Recent zbMATH articles in MSC 22https://zbmath.org/atom/cc/222022-11-17T18:59:28.764376ZWerkzeugQuasi-invariant measures for continuous group actionshttps://zbmath.org/1496.031422022-11-17T18:59:28.764376Z"Kechris, Alexander S."https://zbmath.org/authors/?q=ai:kechris.alexander-sSummary: The class of ergodic, invariant probability Borel measure for the shift action of a countable group is a \(G_\delta\) set in the compact, metrizable space of probability Borel measures. We study in this paper the descriptive complexity of the class of ergodic, quasi-invariant probability Borel measures and show that for any infinite countable group \(\Gamma\) it is \(\boldsymbol{\Pi^0_3}\)-hard, for the group \(\mathbb{Z}\) it is \(\boldsymbol{\Pi^0_3}\)-complete, while for the free group \(\mathbb{F}_\infty\) with infinite, countably many generators it is \(\boldsymbol{\Pi^0_\alpha}\)-complete, for some ordinal \(\alpha\) with \(3\leq\alpha\leq\omega+2\). The exact value of this ordinal is unknown.
For the entire collection see [Zbl 1454.03009].Automorphism groups of countable stable structureshttps://zbmath.org/1496.031452022-11-17T18:59:28.764376Z"Paolini, Gianluca"https://zbmath.org/authors/?q=ai:paolini.gianluca"Shelah, Saharon"https://zbmath.org/authors/?q=ai:shelah.saharonSummary: For every countable structure \(M\) we construct an \(\aleph_0\)-stable countable structure \(N\) such that \(\Aut(M)\) and \(\Aut(N)\) are topologically isomorphic. This shows that it is impossible to detect any form of stability of a countable structure \(M\) from the topological properties of the Polish group \(\Aut(M)\).Sato-Tate equidistribution for families of automorphic representations through the stable trace formulahttps://zbmath.org/1496.110712022-11-17T18:59:28.764376Z"Dalal, Rahul"https://zbmath.org/authors/?q=ai:dalal.rahulSato-Tate equidistribution questions are a local version of Plancherel equidistribution theorems: they ask what can be said about the distribution of the local components of global (or adelic) objects, more precisely which limiting statistics of these local components can be established. These studies originated with Sato and Tate on the distribution of the Frobenius ``eigenangles'' of an elliptic curve. In the realm of classical Hecke eigenforms, \textit{P. Sarnak} [Clay Math. Proc. 4, 659--685 (2005; Zbl 1146.11031)] proved such equidistribution for Maass forms and \textit{J.-P. Serre} [J. Am. Math. Soc. 10, No. 1, 75--102 (1997; Zbl 0871.11032)] for modular forms. More recently, such results have been obtained in the realm of automorphic representations by \textit{S. W. Shin} and \textit{N. Templier} [Invent. Math. 203, No. 1, 1--177 (2016; Zbl 1408.11042)] for families of automorphic forms having local restrictions, precisely being in a fixed discrete series L-packet at the Archimedean place and having bounded non-Archimedean ramification.
The author establishes analogous results when restricting the Archimedean component to a fixed discrete series representation instead of a whole L-packet as in Shin-Templier, and also allows for a nontrivial center. These features make possible to single out certain specific aspects of automorphic representations in applications of statistical results, for instance distinguishing between holomorphic and nonholomorphic discrete series.
While the main tool in Shin-Templier work was the use of the invariant trace formula with a Euler-Poincaré function at the Archimedean place in order to write the Plancherel distribution as the spectral side of a simple trace formula, the author appeals to a hyperendoscopic version of the stable trace formula with a pseudocoefficient at the Archimedean place.
For a more precise statement of the main result of the paper, let \(G\) be a reductive group over \(F\).
Let \(U\) be an open compact subgroup of \(G(\mathbb{A}^{\infty, S})\) for some finite set of places \(S\), \(\xi\) a regular weight of \(G_{\mathbb{C}}\), \(\Pi_\mathrm{disc}(\xi)\) the discrete series L-packet corresponding to \(\xi\) and \(\pi_0 \in \Pi_\mathrm{disc}(\xi)\).
The family of automorphic forms under consideration is
\[
\mathcal{F}_{U, \infty} = \{\pi \in \mathcal{A}_{\mathrm{disc}}(G) \ : \ \pi_\infty \simeq \pi_0, \ \dim(\pi^\infty)^U \geq 1\}
\]
where \(\mathcal{A}_{\mathrm{disc}}(G)\) denotes the space of all the discrete automorphic representations of \(G\).
Equidistribution questions can be phrased as the study of the distribution measure
\[
\mu_{\mathcal{F}_{U, \infty}} := \sum_{\pi \in \mathcal{F}_{U, \infty}} a_\pi \delta_{\pi_S}\tag{1}
\]
where the \(a_\pi\) are the natural spectral weights
\[
a_\pi = m_{\mathrm{disc}}(\pi) \dim(\pi^{S, \infty})^U.
