Recent zbMATH articles in MSC 22Ehttps://www.zbmath.org/atom/cc/22E2021-04-16T16:22:00+00:00WerkzeugAn 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-pShintani 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)Lie 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.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)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.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)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\(\mathrm{Isom}(\mathbb H^n_\mathbb C)\).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)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-bernardTheta 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).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)\(L^1\)-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.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 \((\widetilde{\operatorname{SL}}_2, \operatorname{PGL}_2)\).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)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)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.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)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.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)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)