codimension condition. [ABSTRACT FROM AUTHOR]

We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between $ (B^{ts/2}A^{(1-t)s} B^{ts/2})^{1/s} $ (B ts / 2 A (1 − t) s B ts / 2 ) 1 / s and the t-spectral mean $ A\natural_t B :=(A^{-1}\sharp B)^{t}A(A^{-1}\sharp B)^{t} $ A ♮ t B := (A − 1 ♯B) t A (A − 1 ♯B) t of two positive semidefinite matrices A and B, where $ A\sharp B $ A ♯B is the geometric mean, and the t-spectral mean is the dominant one. The limit involving t-spectral mean is also studied. We then extend all the results in the context of symmetric spaces of negative curvature. [ABSTRACT FROM AUTHOR]

Let M = G / H be a compact, simply connected, Riemannian homogeneous space, where G is (almost) effective and H is a simple Lie group. In this paper, we first classify all G-naturally reductive metrics on M, and then all G-geodesic orbit metrics on M. [ABSTRACT FROM AUTHOR]

We establish some new properties of spectral geometric mean. In particular, we prove a log majorization relation between ( B t s / 2 A ( 1 − t ) s B t s / 2 ) 1 / s and the t-spectral mean A ♮ t B := ( A − 1 ♯ B ) t A ( A − 1 ♯ B ) t of two positive semidefinite matrices A and B, where A ♯ B is the geometric mean, and the t-spectral mean is the dominant one. The limit involving t-spectral mean is also studied. We then extend all the results in the context of symmetric spaces of negative curvature. [ABSTRACT FROM AUTHOR]

In our earlier paper, we generalize the one-parameter (c) family of inhomogeneous first-order differential operator representations of the orthogonal Lie algebras arising from conformal transformations to those of orthosymplectic Lie superalgebras, and determine the irreducible condition. This paper deals with the cases when the irreducible condition fails. We prove that if n − m − 1 > 0 and c is an integer satisfying 1 ≤ c ≤ n − m − 1, the representation of osp(2n + 2|2m) has a composition series of length 2, and when n − m − 1 ≥ 0 and c ∈ −ℕ, the representation of osp(2n + 2|2m) has a composition series of length 3, where ℕ is the set of nonnegative integers. Moreover, we show that if c ∈ (max{n − m, 0} − 1/2 − ℕ) ∪ (−ℕ), the representation of osp(2n + 3|2m) has a composition series of length 2. In particular, we obtain an explicit presentation of the irreducible module with highest weight ℓλ2 − λ1, where ℓ is any positive integer and it is not a generalized Verma module. [ABSTRACT FROM AUTHOR]

Let 픤 be a complex semisimple Lie algebra and let θ be a finite-order automorphism of 픤 . Let 픤 0 be the subalgebra { X ∈ 픤 : θ ⁢ (X) = X } . In this article, we study for which pairs (V 1 , V 2) , consisting of two irreducible finite-dimensional representations of 픤 , we have res 픤 0 ⁡ V 1 ≃ res 픤 0 ⁡ V 2 . In many cases, we show that V 1 and V 2 have isomorphic restrictions to 픤 0 if and only if V 1 is isomorphic to V 2 σ for some outer automorphism σ of 픤 . [ABSTRACT FROM AUTHOR]

A linear étale representation of a complex algebraic group G is given by a complex algebraic G-module V such that G has a Zariski-open orbit in V and $\dim G=\dim V$. A current line of research investigates which reductive algebraic groups admit such étale representations, with a focus on understanding common features of étale representations. One source of new examples arises from the classification theory of nilpotent orbits in semisimple Lie algebras. We survey what is known about reductive algebraic groups with étale representations and then discuss two classical constructions for nilpotent orbit classifications due to Vinberg and to Bala and Carter. We determine which reductive groups and étale representations arise in these constructions and we work out in detail the relation between these two constructions. [ABSTRACT FROM AUTHOR]

This paper classifies the Dirac series for complex E 7 . As applications, we verify a few conjectures raised in 2011, 2019 and 2020 for this exceptional Lie group. In particular, according to Conjecture 1.1 of Barbasch and Pandžić [Dirac cohomology and unipotent representations of complex groups, Noncommutative Geometry and Global Analysis, Contemp. Math. 546, American Mathematical Society, Providence (2011), 1–22], our classification should be helpful for understanding the unitary dual of complex E 7 . [ABSTRACT FROM AUTHOR]