\]
Shin and Templier studied analogous measures (but with the prescribed condition \(\pi_\infty \in \Pi_{\mathrm{disc}}(\xi)\)) in the weight aspect (\(\xi \to \infty\)) and in the level aspect (\(U \to 1\)) separately, and proved that the limiting measure is the Plancherel measure on the dual group \(\hat{G}_S\). In fact they even obtained a quantitative version of this statement, including an explicit error term.
The author of the present article gives a variation of this result, essentially by allowing for a nontrivial center and requiring that \(\pi_\infty \simeq \pi_0\). His result is the following statement, concerning the weight aspect (\(\xi \to \infty\)).
Theorem. Let \((\pi_k)_k\) be a sequence of discrete series representations of \(G_\infty\) such that the Archimedean components have regular weights going to infinity. Denote \(S\) a certain finite sets of finite places.
There are constants \(A\) and \(B\) such that for any \(\phi_{S}\) in the Hecke algebra \(\mathcal{H}(G_{S}, K_{S}, \chi_{S})^{\leq \kappa}\), we have
\[
\frac{\bar{\mu}^{\mathrm{can}}(U_{\mathfrak{X}}^{S, \infty}) |\Pi_{\mathrm{disc}(\xi_k)}|}{\tau'(G) \dim(\xi_k)} \sum_{\pi \in \mathcal{A}_{\mathrm{disc}}(G, \chi)} a_{\mathcal{F}_k}(\pi) \widehat{\phi}_S(\pi) = E(\widehat{\phi}_S | \omega_\xi, L, \chi_S) + O(q_{S_1}^{A+B\kappa} m(\xi_k)^{-1})
\]
and the implied constants only depend on \(G\), \(\mathfrak{X}\), \(\chi\), \(\phi_{S}\) and \(U^{S, \infty}\).
Here, the main term \(E(\widehat{\phi}_S | \omega_\xi, L, \chi_S)\) is a conditional Plancherel expectation, i.e., an average of measures of the form \(\mathrm{d}\mu_{\hat{f}}/\mathrm{d}\mu^{\mathrm{pl}}_{Z_G}\) where \(\mathrm{d}\mu_{\hat{f}}\) is the pushed forward of \(\hat{f}\mathrm{d}\mu^{\mathrm{pl}}\) to \(\hat{Z_G}\), and \(\mathrm{d}\mu^{\mathrm{pl}}\) denotes the Plancherel measure. Also, \(S_1\) denotes a subset of finite places in \(S\) such that \(U_{S_1}\) is a maximal compact subgroup, therefore ensuring the selected representations to be unramified at \(S_1\).
The error term in particular has explicit dependence upon both the Archimedean weight \(m(\xi_k)\) and the prescribed unramified finite places \(S_1\) of the test-function.
This result leads to obtain various consequences, namely Plancherel and Sato-Tate equidistribution results under various restrictions.
The approach to prove these results is via trace formula methods. A rough outline of the strategy is as follows. The trace formula is a distributional equality between two distributions
\[
I_{\mathrm{spec}}(\phi) = I_{\mathrm{geom}}(\phi)
\]
where \(\phi\) is in a certain space of functions \(\mathcal{H}(G)\), \(I_{\mathrm{spec}}\) is a sum over the automorphic spectrum of modified traces of \(\phi\) against components of \(L^2(G(\mathbb{Q})\backslash G(\mathbb{A}))\), and \(I_{\mathrm{geom}}\) is a sum over conjugacy classes of rational points of certain modified orbital integrals.
The distributing measure (1) is then studied in two steps:
\begin{itemize}
\item identifying or approaching the measure (1) by a spectral side \(I_{\mathrm{spec}}(\phi)\) for a suitable function \(\phi \in \mathcal{H}(G)\) ;
\item studying the geometric terms in \(I_{\mathrm{geom}}(\phi)\), that are expected to behave mostly like the sole contribution of the central elements, and where the other terms contribute as an error term.
\end{itemize}
Both steps are generally difficult and involve deep results about harmonic analysis on reductive groups, representation theoretic and automorphic results about the involved automorphic representations, and a fine geometric, arithmetic and/or analytic study of the geometric side. This strategy is indeed the heart of the results of both Shin-Templier and the author.
The invariant trace formula of Arthur is difficult to state involving both a continuous part of the spectrum (splitting into a continuous integral along Eisenstein series) and weighted unipotent orbital integrals (multiplied by global coefficients). With some extra conditions (called \textit{stable cuspidality}) on the test function \(\phi\) (hence, inducing conditions on the automorphic family effectively selected by \(I_{\mathrm{spec}}(\phi)\)), these two more difficult parts disappear: this is the so-called \textit{simple trace formula} of \textit{J. Arthur} [J. Am. Math. Soc. 1, No. 3, 501--554 (1988; Zbl 0667.10019)]. Such restrictions are chosen in the work of Shin-Templier as well as in the author's work and is the reason of the two strong assumptions in the family under consideration:
\begin{itemize}
\item being a discrete series at the Archimedean place;
\item having prescribed ramification at some finite places.
