Recent zbMATH articles in MSC 22https://www.zbmath.org/atom/cc/222021-04-16T16:22:00+00:00WerkzeugLie group methods for eigenvalue function.https://www.zbmath.org/1456.580232021-04-16T16:22:00+00:00"Nazarkandi, H. A."https://www.zbmath.org/authors/?q=ai:nazarkandi.hossain-alizadehSummary: By considering a \(C^\infty\) structure on the ordered non-increasing of elements of \(\mathbb R^n\), we show that it is a differentiable manifold. By using of Lie groups, we show that eigenvalue function is a submersion. This fact is used to prove some results. These results is applied to prove a few facts about spectral manifolds and spectral functions. Orthogonal matrices act on the real symmetric matrices as a Lie transformation group. This fact, also, is used to prove the results.Theta functions and Brownian motion.https://www.zbmath.org/1456.580252021-04-16T16:22:00+00:00"Duncan, Tyrone E."https://www.zbmath.org/authors/?q=ai:duncan.tyrone-eSummary: A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with \textit{su}(2).Derived Langlands. Monomial resolutions of admissible representations.https://www.zbmath.org/1456.110032021-04-16T16:22:00+00:00"Snaith, Victor"https://www.zbmath.org/authors/?q=ai:snaith.victor-pIn this monograph, the author fits the theory of monomial resolutions in the subjects of Langlands programme and closely related topics. The fundamentals of the theory are developed in the first chapter, although the reader might want to consult [\textit{R. Boltje}, J. Algebra 246, No. 2, 811--848 (2001; Zbl 1006.20005)] to get a broader picture. This paper describes a category whose derived category is suitable environment for monomial resolutions when \(G\) is a finite group.
For a locally profinite group, the author of this monograph constructs monomial resolutions of its admissible \(k\)-representations which are recognized by the Langlands programme as objects related to questions arising in number theory. Monomial resolutions for \(\mathrm{GL}_2\) are described in both local and adélic case. They are motivated by the fact that certain subspace of automorphic representations appearing in its monomial resolution includes the classical spaces of modular forms. Additionally, the author poses certain conditions on Hecke operators under which they extend to the monomial resolution and he gives an example of a classical Hecke operator for which they are satisfied. Also, the monomial resolutions are constructed for \(\mathrm{GL}_n\) over a local field \(K\) gradually going from \(\mathrm{GL}_2 (K)\) to \(\mathrm{GL}_3(K)\) and eventually to a general case. The monograph also engages with Deligne representations, \(\varepsilon\)-factors, \(L\)-functions, Kondo-style invariants and Galois base change. Indications showing the utility of connecting these topics with monomial resolutions are given through examples. They suggest that one may be able to construct \(\varepsilon\)-factors and \(L\)-functions of [\textit{R. Godement} and \textit{H. Jacquet}, Zeta functions of simple algebras. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.12011)] in a simpler manner. Also, he illustrates the possibility of functoriality of Galois base change in context of finite general linear groups.
Throughout the monograph, the author explains the essential claims in detail and gives enough instructions for a reader to prove the other ones. Overviews of studied topics from the Langlands programme could be convenient for a reader interested in the results given in this book. On the other hand, a reader interested in a connection of monomial resolutions with topics of the Langlands programme has a motivation for further research.
Reviewer: Barbara Bošnjak (Zagreb)On the use of the rotation minimizing frame for variational systems with Euclidean symmetry.https://www.zbmath.org/1456.829592021-04-16T16:22:00+00:00"Mansfield, E. L."https://www.zbmath.org/authors/?q=ai:mansfield.elizabeth-louise"Rojo-Echeburúa, A."https://www.zbmath.org/authors/?q=ai:rojo-echeburua.aIn this paper, the authors consider variational problems for curves in 3-space for which the Lagrangian is invariant
under the special Euclidean group \(\mathrm{SE}(3)=\mathrm{SO}(3)\ltimes\mathbb{R}^3\) acting linearly in the standard way. They use the rotation minimizing frame, known as the normal, parallel, or Bishop frame. The authors derive the recurrence formulae for the symbolic invariant differentiation of the symbolic invariants and syzygy operator for variational problems with a Euclidean symmetry. As application the author use variational problems in the study of stands of proteins, nucleid acids, and polymers.