We give small deformations of a left-invariant complex structure on each simply connected semisimple compact Lie group of even dimension which are not biholomorphic to any left-invariant (right-invariant) complex structure by using the Kuranishi space. On such deformed complex manifolds, we prove the Borel–Weil–Bott type theorem, and we compute the cohomology of holomorphic tangent bundles. [ABSTRACT FROM AUTHOR]

We prove a uniqueness result of weak solutions to the nD (n ⩾ 3) Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term. [ABSTRACT FROM AUTHOR]

We prove that the local Rankin–Selberg integrals for principal series representations of the general linear groups agree with certain simple integrals over the Rankin–Selberg subgroups, up to certain constants given by the local gamma factors. [ABSTRACT FROM AUTHOR]

We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric. [ABSTRACT FROM AUTHOR]

We show that every sequence of torsion-free arithmetic congruence lattices in PGL (2 , R) or PGL (2 , C) satisfies a strong quantitative version of the limit multiplicity property. We deduce that for R > 0 in certain range, growing linearly in the degree of the invariant trace field, the volume of the R-thin part of any congruence arithmetic hyperbolic surface or congruence arithmetic hyperbolic 3-manifold M is of order at most Vol (M) 11 / 12 . As an application we prove Gelander's conjecture on homotopy type of arithmetic hyperbolic 3-manifolds: we show that there are constants A, B such that every such manifold M is homotopy equivalent to a simplicial complex with at most A Vol (M) vertices, all of degrees bounded by B. [ABSTRACT FROM AUTHOR]

The endoscopic transfer factor is expressed as difference of characters for the even and odd parts of the spin modules, or Dirac index of the trivial representation. The lifting of tempered characters in terms of index of Dirac cohomology is calculated explicitly. [ABSTRACT FROM AUTHOR]

We provide a microlocal necessary condition for distinction of admissible representations of real reductive groups in the context of spherical pairs. Let ${\mathbf {G}}$ be a complex algebraic reductive group and ${\mathbf {H}}\subset {\mathbf {G}}$ be a spherical algebraic subgroup. Let ${\mathfrak {g}},{\mathfrak {h}}$ denote the Lie algebras of ${\mathbf {G}}$ and ${\mathbf {H}}$ , and let ${\mathfrak {h}}^{\bot }$ denote the orthogonal complement to ${\mathfrak {h}}$ in ${\mathfrak {g}}^*$. A ${\mathfrak {g}}$ -module is called ${\mathfrak {h}}$ -distinguished if it admits a nonzero ${\mathfrak {h}}$ -invariant functional. We show that the maximal ${\mathbf {G}}$ -orbit in the annihilator variety of any irreducible ${\mathfrak {h}}$ -distinguished ${\mathfrak {g}}$ -module intersects ${\mathfrak {h}}^{\bot }$. This generalises a result of Vogan [Vog91]. We apply this to Casselman–Wallach representations of real reductive groups to obtain information on branching problems, translation functors and Jacquet modules. Further, we prove in many cases that – as suggested by [Pra19, Question 1] – when H is a symmetric subgroup of a real reductive group G, the existence of a tempered H-distinguished representation of G implies the existence of a generic H-distinguished representation of G. Many of the models studied in the theory of automorphic forms involve an additive character on the unipotent radical of the subgroup $\bf H$ , and we have devised a twisted version of our theorem that yields necessary conditions for the existence of those mixed models. Our method of proof here is inspired by the theory of modules over W-algebras. As an application of our theorem we derive necessary conditions for the existence of Rankin–Selberg, Bessel, Klyachko and Shalika models. Our results are compatible with the recent Gan–Gross–Prasad conjectures for nongeneric representations [GGP20]. Finally, we provide more general results that ease the sphericity assumption on the subgroups, and apply them to local theta correspondence in type II and to degenerate Whittaker models. [ABSTRACT FROM AUTHOR]

Let G be a connected complex simple Lie group, and let G ^ d {\widehat{G}^{\mathrm{d}}} be the set of all equivalence classes of irreducible unitary representations with non-vanishing Dirac cohomology. We show that G ^ d {\widehat{G}^{\mathrm{d}}} consists of two parts: finitely many scattered representations, and finitely many strings of representations. Moreover, the strings of G ^ d {\widehat{G}^{\mathrm{d}}} come from L ^ d {\widehat{L}^{\mathrm{d}}} via cohomological induction and they are all in the good range. Here L runs over the Levi factors of proper θ-stable parabolic subgroups of G. It follows that figuring out G ^ d {\widehat{G}^{\mathrm{d}}} requires a finite calculation in total. As an application, we report a complete description of F ^ 4 d {\widehat{F}_{4}^{\mathrm{d}}}. [ABSTRACT FROM AUTHOR]