\end{itemize}
We emphasize some differences, difficulties and points of interest (mainly compared to Shin-Templier). An easier entrance door towards these methods may be the article of \textit{J. Binder} [Isr. J. Math. 222, No. 2, 973--1028 (2017; Zbl 1431.11063)] that we suggest to those unfamiliar with the methods and willing a guide to dive into the details.
An important difference between this article and the work of Shin and Templier arises in the fact that the author selects a single representation at the Archimedean place instead of an L-packet. This requires to take a pseudocoefficient for the Archimedean test-function in the trace formula, while Shin and Templier take the Euler-Poincaré test-function of \textit{L. Clozel} and \textit{P. Delorme} [Ann. Sci. Éc. Norm. Supér. (4) 23, No. 2, 193--228 (1990; Zbl 0724.22012)]. A key feature in Shin-Templier is that the Euler-Poincaré function satisfies the stable cuspidality conditions, reducing the Arthur trace formula to its simple form, which is not the case with the pseudocoefficient in the author's work.
To address this difficulty, the idea is to use the stable trace formula to write the invariant distribution \(I^G(f)\) attached to such a function as a linear combination of \textit{stable} distributions of (transfers of) this function on smaller endoscopic subgroups, in the form
\[
I^G(f) = \sum_{H \in \mathcal{E}_{\mathrm{ell}}} S^H(f^H),
\]
where \(\mathcal{E}_{\mathrm{ell}}\) is a set of certain endoscopic subgroups. A crucial fact is to notice that the pseudocoefficient and the Euler-Poincaré function have the same stable orbital integral, so that in the above equation the transfer \(f^H\) can be modified to have Euler-Poincaré component at the Archimedean place.
The next step is to appeal to (a generalization of) \textit{A. Ferrari}'s hyperendoscopic formula [Manuscr. Math. 124, No. 3, 363--390 (2007; Zbl 1146.58012)]. The hyperendoscopic formula is a rearrangement of the stabilized trace formula featuring usual \textit{invariant} distributions. It takes the form
\[
I^G(f) = I^G(f^\star) + \sum_{\mathcal{H} \in \mathcal{HE}_{\mathrm{ell}}} \iota(G, \mathcal{H}) I^{\mathcal{H}}((f-f^\star)^{\mathcal{H}}),
\]
where \(\mathcal{HE}_{\mathrm{ell}}\) is a set of certain endoscopic subgroups, \(f^\star\) is a function having the same stable orbital integrals as \(f\), and the \(\iota(G, \mathcal{H}) \) are harmless constants. The interest of appealing to the hyperendoscopic trace formula is that it features invariant orbital integrals on the geometric side, so that the machinery developed by Shin and Templier remains available. Moreover, the transfers \((f-f^\star)^{\mathcal{H}}\) can in fact be taken to be explicit linear combinations of Euler-Poincaré functions, for which bounds established by Shin and Templier can be used.
It has to be noted that fine work is still required in order to obtain estimates that are uniform in the endoscopic subgroup \(\mathcal{H}\) and to deal with the existence of central characters.
Reviewer: Didier Lesesvre (Lille)Two-dimensional locally Nash groupshttps://zbmath.org/1496.140472022-11-17T18:59:28.764376Z"Baro, ElÍas"https://zbmath.org/authors/?q=ai:baro.elias"de Vicente, Juan"https://zbmath.org/authors/?q=ai:de-vicente.juan"Otero, Margarita"https://zbmath.org/authors/?q=ai:otero.margaritaThe article under review deals with locally Nash groups. A locally Nash group is a Lie group equipped with a compatible real algebraic structure. The one-dimensional locally Nash groups were classified by \textit{J. Madden} and \textit{C. Stanton} in [Pac. J. Math. 154, No. 2, 331--344 (1992; Zbl 0723.14042)]. The main goal of the present paper is to classify the two-dimensional simply connected abelian locally Nash groups. In order to get it, the authors use the category of locally \(\mathbb{C}\)-Nash groups, which they introduced in [Rev. Mat. Complut. 32, no. 2, 531--558 (2019; Zbl 1440.14263)].
Theorem 4.1 gives the classification of two-dimensional simply connected abelian locally \(\mathbb{C}\)-Nash groups, by being isomorphic to a group of just one of four classes, while isomorphisms inside each class are also determined. Furthermore, the automorphism group of each of these groups (except for one of the classes) are obtained in Proposition 4.2. Then, the corresponding results for the real case are given in Theorem 5.2, where the groups are classified in five classes, and Proposition 5.4 which gives their automorphism group except for one class.