Reviewer: Nasir N. Ganikhodjaev (Tashkent)Revisiting Horn's problem.https://www.zbmath.org/1456.150342021-04-16T16:22:00+00:00"Coquereaux, Robert"https://www.zbmath.org/authors/?q=ai:coquereaux.robert"McSwiggen, Colin"https://www.zbmath.org/authors/?q=ai:mcswiggen.colin"Zuber, Jean-Bernard"https://www.zbmath.org/authors/?q=ai:zuber.jean-bernardAn operational approach to spacetime symmetries: Lorentz transformations from quantum communication.https://www.zbmath.org/1456.811092021-04-16T16:22:00+00:00"Höhn, Philipp A."https://www.zbmath.org/authors/?q=ai:hohn.philipp-a"Müller, Markus P."https://www.zbmath.org/authors/?q=ai:muller.markus-pErgodic actions of compact quantum groups from solutions of the conjugate equations.https://www.zbmath.org/1456.370112021-04-16T16:22:00+00:00"Pinzari, Claudia"https://www.zbmath.org/authors/?q=ai:pinzari.claudia"Roberts, John E."https://www.zbmath.org/authors/?q=ai:roberts.john-eliasSummary: We use a tensor \(C^{\ast}\)-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a \({}^{\ast}\)-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal \(C^{\ast}\)-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the \(C^{\ast}\)-algebra in question. The quantum groups involved are \(A_{u}(Q)\) and \(B_{u}(Q)\).Spinor modules for Hamiltonian loop group spaces.https://www.zbmath.org/1456.580052021-04-16T16:22:00+00:00"Loizides, Yiannis"https://www.zbmath.org/authors/?q=ai:loizides.yiannis"Meinrenken, Eckhard"https://www.zbmath.org/authors/?q=ai:meinrenken.eckhard"Song, Yanli"https://www.zbmath.org/authors/?q=ai:song.yanli.1|song.yanliThis paper studies the spinor modules theory of loop groups.
Let \( G \) be a compact, connected Lie group and let the loop group \( LG \) be the Banach Lie group of \(G\)-valued loops of a fixed Sobolev class \( S > 1/2 \). The authors prove that the tangent bundle of any proper Hamiltonian loop group space \(M\) possesses a canonically defined \(LG-\)equivariant completion \(\overline{T}M\), such
that any weakly symplectic 2-form \(\omega\) of any proper Hamiltonian loop group space extends to a strongly symplectic 2-form on \(\overline{T}M\).
Furthermore, it is proved that the bundle \(\overline{T}M\) possesses a distinguished \(LG-\)invariant polarization and a global \(LG-\)invariant \(\omega-\)compatible complex structure \(J\)
within this polarization class, unique up to homotopy. This leads to the definition
of \( LG-\)equivariant spinor bundle \( \mathrm{S}_{\overline{T}M} \),
which is used to construct the twisted \( \mathrm{Spin}_c \)-structure for the associated quasi-Hamiltonian \(G\)-space \(M\). This is is a way to get a finite-dimensional version of the spinor module \( \mathrm{S}_{\overline{T}M} \).
The authors also discuss \textquoteleft abelianization procedure\textquoteright\, which is another way to get a finite-dimensional version of \( \mathrm{S}_{\overline{T}M} \). The idea is to shift
to a finite-dimensional maximal torus \(T \subseteq LG-\)invariant submanifold of \(M,\) and construct an equivalent
\(\mathrm{Spin}_c \)-structure on that
submanifold. More precisely, if the moment map \(\Phi\) of a proper Hamiltonian
\(LG\)-space is transverse to the Lie algebra \( \mathfrak{t}^* \) (as a space of constant connections valued
in the Lie algebra of the maximal torus \( T \)), then the pre-image \(\Phi^{-1} (\mathfrak{t}^*)\)
is a finite-dimensional pre-symplectic manifold that
inherits a \(T\)-equivalent \(\mathrm{Spin}_c \)-structure.
Reviewer: Kaveh Eftekharinasab (Kyiv)Special functions associated with \(K\)-types of degenerate principal series of \(\mathrm{Sp}(n,\mathbb{C})\).https://www.zbmath.org/1456.220052021-04-16T16:22:00+00:00"Mendousse, Grégory"https://www.zbmath.org/authors/?q=ai:mendousse.gregoryThis article is devoted to the study of special vectors contained in various incarnations of generalized principal series representations induced from maximal parabolic subgroups of the complex symplectic group \(\mathrm{Sp}(n,\mathbf{C})\). After reviewing the well-known decomposition of the Hilbert space \(L^2(S^{4n-1})\) under the natural action of \(\mathrm{Sp}(n)\times\mathrm{Sp}(1)\) in terms of spherical harmonics, the author uses quaternionic geometry to establish the existence and uniqueness of bi-invariant spherical harmonics and determines an explicit hypergeometric equation that they satisfy.
The other main result in the paper concerns certain vectors in the so-called \textit{non-standard} picture of the degenerate principal representations. This picture was introduced by \textit{T. Kobayashi} et al. [J. Funct. Anal. 260, No. 6, 1682--1720 (2011; Zbl 1217.22003)] for real symplectic groups and adapted to the complex case by the reviewer in [J. Funct. Anal. 262, No. 9, 4160--4180 (2012; Zbl 1242.22017)]. It is the image of the the classical non-compact picture under a partial Fourier transform afforded by the fact that the unipotent radicals of the inducing parabolic subgroups are Heisenberg groups. The author calculates the image in this picture of particular highest weight vectors, showing that they can be expressed in terms of modified Bessel functions.