Let Z = G / H be the homogeneous space of a real reductive group and a unimodular real spherical subgroup, and consider the regular representation of G on L 2 (Z) . It is shown that all representations of the discrete series, that is, the irreducible subrepresentations of L 2 (Z) , have infinitesimal characters which are real and belong to a lattice. Moreover, let K be a maximal compact subgroup of G. Then each irreducible representation of K occurs in a finite set of such discrete series representations only. Similar results are obtained for the twisted discrete series, that is, the discrete components of the space of square integrable sections of a line bundle, given by a unitary character on an abelian extension of H. [ABSTRACT FROM AUTHOR]

We construct various systems of coherent states (SCS) on the O(D)-equivariant fuzzy spheres S Λ d ( d = 1 , 2 , D = d + 1 ) constructed in Fiore and Pisacane (J Geom Phys 132:423–451, 2018) and study their localizations in configuration space as well as angular momentum space. These localizations are best expressed through the O(D)-invariant square space and angular momentum uncertainties (Δ x) 2 , (Δ L) 2 in the ambient Euclidean space R D . We also determine general bounds (e.g., uncertainty relations from commutation relations) for (Δ x) 2 , (Δ L) 2 , and partly investigate which SCS may saturate these bounds. In particular, we determine O(D)-equivariant systems of optimally localized coherent states, which are the closest quantum states to the classical states (i.e., points) of S d . We compare the results with their analogs on commutative S d . We also show that on S Λ 2 our optimally localized states are better localized than those on the Madore–Hoppe fuzzy sphere with the same cutoff Λ . [ABSTRACT FROM AUTHOR]

Stoichiometric producer–grazer models are nonsmooth due to the Liebig’s Law of Minimum and can generate new dynamics such as bistability for producer–grazer interactions. Environmental noises can be extremely important and change dynamical behaviors of a stoichiometric producer–grazer model. In this paper, we consider a stochastically forced producer–grazer model and study the phenomena of noise-induced state switching between two stochastic attractors in the bistable zone. Namely, there is a frequent random hopping of phase trajectories between attracting basins of the attractors. In addition, by applying the stochastic sensitivity function technique, we construct the confidence ellipse and confidence band to find the configurational arrangement of equilibria and a limit cycle, respectively. [ABSTRACT FROM AUTHOR]

We prove differentiability of certain linear combinations of the Lyapunov spectra of a flow on a principal bundle of a semi-simple Lie group. The specific linear combinations that yield differentiability are determined by the finest Morse decomposition on the flag bundles. Differentiability is taken with respect to a differentiable structure on the gauge group, which is a Banach-Lie group. [ABSTRACT FROM AUTHOR]

In this paper we will discuss the orbits of the fixed-point group on the tori of the generalized symmetric spaces of SL 2 (k) where k is a finite field. Specifically, we will provide a characterization and classification of the maximal k-split and k-anisotropic tori. [ABSTRACT FROM AUTHOR]

We prove an Lp spectral multiplier theorem for functions of the K‐invariant sublaplacian L acting on the space of functions of fixed K‐type on the group SL(2,R). As an application we compute the joint Lp(SL(2,R)) spectrum of L and the derivative along K. [ABSTRACT FROM AUTHOR]

In this paper, we present a new fractional Tikhonov regularization method for solving an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting. Fractional Tikhonov regularization method not only retains the advantage of classical Tikhonov method, but also overcomes the effect of over-smoothing of classical Tikhonov method. We give the selection of regularization parameters of the new method and the corresponding error estimation. Furthermore, numerical results show that the fractional Tikhonov method outperforms the classical one. [ABSTRACT FROM AUTHOR]

We show that every effective action of a compact Lie group K on a unit sphere admits an explicit orbit whose principal curvatures are bounded from above by. [ABSTRACT FROM AUTHOR]