Reviewer: José Javier Etayo (Madrid)Descents of unipotent representations of finite unitary groupshttps://zbmath.org/1496.200162022-11-17T18:59:28.764376Z"Liu, Dongwen"https://zbmath.org/authors/?q=ai:liu.dongwen"Wang, Zhicheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.2Summary: Inspired by the Gan-Gross-Prasad conjecture and the descent problem for classical groups, in this paper we study the descents of unipotent representations of unitary groups over finite fields. We give the first descents of unipotent representations explicitly, which are unipotent as well. Our results include both the Bessel case and Fourier-Jacobi case, which are related via theta correspondence.Remarks on the theta correspondence over finite fieldshttps://zbmath.org/1496.200172022-11-17T18:59:28.764376Z"Liu, Dongwen"https://zbmath.org/authors/?q=ai:liu.dongwen"Wang, Zhicheng"https://zbmath.org/authors/?q=ai:wang.zhi-cheng.2Summary: S.-Y. Pan decomposed the uniform projection of the Weil representation of a finite symplectic-odd orthogonal dual pair, in terms of Deligne-Lusztig virtual characters, assuming that the order of the finite field is large enough. We use Pan's decomposition to study the theta correspondence for dual pairs of this kind, following the approach of Adams and Moy and Aubert, Michel and Rouquier. Our results give the theta correspondence between unipotent representations and certain quadratic unipotent representations.Commensurated subgroups in tree almost automorphism groupshttps://zbmath.org/1496.200482022-11-17T18:59:28.764376Z"Le Boudec, Adrien"https://zbmath.org/authors/?q=ai:le-boudec.adrien"Wesolek, Phillip"https://zbmath.org/authors/?q=ai:wesolek.phillip-rSummary: We prove that the almost automorphism groups \(\operatorname{AAut}(\mathcal T_{d,k})\) admit exactly three commensurability classes of closed commensurated subgroups. Our proof utilizes an independently interesting characterization of subgroups of \(\operatorname{AAut}(\mathcal T_{d,k})\) which contain only periodic elements in terms of the dynamics of the action on the boundary of the tree.
Our results further cover several interesting finitely generated subgroups of the almost automorphism groups, including the Thompson groups \(F, T\), and \(V\). We show in particular that Thompson's group \(T\) has no commensurated subgroups other than the finite subgroups and the entire group. As a consequence, we derive several rigidity results for the possible embeddings of these groups into locally compact groups.Sofic boundaries of groups and coarse geometry of sofic approximationshttps://zbmath.org/1496.200692022-11-17T18:59:28.764376Z"Alekseev, Vadim"https://zbmath.org/authors/?q=ai:alekseev.vadim"Finn-Sell, Martin"https://zbmath.org/authors/?q=ai:finn-sell.martinSummary: Sofic groups generalise both residually finite and amenable groups, and the concept is central to many important results and conjectures in measured group theory. We introduce a topological notion of a sofic boundary attached to a given sofic approximation of a finitely generated group and use it to prove that coarse properties of the approximation (property A, asymptotic coarse embeddability into Hilbert space, geometric property (T)) imply corresponding analytic properties of the group (amenability, a-T-menability and property (T)), thus generalising ideas and results present in the literature for residually finite groups and their box spaces. Moreover, we generalise coarse rigidity results for box spaces due to Kajal Das, proving that coarsely equivalent sofic approximations of two groups give rise to a uniform measure equivalence between those groups. Along the way, we bring to light a coarse geometric viewpoint on ultralimits of a sequence of finite graphs first exposed by Ján Špakula and Rufus Willett, as well as proving some bridging results concerning measure structures on topological groupoid Morita equivalences that will be of interest to groupoid specialists.Deligne-Lusztig duality and wonderful compactificationhttps://zbmath.org/1496.200752022-11-17T18:59:28.764376Z"Bernstein, Joseph"https://zbmath.org/authors/?q=ai:bernstein.joseph"Bezrukavnikov, Roman"https://zbmath.org/authors/?q=ai:bezrukavnikov.roman"Kazhdan, David"https://zbmath.org/authors/?q=ai:kazhdan.david-aSummary: ``We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne-Lusztig (or Alvis-Curtis) duality for \(p\)-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for \(G=GL(n)\) and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a \(p\)-adic group.''
From the Introduction: Let \(G\) be a reductive \(p\)-adic group. For every smooth \(G\)-module \(M\) one can form a complex
\[
0\to M\to\bigoplus_P i_P^G r_P^G(M)\to\cdots\to i_B^G r_B^G\to 0,
\]
where \(i_P^G \), \(r_P^G\) denote, respectively, the parabolic induction and Jacquet functors, and summation in the \(i\)-th term runs over conjugacy classes of parabolic subgroups of corank \(i\). The complex is called the Deligne-Lusztig complex \(DL(M)\) associated to \(M\). One extends \(DL\) to the bounded derived category of smooth representations.
Theorem. For a complex \(M\) with admissible cohomology we have a canonical quasi-isomorphism
\[
DL( \check{M})\cong \mathrm{RHom}_H(M,H)[r],
\]
where \(\check{M}\) denotes the contragradient representation, \(H\cong C_c^\infty\) is the regular bimodule for the Hecke algebra and \(r\) is the split rank of the center of \(G\).