Reviewer: Pierre Clare (Williamsburg)No Tits alternative for cellular automata.https://www.zbmath.org/1456.200482021-04-16T16:22:00+00:00"Salo, Ville"https://www.zbmath.org/authors/?q=ai:salo.ville-o|salo.villeSummary: We show that the automorphism group of a one-dimensional full shift (the group of reversible cellular automata) does not satisfy the Tits alternative. That is, we construct a finitely-generated subgroup which is not virtually solvable yet does not contain a free group on two generators. We give constructions both in the two-sided case (spatially acting group \(\mathbb Z)\) and the one-sided case (spatially acting monoid \(\mathbb N\), alphabet size at least eight). Lack of Tits alternative follows for several groups of symbolic (dynamical) origin: automorphism groups of two-sided one-dimensional uncountable sofic shifts, automorphism groups of multidimensional subshifts of finite type with positive entropy and dense minimal points, automorphism groups of full shifts over non-periodic groups, and the mapping class groups of two-sided one-dimensional transitive SFTs. We also show that the classical Tits alternative applies to one-dimensional (multi-track) reversible linear cellular automata over a finite field.On graph products of multipliers and the Haagerup property for \(C^{\ast}\)-dynamical systems.https://www.zbmath.org/1456.460542021-04-16T16:22:00+00:00"Atkinson, Scott"https://www.zbmath.org/authors/?q=ai:atkinson.scott-eThe following paragraphs are essentially taken from the author's abstract and introduction.
The Haagerup property is an important approximation property for groups and for self-adjoint operator algebras. Since its appearance in Haagerup's seminal article, this property has been the subject of intense study. In 2012, \textit{Z. Dong} and \textit{Z.-J. Ruan} [Integral Equations Oper. Theory 73, No. 3, 431--454 (2012; Zbl 1263.46043)] introduced the Haagerup property for the action of a discrete group \(G\) on a unital \(C^*\)-algebra \(A\).
The author considers the notion of the graph product of actions of discrete groups \(\{G_v\}\) on a \(C^*\)-algebra \(A\) and shows that, under suitable commutativity conditions, the graph product action \(\bigstar_\Gamma \alpha_v:\bigstar_\Gamma G_v\curvearrowright A\) has the Haagerup property if each action \(\alpha_v: G_v\curvearrowright A\) possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, the author introduces the graph product of multipliers associated to the actions and shows that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. The author also records a cohomological characterization of the Haagerup property for group actions.
Reviewer: Qing Meng (Qufu)A tool kit for groupoid $C^*$-algebras.https://www.zbmath.org/1456.460022021-04-16T16:22:00+00:00"Williams, Dana P."https://www.zbmath.org/authors/?q=ai:williams.dana-pGroupoids are small categories in which every morphism is an isomorphism.
They appear as early as in the work of \textit{H. Brandt} [Mat. Ann. 96, 360--366 (1926; JFM 52.0110.09)] on generalising a composition of binary quadratic forms (due to Gauss) to quaternary quadratic forms.
Since then, groupoids have been used in a wide variety of areas of mathematics including ergodic theory, functional analysis, homotopy theory, algebraic geometry, differential geometry, differential topology, and group theory.
The book under review focuses on the interactions with operator algebras that span more than four decades of research.
In the late 1970s, Renault initiated the idea of associating a \(C^*\)-algebra to a locally compact groupoid in analogy to what is done for groups.
Since then, a large amount of research has emerged concerning the structure and properties of such a \(C^*\)-algebra.
Starting with the foundational work of \textit{J. Renault} [A groupoid approach to \(C^*\)-algebras. Berlin: Springer (1980; Zbl 0433.46049)] and \textit{A. Kumjian} [Can. J. Math. 38, 969--1008 (1986; Zbl 0627.46071)], the current book aims to present the core theory of groupoid \(C^*\)-algebras by taking advantage of recent developments covered by the book of \textit{A. L. T. Paterson} [Groupoids, inverse semigroups, and their operator algebras. Boston, MA: Birkhäuser (1999; Zbl 0913.22001)], \textit{P. Muhly}'s unpublished CBMS lecture notes [``Coordinates in operator algebra'', CBMS Conference Lecture Notes (Texas Christian University 1990), unfinished manuscript (1999), \url{https://operatoralgebras.org/resources-resources/Groupoids-Book-Muhly.pdf}], and the monograph by \textit{C. Anantharaman-Delaroche} and \textit{J. Renault} [Amenable groupoids. Genève: L'Enseignement Mathématique; Université de Genève (2000; Zbl 0960.43003)].
The book covers four topics for second countable, locally compact Hausdorff groupoids that are at the epicentre in this endeavour:
\begin{itemize}
\item[(a)] the disintegration theorem;
\item[(b)] the equivalence theorem;
\item[(c)] three notions of amenability (topological amenability, measurewise amenability and metric amenability) and their relations;
\item[(d)] simplicity (in the sense of Effros-Hahn-type results).
\end{itemize}
The author uses current developments to provide simplified proofs with a clear exposition. In the process, all required tools are carefully laid down. Some technical parts of proofs are included as exercises with solutions at the end that allow the reader to dig into the fine details of the material.