Recognised as one of the hallmarks of cancer, local cancer cell invasion is a complex multiscale process that combines the secretion of matrix-degrading enzymes with a series of altered key cell processes (such as abnormal cell proliferation and changes in cell–cell and cell–matrix adhesion leading to enhanced migration) to degrade important components of the surrounding extracellular matrix (ECM) and this way spread further in the human tissue. In order to gain a deeper understanding of the invasion process, we pay special attention to the interacting dynamics between the cancer cell population and various constituents of the surrounding tumour microenvironment. To that end, we consider the key role that ECM plays within the human body tissue, and in particular we focus on the special contribution of its fibrous proteins components, such as collagen and fibronectin, which play an important part in cell proliferation and migration. In this work, we consider the two-scale dynamic cross-talk between cancer cells and a two-component ECM (consisting of both a fibre and a non-fibre phase). To that end, we incorporate the interlinked two-scale dynamics of cell–ECM interactions within the tumour support that contributes simultaneously both to cell adhesion and to the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is embedded within a multiscale moving boundary approach for the invading cancer cell population, in the presence of cell adhesion at the tissue scale and cell-scale fibre redistribution activity and leading edge matrix-degrading enzyme molecular proteolytic processes. The overall modelling framework will be accompanied by computational results that will explore the impact on cancer invasion patterns of different levels of cell adhesion in conjunction with the continuous ECM fibres rearrangement. [ABSTRACT FROM AUTHOR]

Motivated by the study of symmetry breaking operators for indefinite orthogonal groups, we give a Gegenbauer expansion of the two variable function in terms of the ultraspherical polynomials and. Generalization, specialization, and limits of the expansion are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, the authors consider the problem of which (generalized) moment-angle manifolds admit Ricci positive metrics. For a simple polytope P, the authors can cut off one vertex v of P to get another simple polytope Pv, and prove that if the generalized moment-angle manifold corresponding to P admits a Ricci positive metric, the generalized moment-angle manifold corresponding to Pv also admits a Ricci positive metric. For a special class of polytope called Fano polytopes, the authors prove that the moment-angle manifolds corresponding to Fano polytopes admit Ricci positive metrics. Finally some conjectures on this problem are given. [ABSTRACT FROM AUTHOR]

Recently filter-based regularization methods have been well investigated for ill-posed problems when the forward operators are compact. There are many ill-posed problems connected with pseudo-differential operators. But there is no uniform method for this kind of problems. The work on generalization of filter-based regularization methods to pseudo-differential operator is necessary. In this paper, we present a regularization framework for solving the mildly ill-posed problems involved pseudo-differential operators. A general regularization method for this kind of problems is given. The order-optimal error estimates are derived under the usual source conditions. As an example, a new fractional Tikhonov regularization method could be cast into the general framework. Numerical experiments are conducted for showing the validity of the new fractional Tikhonov method. [ABSTRACT FROM AUTHOR]

Abstract: Shelstad's character identity is an equality between sums of characters of tempered representations in corresponding L‐packets of two real, semisimple, linear, algebraic groups that are inner forms to each other. We reconstruct this character identity in the case of the discrete series, using index theory of elliptic operators in the framework of K‐theory. Our geometric proof of the character identity is evidence that index theory can play a role in the classification of group representations via the Langlands program. [ABSTRACT FROM AUTHOR]

Sensing and reciprocating cellular systems (SARs) are important for the operation of many biological systems. Production in interferon (IFN) SARs is achieved through activation of the Jak-Stat pathway, and downstream upregulation of IFN regulatory factor (IRF)-7 and IFN transcription, but the role that high- and low-affinity IFNs play in this process remains unclear. We present a comparative between a minimal spatio-temporal partial differential equation model and a novel spatio-structural-temporal (SST) model for the consideration of receptor, binding, and metabolic aspects of SAR behaviour. Using the SST framework, we simulate single- and multi-cluster paradigms of IFN communication. Simulations reveal a cyclic process between the binding of IFN to the receptor, and the consequent increase in metabolism, decreasing the propensity for binding due to the internal feedback mechanism. One observes the effect of heterogeneity between cellular clusters, allowing them to individualise and increase local production, and within clusters, where we observe ‘subpopular quiescence’; a process whereby intra-cluster subpopulations reduce their binding and metabolism such that other such subpopulations may augment their production. Finally, we observe the ability for low-affinity IFN to communicate a long range signal, where high affinity cannot, and the breakdown of this relationship through the introduction of cell motility. Biological systems may utilise cell motility where environments are unrestrictive and may use fixed system, with low-affinity communication, where a localised response is desirable. [ABSTRACT FROM AUTHOR]