This theorem is proved by using an explicit resolution for the regular bimodule over \(H\) coming from geometry of the wonderful compactification \(\bar G\) of \(G\).
The main application says that for an irreducible module \(M\) the complex \(DL(M)\) has cohomology in only one degree \(d\), where \(d\) is dimension of the component of the Bernstein center containing \(M\). This implies that in fact \(H^d (DL(M))\) is an irreducible representation. Thus one gets a ``new way to introduce an involution on the set of irreducible representations'', as mentioned in the summary.
Reviewer: Wilberd van der Kallen (Utrecht)Algebraic entropies of commuting endomorphisms of torsion abelian groupshttps://zbmath.org/1496.200862022-11-17T18:59:28.764376Z"Biś, Andrzej"https://zbmath.org/authors/?q=ai:bis.andrzej"Dikranjan, Dikran"https://zbmath.org/authors/?q=ai:dikranjan.dikran-n"Bruno, Anna Giordano"https://zbmath.org/authors/?q=ai:giordano-bruno.anna"Stoyanov, Luchezar"https://zbmath.org/authors/?q=ai:stoyanov.luchezar-nA left semigroup action of a semigroup \(S\) on an abelian group \(A\) by group endomorphisms is defined by \(\alpha: S\times A \to A, (s, x) \mapsto \alpha(s)(x)\) with \(\alpha(st)=\alpha(s) \circ \alpha(t)\) and \(\alpha(s)(x+y)=\alpha(s)(x)+\alpha(s)(y)\) for every \(s, t\in S\) and \(x,y\in A\). The concept of the algebraic entropy of \(\alpha\), developed in [\textit{D. Dikranjan} et al., Trans. Am. Math. Soc. 361, No. 7, 3401--3434 (2009; Zbl 1176.20057)], has some nice properties, but also has some shortcomings. For instance, if \(S\) is finite and \(A\) is infinite then ent(\(\alpha)=\infty\). If \(S\) is a group and \(T\) is a subgroup of \(S\) there are many examples where the restricted action \(\alpha \upharpoonright_T\) has ent(\(\alpha \upharpoonright_T )=\infty\) and ent(\(\alpha\))=0. This led the authors to define \(\widetilde{\mathrm{ent}}^\Gamma(\alpha)\), the algebraic receptive entropy of \(\alpha\) with respect to the regular system \(\Gamma\) of \(S\).
\par The authors will show in Section 2 that the algebraic receptive entropy naturally extends the algebraic entropy for \(\mathbb{N}\)-actions. Section 3 makes use of the correspondence between \(\mathbb{N}^m\)-actions on abelian groups \(A\) with prime exponent \(p\) and \(R_p\)-module structures on \(A\), where \(R_p=\mathbb{F}_p[X_1, \ldots , X_m]\). This leads to the computation of the algebraic entropy and the algebraic receptive entropy of \(R_P\) in Theorem 3.3.
Section 4 includes the computations of the algebraic entropies. Theorem 4.1 shows that when \(\mathfrak{a}\) is a non-zero ideal of \(R_p\) and \(A=R_p/\mathfrak{a}\) is an infinite cyclic \(R_p\)-module then ent(\(A\))=0. This leads to Theorem 4.4 showing that ent(\(A)=\mathrm{rank}_{R_P}(A) \log p\) for an arbitrary \(R_p\) module \(A\). Proposition 4.7 is a characterization of \(\mathbb{N}^m\)-actions on torsion abelian groups with zero algebraic entropy using Bernoulli shifts.
In Section 5, Theorem 5.1 establishes that the receptive algebraic entropy of an infinite cyclic \(R_P\)-module \(A =R_p/\mathfrak{a}\) is infinite when \(0\not= \mathfrak{a}\) is a principal ideal in \(R_p\). When \(m=2\) Theorem 5.5 and Corollary 5.7 allow the computation of the receptive algebraic entropy of \(A\) whether \(\mathfrak{a}\) is principal or not. In Corollary 5.8, keeping \(m=2\), we have for an infinite finitely generated \(R_p\)-module \(A\) the following are equivalent: (a) ent(\(A\))=0; (b) \(0< \widetilde{\mathrm{ent}}(A) <\infty\); rank\(_{R_p}(A)=0\).
Reviewer: Ross P. Abraham (Brookings)Local formulas for multiplicative formshttps://zbmath.org/1496.220012022-11-17T18:59:28.764376Z"Cabrera, A."https://zbmath.org/authors/?q=ai:cabrera.alejandro"Mărcuţ, I."https://zbmath.org/authors/?q=ai:marcut.ioan"Salazar, M. A."https://zbmath.org/authors/?q=ai:salazar.maria-ameliaIn a previous paper, the authors gave an explicit construction of a local Lie groupoid, for a given Lie algebroid, which is not necessarily integrable. In the paper under review, they extend this integration process to Lie algebroids endowed with extra infinitesimal data. This way, they give explicit formulas for the local version of various multiplicative forms on groupoids. Explicitly, they obtain local integrations and non-degenerate realizations of Poisson, Nijenhuis-Poisson, Dirac, and Jacobi structures by local symplectic, symplectic-Nijenhuis, presymplectic, and contact groupoids.