The book is self-contained for a reader with some knowledge in functional analysis, and while experience in working with group \(C^*\)-algebras is -- always -- helpful, it is not much required. The material is accessible to graduate students that wish to undertake work in this direction.
Reviewer: Evgenios Kakariadis (Newcastle upon Tyne)Hausdorff dimension of limsup sets of rectangles in the Heisenberg group.https://www.zbmath.org/1456.600382021-04-16T16:22:00+00:00"Ekström, Fredrik"https://www.zbmath.org/authors/?q=ai:ekstrom.fredrik"Järvenpää, Esa"https://www.zbmath.org/authors/?q=ai:jarvenpaa.esa"Järvenpää, Maarit"https://www.zbmath.org/authors/?q=ai:jarvenpaa.maaritThe main findings of the paper refer to computing the almost sure value of the Hausdorff dimension of limsup sets generated by randomly distributed rectangles in the Heisenberg group, in terms of directed singular value functions.
Reviewer: George Stoica (Saint John)Coloured Neretin groups.https://www.zbmath.org/1456.220082021-04-16T16:22:00+00:00"Lederle, Waltraud"https://www.zbmath.org/authors/?q=ai:lederle.waltraudLederle establishes and utilises a connection between topological full groups of étale groupoids associated to one-sided shifts after \textit{H. Matui} [Proc. Lond. Math. Soc. (3) 104, No. 1, 27--56 (2012; Zbl 1325.19001); J. Reine Angew. Math. 705, 35--84 (2015; Zbl 1372.22006)] and almost automorphism groups of trees after \textit{Yu. A. Neretin} [Russ. Acad. Sci., Izv., Math. 41, No. 2, 1072--1085 (1992; Zbl 0789.22036); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 5, 1072--1085 (1992)] to construct compactly generated, locally compact totally disconnected groups that are virtually simple and do not have lattices.
An étale groupoid is a small category in which all morphisms are isomorphisms and whose object and morphism space are topologised in such a way that all structure maps are continuous, and source and range are local homeomorphisms. The associated topological full group is defined as a subgroup of the homeomorphism group of the object space, which is typically a Cantor space.
An almost automorphism of a locally finite tree \(T\) is an equivalence class of forest isomorphisms that arise from removing finite subtrees from \(T\). It can be viewed as a homeomorphism of the boundary of \(T\), which is also a Cantor space.
Lederle shows, using the newly established connection above and Matui's theory, that the groups of almost automorphisms arising from certain Burger-Mozes universal groups [\textit{M. Burger} and \textit{S. Mozes}, Publ. Math., Inst. Hautes Étud. Sci. 92, 113--150 (2000; Zbl 1007.22012)] have the desired, rare properties. The no-lattice argument follows the approach of [\textit{U. Bader} et al., Bull. Lond. Math. Soc. 44, No. 1, 55--67 (2012; Zbl 1239.22007)].
In particular, Lederle provides a family of candidates of groups without invariant random subgroups.
Reviewer: Stephan Tornier (Newcastle)Dynamics near an idempotent.https://www.zbmath.org/1456.370232021-04-16T16:22:00+00:00"Shaikh, Md. Moid"https://www.zbmath.org/authors/?q=ai:shaikh.md-moid"Patra, Sourav Kanti"https://www.zbmath.org/authors/?q=ai:patra.sourav-kanti"Ram, Mahesh Kumar"https://www.zbmath.org/authors/?q=ai:ram.mahesh-kumarSummary: \textit{N. Hindman} and \textit{I. Leader} [Semigroup Forum 59, No. 1, 33--55 (1999; Zbl 0942.22003)]
first introduced the notion of the semigroup of ultrafilters converging to zero for a dense subsemigroup of \(((0, \infty), +)\). Using the algebraic structure of the Stone-Čech compactification, \textit{M. A. Tootkaboni} and \textit{T. Vahed} [Topology Appl. 159, No. 16, 3494--3503 (2012; Zbl 1285.54017)]
generalized and extended this notion to an idempotent instead of zero, that is a semigroup of ultrafilters converging to an idempotent \(e\) for a dense subsemigroup of a semitopological semigroup \((R, +)\) and they gave the combinatorial proof of the Central Sets Theorem near \(e\). Algebraically one can define quasi-central sets near \(e\) for dense subsemigroups of \((R, +)\). In a dense subsemigroup of \((R, +)\), C-sets near \(e\) are the sets, which satisfy the conclusions of the Central Sets Theorem near \(e\). \textit{S. K. Patra} [Topology Appl. 240, 173--182 (2018; Zbl 1392.37008)]
gave dynamical characterizations of these combinatorially rich sets near zero. In this paper, we shall establish these dynamical characterizations for these combinatorially rich sets near \(e\). We also study minimal systems near \(e\) in the last section of this paper.Barnes-Ismagilov integrals and hypergeometric functions of the complex field.https://www.zbmath.org/1456.330082021-04-16T16:22:00+00:00"Neretin, Yury A."https://www.zbmath.org/authors/?q=ai:neretin.yuri-aThe purpose of this paper is first to extend Eulerian integrals for generalized hypergeometric functions \(\ _{p }\textup{F}_{q}\) to complex integrals. Several special cases are indicated and there are many similarities with the Meijer G-function. It seems that there are more than one definition available and the usual stringent structure ``definition, theorem, proof'' is unfortunately missing. It would be better not to have so many references, the paper has kind of a physics style. The concept Gamma-function of the complex field on page 4 is not properly defined.