We provide an explicit combinatorial description of the embedding of the crystal of Kashiwara–Nakashima tableaux in types B and C into that of i -Lusztig data associated to a family of reduced expressions i of the longest element w 0 . The description of the embedding is simple and elementary using only the Schützenberger's jeu de taquin and RSK algorithm. A spinor model for classical crystals plays an important role as an intermediate object connecting Kashiwara–Nakashima tableaux and Lusztig data. [ABSTRACT FROM AUTHOR]

Consider the iteration of an invertible matrix on the projective space: are the Morse components normally hyperbolic? As far as we know, this was only stablished when the matrix is diagonalizable over the complex numbers. In this article we prove that this is true in the far more general context of an arbitrary element of a semisimple Lie group acting on its generalized flag manifolds: the so called translations on flag manifolds. This context encompasses the iteration of an invertible non-diagonazible matrix on the real or complex projective space, the classical flag manifolds of real or complex nested subspaces and also symplectic grassmanians. Without these tools from Lie theory we do not know how to solve this problem even for the projective space. [ABSTRACT FROM AUTHOR]

Spatio-temporal models have long been used to describe biological systems of cancer, but it has not been until very recently that increased attention has been paid to structural dynamics of the interaction between cancer populations and the molecular mechanisms associated with local invasion. One system that is of particular interest is that of the urokinase plasminogen activator (uPA) wherein uPA binds uPA receptors on the cancer cell surface, allowing plasminogen to be cleaved into plasmin, which degrades the extracellular matrix and this way leads to enhanced cancer cell migration. In this paper, we develop a novel numerical approach and associated analysis for spatio-structuro-temporal modelling of the uPA system for up to two-spatial and two-structural dimensions. This is accompanied by analytical exploration of the numerical techniques used in simulating this system, with special consideration being given to the proof of stability within numerical regimes encapsulating a central differences approach to approximating numerical gradients. The stability analysis performed here reveals instabilities induced by the coupling of the structural binding and proliferative processes. The numerical results expound how the uPA system aids the tumour in invading the local stroma, whilst the inhibitor to this system may impede this behaviour and encourage a more sporadic pattern of invasion. [ABSTRACT FROM AUTHOR]

We provide L -versus L -bounds for eigenfunctions on a real spherical space Z of wavefront type. It is shown that these bounds imply a non-trivial error term estimate for lattice counting on Z. The paper also serves as an introduction to geometric counting on spaces of the mentioned type. [ABSTRACT FROM AUTHOR]

We give an introduction to basic harmonic analysis and representation theory for homogeneous spaces Z = G/H attached to a real reductive Lie group G. A special emphasis is made to the case where Z is real spherical. [ABSTRACT FROM AUTHOR]

During the last two decades, great efforts have been devoted to the calculation of the local theta correspondence for reductive dual pairs. However, uniform formulas remain elusive for real dual pairs of type I. The purpose of this paper is twofold: to formulate an explicit version of induction principle for dual pairs ( O ( p , q ) , S p ( 2 n , R ) ) with p + q even, and to apply it to obtain a complete and explicit description of the local theta correspondence when p + q = 4 . Our approach is very elementary by analysis on the infinitesimal characters and K -types under the theta correspondence. [ABSTRACT FROM AUTHOR]

In this paper we extend previous results regarding SL2(k) over any finite fieldkby investigating the structure of the symmetric spaces for the family of special linear groups SLn(k) for any integern>2. Specifically, we discuss the generalized and extended symmetric spaces of SLn(k) for all conjugacy classes of involutions over a finite field of odd or even characteristic. We characterize the structure of these spaces and provide an explicit difference set in cases where the two spaces are not equal. [ABSTRACT FROM PUBLISHER]

Let Q be a connected algebraic group with Lie algebra q . Symmetric invariants of q , i.e., the Q -invariants in the symmetric algebra S ( q ) of q , is a first approximation to the understanding of the coadjoint action ( Q : q ⁎ ) and coadjoint Q -orbits. In this article, we study a class of non-reductive Lie algebras, where the description of the symmetric invariants is possible and the coadjoint representation has a number of nice invariant-theoretic properties. If G is a semisimple group with Lie algebra g and V is G -module, then we define q to be the semi-direct product of g and V . Then we are interested in the case, where the generic isotropy group for the G -action on V is reductive and commutative. It turns out that in this case symmetric invariants of q can be constructed via certain G -equivariant maps from g to V (“covariants”). [ABSTRACT FROM AUTHOR]