Reviewer: Iakovos Androulidakis (Athína)Sufficiently close one-dimensional pseudorepresentations are equalhttps://zbmath.org/1496.220022022-11-17T18:59:28.764376Z"Shtern, A. I."https://zbmath.org/authors/?q=ai:shtern.alexander-iIn this interesting short paper, the author gives a sufficient condition such that two one-dimensional pseudorepresentations of a group coincide. This result does not hold for pseudorepresentations whose dimension is greater than one.
Finally the following is shown: Let \(\pi\) and \(\rho\) be bounded pseudorepresentations of a group \(G\) into finite-dimensional Banach spaces \(E\) and \(F\). If the characters \(\chi_\pi\) and \(\chi_\rho\) of \(\pi\) and \(\rho\), respectively, coincide, then \(\pi\) and \(\rho\) are pointwise equivalent.
Reviewer: Dieter Remus (Paderborn)Locally pro-\(p\) contraction groups are nilpotenthttps://zbmath.org/1496.220032022-11-17T18:59:28.764376Z"Glöckner, Helge"https://zbmath.org/authors/?q=ai:glockner.helge"Willis, George A."https://zbmath.org/authors/?q=ai:willis.george-aA \textit{contraction group} is a pair \((G,\tau )\), where \(G\) is a locally compact group and \(\tau\) is an automorphism of \(G\) such that \(\tau^n(x)\rightarrow e\) as \(n\rightarrow\infty\) for all \(x\in G\). A topological group is \textit{locally pro-p} if it has an open subgroup which is a pro-\(p\) group. In this paper the authors answer an open question showing that every locally pro-\(p\) contraction group is nilpotent.
Any locally pro-\(p\) contraction group is the direct product of a torsion subgroup and a nilpotent \(p\)-adic Lie group. It is natural to ask whether the torsion factor is nilpotent as well. A torsion locally pro-\(p\) contraction group has a composition series in which each factor is isomorphic to \((\mathbb{F}_p((t)),[t\cdot\,])\), the additive group of the field of formal Laurent series over the finite field \(\mathbb{F}_p\) (and thus is a torsion group of exponent \(p\)), and the map \([t\cdot\,]:f\mapsto tf\) is an automorphism of \(\mathbb{F}_p((t))\) such that \([t\cdot\,]^k f\rightarrow 0\) as \(k\rightarrow\infty\) for every \(f\in\mathbb{F}_p((t))\). In this case the group is called \(p\)-\textit{power contraction group}.
\textbf{Theorem A}: Every \(p\)-power contraction group \((G,\tau)\) is nilpotent.
The proof of this theorem is by induction on the number of composition factors and follows from the next result.
\textbf{Theorem B}: Suppose that \(\phi:\mathbb{F}_p((t))\rightarrow\Aut(\mathbb{F}_p((t))^d)\) is a continuous homomorphism of groups and satisfies \(\phi\circ[t\cdot\,]=\text{ad}([t\cdot\,])\circ\phi\), where ad(\([t\cdot\,](\psi)=[t^{\oplus \,d}\cdot\,]\circ\psi\circ[t^{\oplus \,d}\cdot\,]^{-1}\) \((\psi\in\Aut(\mathbb{F}_p((t))^d))\), the automorphism \([t^{\oplus \,d}\cdot\,]\) of \(\mathbb{F}_p((t))^d\) is given by \([t^{\oplus \,d}\cdot\,](f_1,\cdots,f_d)=(tf_1,\cdots,tf_d)\), and \(\Aut(\mathbb{F}_p((t))^d)\) has the Braconnier topology. Then there is \(\xi\in\mathbb{F}_p((t))^d\setminus\{0\}\) such that \(\phi(f)(\xi)=\xi\) for every \(f\in \mathbb{F}_p((t))\).
The proof of the second theorem goes by representing endomorphisms in End(\(\mathbb{F}_p((t))^d)\) as infinite block matrices with the blocks being \(d\times d\) matrices over \(\mathbb{F}_p\). Conditions on block matrices for them to correspond to elements of End(\(\mathbb{F}_p((t))^d\)) are determined, and then additional conditions for them to belong to the image of \(\phi\) are derived.
Reviewer: María Vicenta Ferrer González (Castelló)Analytic properties of approximate latticeshttps://zbmath.org/1496.220042022-11-17T18:59:28.764376Z"Björklund, Michael"https://zbmath.org/authors/?q=ai:bjorklund.michael"Hartnick, Tobias"https://zbmath.org/authors/?q=ai:hartnick.tobiasThis paper is concerned with analytic properties of approximate lattices in locally compact second countable (lcsc) groups, in particular with properties of Kazhdan and Haagerup type.