Reviewer: Thomas Ernst (Uppsala)Lie groups of controlled characters of combinatorial Hopf algebras.https://www.zbmath.org/1456.220072021-04-16T16:22:00+00:00"Dahmen, Rafael"https://www.zbmath.org/authors/?q=ai:dahmen.rafael"Schmeding, Alexander"https://www.zbmath.org/authors/?q=ai:schmeding.alexanderA theory of controlled characters of a combinatorial Hopf algebras is introduced, given subgroups of the groups of characters. The model is
the tame Butcher group, seen as a subgroup of the Butcher-Connes-Kreimer group.
A combinatorial Hopf algebra is here a graded connected Hopf algebra, isomorphic to a polynomial algebra, with a particular basis, and the characters take their value in a fixed Banach algebra. A controlled character satisfies a growth condition given by a particular bound. If this bound is compatible with the combinatorial structure of the Hopf algebra, then the set of controlled characters is a subgroup of the group of all characters. It is proved that the group of controlled characters is an infinite-dimensional Lie group and that the underlying group is the Lie algebra of infinitesimal controlled characters. When the Hopf algebra is right-handed, it is shown that the group of controlled characters is regular in Milnor's sense.
Reviewer: Loïc Foissy (Calais)D sets and IP rich sets in countable, cancellative abelian semigroups.https://www.zbmath.org/1456.200632021-04-16T16:22:00+00:00"Campbell, James T."https://www.zbmath.org/authors/?q=ai:campbell.james-t"McCutcheon, Randall"https://www.zbmath.org/authors/?q=ai:mccutcheon.randallSummary: We give combinatorial characterizations of IP rich sets (IP sets that remain IP upon removal of any set of zero upper Banach density) and D sets (members of idempotent ultrafilters, all of whose members have positive upper Banach density) in a general countable, cancellative abelian semigroup. We then show that the family of IP rich sets strictly contains the family of D sets.Equidistribution of expanding translates of curves and Diophantine approximation on matrices.https://www.zbmath.org/1456.220032021-04-16T16:22:00+00:00"Yang, Pengyu"https://www.zbmath.org/authors/?q=ai:yang.pengyuOne can begin with author's abstract:
``We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space \(G/\Gamma\) of a semisimple algebraic group \(G\). We define two families of algebraic subvarieties of the associated partial flag variety \(G/\Gamma\), which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of \(m\times n\) real matrices whose image is not contained in any subvariety coming from these two families, Dirichlet's theorem on simultaneous Diophantine approximation cannot be improved. The proof combines geometric invariant theory, Ratner's theorem on measure rigidity for unipotent flows, and linearization technique.''
It is noted that many problems in number theory can be recast in the language of homogeneous dynamics. A survey is devoted to this fact, to the main problems of this research, and to the motivation of the present investigations. Notions that are useful for proving the main statements are recalled and explained.
The main results and several auxiliary statements are proven with explanations. Applications of the main results and also connections between these results and known researches are noted.
Reviewer: Symon Serbenyuk (Kyïv)\(L^1\)-determined primitive ideals in the \(C^\ast\)-algebra of an exponential Lie group with closed non-\(\ast\)-regular orbits.https://www.zbmath.org/1456.430012021-04-16T16:22:00+00:00"Inoue, Junko"https://www.zbmath.org/authors/?q=ai:inoue.junko"Ludwig, Jean"https://www.zbmath.org/authors/?q=ai:ludwig.jeanSummary: Let \(G = \exp (\mathfrak{g})\) be an exponential solvable Lie group and Ad\((G) \subset \mathbb{D}\) an exponential solvable Lie group of automorphisms of \(G\). Assume that for every non-\(\ast\)-regular orbit \(\mathbb{D} \cdot q$, $q \in \mathfrak{g}^\ast\), of \(\mathbb{D} = \exp(\mathfrak{d})\) in \(g^\ast\), there exists a nilpotent ideal \(\mathfrak{n}\) of \(\mathfrak{g}\) containing \(\mathfrak{d} \cdot \mathfrak{g}\) such that \(\mathbb{D} \cdot q_{|\mathfrak{n}}\) is closed in \(\mathfrak{n}^\ast\). We then show that for every \(\mathbb{D}\)-orbit \(\Omega\) in \(g^\ast\) the kernel \(\ker_{C^\ast} (\Omega)\) of \(\Omega\) in the \(C^\ast\)-algebra of \(G\) is \(L^1\)-determined, which means that \(\ker_{C^\ast} (\Omega)\) is the closure of the kernel \(\ker L^1(\Omega )\) of \(\Omega\) in the group algebra \(L^1(G)\). This establishes also a new proof of a result of Ungermann, who obtained the same result for the trivial group \(\mathbb{D} = \text{Ad}(G)\). We finally give an example of a non-closed non-\(\ast\)-regular orbit of an exponential solvable group \(G\) and of a coadjoint orbit \(O \subset \mathfrak{g}^\ast\), for which the corresponding kernel \(\text{ker}_{C^\ast}(\pi_O)\) in \(C^\ast (G)\) is not \(L^1\)-determined.Differential geometry and Lie groups. A second course.https://www.zbmath.org/1456.530012021-04-16T16:22:00+00:00"Gallier, Jean"https://www.zbmath.org/authors/?q=ai:gallier.jean-h"Quaintance, Jocelyn"https://www.zbmath.org/authors/?q=ai:quaintance.jocelynThis book is written as a second course on differential geometry. So the reader is supposed to be familiar with some themes from the first course on differential geometry -- the theory of manifolds and some elements of Riemannian geometry.