We consider the cohomology group H 1 ( Γ , ρ ) of a discrete subgroup Γ ⊂ G = S U ( n , 1 ) and the symmetric tensor representation ρ on S m ( C n + 1 ) . We give an elementary proof of the Eichler–Shimura isomorphism that harmonic forms H 1 ( Γ ∖ G ∕ K , ρ ) are ( 0 , 1 ) -forms for the automorphic holomorphic bundle induced by the representation S m ( C n ) of K . [ABSTRACT FROM AUTHOR]

In the recent paper (Achab and Faraut, 2012), we introduced an analysis of the Brylinski–Kostant model for spherical minimal representations for simple real Lie groups of non Hermitian type. We generalize here that analysis and give a unified geometric realization to a family of unitary irreducible representations of such groups. [ABSTRACT FROM AUTHOR]

If G is a compact Lie group endowed with a left invariant metric g, then G acts via pullback by isometries on each eigenspace of the associated Laplace operator $$\Delta _g$$ . We establish algebraic criteria for the existence of left invariant metrics g on G such that each eigenspace of $$\Delta _g$$ , regarded as the real vector space of the corresponding real eigenfunctions, is irreducible under the action of G. We prove that generic left invariant metrics on the Lie groups $$G={ SU}(2)\times \cdots \times { SU}(2)\times T$$ , where T is a (possibly trivial) torus, have the property just described. The same holds for quotients of such groups G by discrete central subgroups. In particular, it also holds for $${ SO}(3)$$ , $${ U}(2)$$ , $${ SO}(4)$$ . [ABSTRACT FROM AUTHOR]

We consider a Keller-Segel-Euler model and prove a regularity criterion of the local strong solutions in a 3D bounded domain Ω. [ABSTRACT FROM AUTHOR]

Let G be a real, reductive algebraic group, and let X be a homogeneous space for G with a non-zero invariant density. We give an explicit description of a Zariski open, dense subset of the asymptotics of the tempered support of L 2 ( X ) . Under additional hypotheses, this result remains true for vector bundle valued harmonic analysis on X . These results follow from an upper bound on the wave front set of an induced Lie group representation under a uniformity condition. [ABSTRACT FROM AUTHOR]

A Finsler space ( M , F ) is called flag-wise positively curved, if for any x ∈ M and any tangent plane P ⊂ T x M , we can find a nonzero vector y ∈ P , such that the flag curvature K F ( x , y , P ) > 0 . Though compact positively curved spaces are very rare in both Riemannian and Finsler geometry, flag-wise positively curved metrics should be easy to be found. A generic Finslerian perturbation for a non-negatively curved homogeneous metric may have a big chance to produce flag-wise positively curved metrics. This observation leads our discovery of these metrics on many compact manifolds. First we prove any Lie group G such that its Lie algebra g is compact non-Abelian and dim ⁡ c ( g ) ≤ 1 admits flag-wise positively curved left invariant Finsler metrics. Similar techniques can be applied to our exploration for more general compact coset spaces. We will prove, whenever G / H is a compact coset space with a finite fundamental group, G / H and S 1 × G / H admit flag-wise positively curved Finsler metrics. This provides abundant examples for this type of metrics, which are not homogeneous in general. These examples implies a significant difference between the flag-wise positively curved condition and the positively curved condition, even though they are reduced to the same one in Riemannian geometry. [ABSTRACT FROM AUTHOR]

It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary group exists on a pseudo-Kählerian manifold; instead of the Weyl tensor one obtains the Bochner tensor. In the present paper, the known decomposition with respect to the pseudo-orthogonal group of the covariant derivative of the curvature tensor of a pseudo-Riemannian manifold is refined. A decomposition with respect to the pseudo-unitary group of the covariant derivative of the curvature tensor for pseudo-Kählerian manifolds is obtained. This defines natural classes of spaces generalizing locally symmetric spaces and Einstein spaces. It is shown that the values of the covariant derivative of the curvature tensor for a non-locally symmetric pseudo-Riemannian manifold with an irreducible connected holonomy group different from the pseudo-orthogonal and pseudo-unitary groups belong to an irreducible module of the holonomy group. [ABSTRACT FROM AUTHOR]

Let [ABSTRACT FROM AUTHOR]