A subset \(\Lambda\) of a group is an \textit{approximate subgroup} if \(\Lambda^{-1}=\Lambda\), \(e\in\Lambda\) and there exists a finite set \(F_\Lambda\subset\Lambda^3\) such that \(\Lambda^2\subset\Lambda F_\Lambda\). If \(\Lambda \) is an approximate subgroup of a lcsc group \(G\), the group \(\Lambda^\infty\) generated by \(\Lambda\) in \(G\) is known as the \textit{enveloping group} of \(\Lambda\) and the pair \((\Lambda,\Lambda^\infty)\) as an \textit{approximate group}. An approximate subgroup \(\Lambda\subset G\) is a \textit{uniform approximate lattice} in \(\Lambda\) if it is a Delone subset of \(G\). If \(\Lambda\) is a uniform approximate lattice in \(G\) which admits a \(G\)-invariant probability measure we call \(\Lambda\) a \textit{strong uniform approximate lattice}. Recall that if \((X,d)\) is a metric space, then a subset \(\Lambda\subset X\) is called a \textit{Delone set} if there exist constants \(R>r>0\) such that \(d(x,y)\geq r\) for all \(x,\,y\in\Lambda\) with \(x\neq y\); and for every \(x\in X\) there is \(y\in\Lambda\) with \(d(x,y)\leq R\). If \(G\) is a lcsc group, a subset \(\Lambda\subset G\) is Delone if it is a Delone set with respect to some left-admissible metric on \(G\).
Let \(G\), \(H\) be lcsc groups and denote by \(\pi_G:G\times H\rightarrow G\) the projection onto \(G\). Let \(\Gamma< G\times H\) be a uniform lattice which projects injectively into \(G\) and densely into \(H\), and let \(W\) be a compact identity neighborhood in \(H\) with dense interior. Then the set \(\Lambda:=\pi_G(\Gamma\cap(G\times W))\) is called a \textit{symmetric model set}. A subset \(\Delta\) of a lcsc group \(G\) is called a \textit{Meyer set} if there exists a symmetric model set \(\Lambda\subset G\) and a finite subset \(F\subset G\) such that \(\Delta\subset \Lambda\subset \Lambda F\).
A lcsc group \(G\) has the \textit{Haagerup Property} if there exists a metrically proper affine isometric action on a Hilbert space. The group \(G\) has the \textit{Property (FH)}, if every affine isometric action of \(G\) on a Hilbert space has bounded orbits. Both properties have been generalized in two different directions. Firstly, replacing Hilbert spaces by \(L^p\)-spaces, \(p>1\), we obtain the notion of a-F\(L^p\)-menability, respectively the notion of Property (F\(L^p\)). Secondly, one can replace affine actions and cocycles by the weaker notions of quasi-cocycles, respectively weak quasi-cocycles. \(L^p\)-versions of Property (FH) and a-(FH)-menability are refered as Kazhdan and Haagerup type properties, respectively, as well a quasified version thereof à la Burguer-Monod and Ozawa. Let \((\Lambda,\Lambda^\infty)\) be an approximate group and let \(\mathcal{P}\) be a Kazhdan-type of Haagerup-type property, then we say that \((\Lambda,\Lambda^\infty)\) has \(\mathcal{P}\) if \(\Lambda^\infty\) has \(\mathcal{P}\) relative to \(\Lambda\).
Some of the main results are the following. Their proofs are based on a version of cocycle induction from a strong uniform approximate lattice of a lcsc group. This construction is general enough to also apply to (weak) quasi-cocycles with values in \(L^p\)-spaces.
Theorem 1.6: Let \(\mathcal{H}\) be a Haaregup-type property, \(G\) be a lcsc group and \(\Sigma\subset G\) be a uniform approximate lattice which is contained in a strong uniform approximate lattice \(\Lambda\subset G\). Then \(G\) has \(\mathcal{H}\) if and only if \((\Sigma,\Sigma^\infty)\) has \(\mathcal{H}\).
Theorem 1.9: Let \(\mathcal{T}\) be a Kazhdan-type property,\(G\) be a lcsc group and \(\Lambda\subset G\) be a uniform approximate lattice. Assume that one of the following holds: (i) \(\Lambda\) is a model set or (ii) \(\Lambda\) is a Meyer set which is contained in a model set of almost conected type. Then \(G\) has \(\mathcal{T}\) if and only if \((\Sigma,\Sigma^\infty)\) has \(\mathcal{T}\).
The authors also list some examples, applications and open problems.