In the first two chapters here some topics from linear algebra are provided -- a detailed exposition of tensor algebra and symmetric algebra, exterior tensor products and exterior algebra. These chapters may be useful when studying the material of this book for those students, who did not study these topics in their algebraic course.
Some themes, which are covered in this book, are rather standard for books on differential geometry - they are differential forms, de Rham cohomology, integration on manifolds, connections and curvature in vector bundles, fibre bundles, principal bundles and metrics on bundles. But a number of topics discussed in this book are not always included in courses on differential geometry and are rarely contained in textbooks on differential geometry. The presence of these topics makes this book especially interesting for modern students. Here is a list of some such topics: an introduction to Pontrjagin
classes, Chern classes, and the Euler class, distributions and the Frobenius theorem. Three chapters need to be highlighted separately. Chapter 7 -- spherical harmonics and an introduction to the representations of compact Lie groups. Chapter 8 -- operators on Riemannian manifolds: Hodge Laplacian, Laplace-Beltrami Laplacian, Bochner
Laplacian. Chapter 11 -- Clifford algebras and groups, groups Pin\((n)\), Spin\((n)\).
Not all statements in this book are given with proofs, for some only links to other textbooks are given. But the most important results are given here with complete proofs and accompanied by examples. Each chapter of this book ends with a list of interesting and sometimes very important problems. At the end of the book there is a very detailed list of the notation used (symbol index) and a detailed list (index) of the terms used.
Reviewer: V. V. Gorbatsevich (Moskva)A simplified proof of the reduction point crossing sign formula for Verma modules.https://www.zbmath.org/1456.220062021-04-16T16:22:00+00:00"St. Denis, Matthew"https://www.zbmath.org/authors/?q=ai:st-denis.matthew"Yee, Wai Ling"https://www.zbmath.org/authors/?q=ai:yee.wai-lingThe article engages with the Unitary Dual Problem for real reductive groups through the simplification of the proof for the reduction point crossing sign formula for Verma modules.
Since the classification of Hermitian representations of real reductive groups, the solution for the Unitary Dual Problem is sought in the calculation of signatures of the invariant Hermitian forms of Hermitian representations.
The second author calculated the signature character in [\textit{W. L. Yee}, Represent. Theory 9, 638--677 (2005; Zbl 1404.22040)] and simplified the signature character formula in [\textit{W. L. Yee}, Math. Z. 292, No. 1--2, 267--305 (2019; Zbl 1416.22021)] and [\textit{J. Lariviere} and \textit{W. L. Yee}, ``Signature characters of invariant Hermitian forms on irreducible Verma modules of singular highest weight and Hall-Littlewood polynomials'', Preprint].
The fact whose proof is simplified in this article played an important role in the former.
The main methods of the paper rely on the results of cited papers and neat calculation with roots, Weyl group and associated objects.
Reviewer: Barbara Bošnjak (Zagreb)Stability, cohomology vanishing, and nonapproximable groups.https://www.zbmath.org/1456.220022021-04-16T16:22:00+00:00"De Chiffre, Marcus"https://www.zbmath.org/authors/?q=ai:de-chiffre.marcus"Glebsky, Lev"https://www.zbmath.org/authors/?q=ai:glebsky.lev-yu"Lubotzky, Alexander"https://www.zbmath.org/authors/?q=ai:lubotzky.alexander"Thom, Andreas"https://www.zbmath.org/authors/?q=ai:thom.andreas-bertholdThere is substantial current interest in finite-dimensional approximation properties of groups. These generally ask whether every discrete groups can be faithfully represented by approximate homomorphisms into matrix groups, with many possible variations on how exactly this approximation is measured. This paper provides the first negative answer to one of these questions, namely where approximation in the unitary groups is measured by distance in the unnormalized Frobenius norm. (The technical details are given in terms of ultrafilter convergence.)
The paper constructs finitely presented groups for which every approximate homomorphism into a finite-dimensional unitary group must map some (fixed) nontrivial group elements to close to the identity (Theorem 1.1). This follows from two separate results: first, it is shown (Theorem 1.2) that every finitely presented group which has vanishing second cohomology with coefficients in a unitary representation is stable, allowing for the deformation of approximate homomorphisms into actual homomorphisms. This is similar in style to \textit{D. Kazhdan}'s work on stability and cohomology [Isr. J. Math. 43, 315--323 (1982; Zbl 0518.22008)]. Second, it is shown that there are finitely presented groups for which these second cohomology groups vanish, but which are not residually finite. The construction technique here is a \(p\)-adic analogue of Deligne's construction of finite central extensions of arithmetic groups which are not residually finite [\textit{P. Deligne}, C. R. Acad. Sci., Paris, Sér. A 287, 203--208 (1978; Zbl 0416.20042)].
The paper is a pleasure to read and also contains a number of further interesting observations on approximation properties and stability. The problem whether every group is hyperlinear -- which coincides with the question resolved here except for the replacement of the unnormalized Frobenius norm by its normalized version -- remains open.
Reviewer: Tobias Fritz (Innsbruck)The boundary model for the continuous cohomology of \(\mathrm{Isom}^+ (\mathbb H^n)\).https://www.zbmath.org/1456.220042021-04-16T16:22:00+00:00"Pieters, Hester"https://www.zbmath.org/authors/?q=ai:pieters.hesterSummary: We prove that the continuous cohomology of \(\mathrm{Isom}^+ (\mathbb H^n)\) can be measurably realized on the boundary of hyperbolic space. This implies in particular that for \(\mathrm{Isom}^+ (\mathbb H^n)\) the comparison map from continuous bounded cohomology to continuous cohomology is injective in degree 3. We furthermore prove a stability result for the continuous bounded cohomology of \(\mathrm{Isom}(\mathbb H^n)\) and\(\mathrm{Isom}(\mathbb H^n_\mathbb C)\).Analogs of Korn's inequality on Heisenberg groups.https://www.zbmath.org/1456.530282021-04-16T16:22:00+00:00"Isangulova, D. V."https://www.zbmath.org/authors/?q=ai:isangulova.d-vSummary: We give two analogs of Korn's inequality on Heisenberg groups. First, the norm of the horizontal differential is estimated in terms of the symmetric part of the differential. Second, Korn's inequality is treated as a coercive estimate for a differential operator whose kernel coincides with the Lie algebra of the isometry group. For this purpose, we construct a differential operator whose kernel coincides with the Lie algebra of the isometry group on Heisenberg groups and prove a coercive estimate for the operator.Translates of functions on the Heisenberg group and the HRT conjecture.https://www.zbmath.org/1456.420382021-04-16T16:22:00+00:00"Currey, B."https://www.zbmath.org/authors/?q=ai:currey.bradley-n-iii|currey.bradley|currey.brad"Oussa, V."https://www.zbmath.org/authors/?q=ai:oussa.vignon|oussa.vignon-sSummary: We prove that the HRT (Heil, Ramanathan, and Topiwala) Conjecture [\textit{C. Heil} et al., Proc. Am. Math. Soc. 124, No. 9, 2787--2795 (1996; Zbl 0859.42023)] is equivalent to the conjecture that co-central translates of square-integrable functions on the Heisenberg group are linearly independent.On self-adjoint extensions and symmetries in quantum mechanics.https://www.zbmath.org/1456.811792021-04-16T16:22:00+00:00"Ibort, Alberto"https://www.zbmath.org/authors/?q=ai:ibort.alberto"Lledó, Fernando"https://www.zbmath.org/authors/?q=ai:lledo.fernando"Pérez-Pardo, Juan Manuel"https://www.zbmath.org/authors/?q=ai:perez-pardo.juan-manuelSummary: Given a unitary representation of a Lie group \(G\) on a Hilbert space \(\mathcal H\), we develop the theory of \(G\)-invariant self-adjoint extensions of symmetric operators using both von Neumann's theorem and the theory of quadratic forms. We also analyze the relation between the reduction theory of the unitary representation and the reduction of the \(G\)-invariant unbounded operator. We also prove a \(G\)-invariant version of the representation theorem for closed and semi-bounded quadratic forms. The previous results are applied to the study of \(G\)-invariant self-adjoint extensions of the Laplace-Beltrami operator on a smooth Riemannian manifold with boundary on which the group \(G\) acts. These extensions are labeled by admissible unitaries \(U\) acting on the \(L^2\)-space at the boundary and having spectral gap at \(-1\). It is shown that if the unitary representation \(V\) of the symmetry group \(G\) is traceable, then the self-adjoint extension of the Laplace-Beltrami operator determined by \(U\) is \(G\)-invariant if \(U\) and \(V\) commute at the boundary. Various significant examples are discussed at the end.Affine structures on Lie groupoids.https://www.zbmath.org/1456.220012021-04-16T16:22:00+00:00"Lang, Honglei"https://www.zbmath.org/authors/?q=ai:lang.honglei"Liu, Zhangju"https://www.zbmath.org/authors/?q=ai:liu.zhangju"Sheng, Yunhe"https://www.zbmath.org/authors/?q=ai:sheng.yunheAuthors' abstract: We study affine structures on a Lie groupoid, including affine \(k\)-vector fields, \(k\)-forms and \((p, q)\)-tensors. We show that the space of affine structures is a 2-vector space over the space of multiplicative structures. Moreover, the space of affine multivector fields with the Schouten bracket and the space of affine vector-valued forms with the Frölicher-Nijenhuis bracket are graded strict Lie 2-algebras, and affine \((1, 1)\)-tensors constitute a strict monoidal category. Such higher structures can be seen as the categorification of multiplicative structures on a Lie groupoid.
Reviewer: Iakovos Androulidakis (Athína)Geometric Waldspurger periods. II.https://www.zbmath.org/1456.110912021-04-16T16:22:00+00:00"Lysenko, Sergey"https://www.zbmath.org/authors/?q=ai:lysenko.sergeySummary: In this paper we extend the calculation of the geometric Waldspurger periods from our paper [Part I, Compos. Math. 144, No. 2, 377--438 (2008; Zbl 1209.14010)] to the case of ramified coverings. We give some applications to the study of Whittaker coeffcients of the theta-lifting of automorphic sheaves from \(\operatorname{PGL}_2\) to the metaplectic group \(\widetilde{\operatorname{SL}}_2\); they agree with our conjectures from [``Geometric Whittaker models and Eisenstein series for \(\mathrm{Mp}_2\)'', Preprint, \url{arXiv:1221.1596}]. In the process of the proof, we construct some new automorphic sheaves for \({\operatorname{GL}_2}\) in the ramified setting. We also formulate stronger conjectures about Waldspurger periods and geometric theta-lifting for the dual pair \((\widetilde{\operatorname{SL}}_2, \operatorname{PGL}_2)\).Vector valued polynomials, exponential polynomials and vector valued harmonic analysis.https://www.zbmath.org/1456.430032021-04-16T16:22:00+00:00"Laczkovich, M."https://www.zbmath.org/authors/?q=ai:laczkovich.miklosLet \(G\) be a topological abelian semigroup with unit, \(E\) be a Banach space, and let \(C(G,E)\) stand for the set of all continuous functions from \(G\) into \(E\). A function \(f\in C(G,E)\) is called a generalized polynomial if there is an \(n\ge 0\) such that \(\Delta_{h_1}\dots\Delta_{h_{n+1}} f=0\) for every \(h_1, \dots,h_{n+1}\in G\) in which \(\Delta_h\) denotes the difference operator. A function \(f\in C(G,E)\) is said to be a polynomial if it is a generalized polynomial and the linear span of its translates is of finite dimension; \(f\) is a \(w\)-polynomial if \(u\circ f\) is a polynomial for every \(u\) in the dual space of \(E\), and \(f\) is a local polynomial if it is a polynomial on every finitely generated subsemigroup.
The author proves that each of the classes of polynomials, \(w\)-polynomials, generalized polynomials, local polynomials is contained in the next class. If \(G\) is an abelian group and has a dense subgroup with finite torsion free rank, then these classes coincide. He also introduces the classes of exponential polynomials and \(w\)-exponential polynomials and investigates their representations and connection with polynomials and \(w\)-polynomials. He establishes spectral synthesis and analysis in the class \(C(G,E)\). He shows that, if \(G\) is an infinite and discrete abelian group and \(E\) is a Banach space of infinite dimension, then spectral analysis fails in \(C(G,E)\). If \(G\) is discrete with finite torsion free rank and \(E\) is a Banach space of finite dimension, then he proves that spectral synthesis holds in \(C(G,E)\).
Reviewer: Mohammad Sal Moslehian (Mashhad)Shintani functions for the holomorphic discrete series representation of \(\mathrm{GSp}_4(\mathbb R)\).https://www.zbmath.org/1456.110892021-04-16T16:22:00+00:00"Gejima, Kohta"https://www.zbmath.org/authors/?q=ai:gejima.kohtaThis paper is concerned with Shintani functions for the real reductive symmetric pair \((\mathrm{GSP}_4 (\mathbb R), (\mathrm{GL}_2 \times_{\mathrm{GL}_1} \mathrm{GL}_2) (\mathbb R))\). The author obtains an explicit formula for the Shintani functions for the holomorphic discrete series representation of \(\mathrm{GSP}_4 (\mathbb R)\) and proves their uniqueness. He also formulates an archimedean zeta integral of the type studied by \textit{A. Murase} and \textit{T. Sugano} [Math. Ann. 299, No. 1, 17--56 (1994; Zbl 0813.11032)] for the above mentioned symmetric pair and proves that the local zeta integral represents the local \(L\)-factor associated to the holomorphic discrete representations of \(\mathrm{GSP}_4 (\mathbb R)\).
Reviewer: Min Ho Lee (Cedar Falls)