Reviewer: María Vicenta Ferrer González (Castelló)Density conditions with stabilizers for lattice orbits of Bergman kernels on bounded symmetric domainshttps://zbmath.org/1496.220052022-11-17T18:59:28.764376Z"Caspers, Martijn"https://zbmath.org/authors/?q=ai:caspers.martijn"van Velthoven, Jordy Timo"https://zbmath.org/authors/?q=ai:van-velthoven.jordy-timoThe present paper studies certain estimates which improve on general density theorems for restricted discrete series through the dependence on the stabilizers, while recovering in part sharp results for \(G=\mathrm{PSU}(1,1)\). More precisely, let \(\pi_\alpha\) be a holomorphic discrete series representation of a connected semisimple Lie group \(G\) with finite center, acting on a weighted Bergman space \(A^2_\alpha (\Omega)\) on a bounded symmetric domain \(\Omega\), of formal dimension \(d_{\pi_\alpha}\). The authors show that if the Bergman kernel \(k _Z^{(\alpha)}\) is a cyclic vector for the restriction \(\pi_\alpha|_{\Gamma}\) to a lattice \(\Gamma\le G\), then \(\mathrm{vol}(G/\Gamma)d_{\pi_\alpha}\le|\Gamma_Z|^{-1}\).
Reviewer: Andreas Arvanitoyeorgos (Patras)Degenerate principal series and nilpotent invariantshttps://zbmath.org/1496.220062022-11-17T18:59:28.764376Z"Li, Ning"https://zbmath.org/authors/?q=ai:li.ningLet \(G\) be a noncompact real semisimple Lie group. The representation theory of \(G\) is a fertile branch of the whole field of representation theory, and the main topic there is to understand infinite dimensional irreducible representations, especially unitary ones (unitary dual problem). To understand them there are two main directions. One is to construct representations (parabolic induction/ orbit methods/ theta correspondence, etc.) and the other is to identify each representations. For the latter problem, Vogan attached a nice geometric invariant to each representation, namely associated cycles [\textit{D. A. Vogan jun.}, Prog. Math. 101, 315--388 (1991; Zbl 0832.22019)]. This is a positive integer linear combination of nilpotent orbits, which are known to be finitely many.
In this paper under review the author computes the associated cycles for relatively small representations which appear in the degenerate principal series for the symplectic group \( Sp(2n, \mathbb{R}) \).
There are several techniques needed, but the main tool here is the machinery of the theta correspondence. Namely, by the work of \textit{S. T. Lee} [Compos. Math. 103, No. 2, 123--151 (1996; Zbl 0857.22010)] (also see the references in the paper; Howe, Kudla-Rallis, Lee-Zhu, ...), the theta lift of the trivial representation can be embedded into a degenerate principal series. On the other hand, for the theta lift of the trivial representation, there are rich techniques to compute the associated cycles [\textit{H. Y. Loke} and \textit{J. Ma}, Compos. Math. 151, No. 1, 179--206 (2015; Zbl 1319.22009)]. Combining various embedding and comparing them, the author can specify the associated cycle. She is lucky because the multiplicity of each cycle is always one, and there is no need to use other techniques.
Finally, the author also calculates the dimension of generalized Whittaker vectors, which is associated to the nilpotent orbits in the associated cycles, and conclude that the multiplicities of the cycles agree with the dimension of generalized Whittaker vectors. Techniques are largely the same as above, using [\textit{R. Gomez} and \textit{C.-B. Zhu}, Geom. Funct. Anal. 24, No. 3, 796--853 (2014; Zbl 1404.22033); \textit{R. Gomez} et al., Compos. Math. 153, No. 2, 223--256 (2017; Zbl 1384.20039)]. The dimensions are all one.
All the results are summarized neatly in Theorems 1.3--1.5, with the parametrization of representations explained in \S~2. The parametrization of nilpotent orbits are given in \S~3.1, related to theta correspondence.
By the work of \textit{H. Matumoto} [Compos. Math. 82, No. 2, 189--244 (1992; Zbl 0797.22005)], for the representations of the largest possible Gelfand-Kirillov dimension, the dimension of the Whittaker vectors agree with the multiplicities in the associated cycles. There are no such results in general, but ultimately desired.
Reviewer: Kyo Nishiyama (Aoyama)Parabolic induction and the Harish-Chandra \(\mathcal{D}\)-modulehttps://zbmath.org/1496.220072022-11-17T18:59:28.764376Z"Ginzburg, Victor"https://zbmath.org/authors/?q=ai:ginzburg.viktor-l|ginzburg.victorLet \(G\) be a reductive group and \(L\) a Levi subgroup. Parabolic induction and restriction are a pair of adjoint functors between Ad-equivariant derived categories of either constructible sheaves or (not necessarily holonomic) \(\mathcal{D}\)-modules on \(G\) and \(L\), respectively. Bezrukavnikov and Yom Din proved, that these functors are exact. In this paper, the author considered a special case where \(L = T\) is a maximal torus. Let \(\mathcal{D}_T\) denote the sheaf of differential operators on \(T\) and \(\mathcal{D}(T) = \Gamma(T, \mathcal{D}_T)\). He gave explicit formulas for parabolic induction and restriction in terms of the Harish-Chandra \(\mathcal{D}\)-module on \(G\times T\). He showed that this module is flat over \(\mathcal{D}(T)\), which easily implies that parabolic induction and restriction are exact functors between the corresponding abelian categories of \(\mathcal{D}\)-modules.
Reviewer: Zhangqiang Bai (Suzhou)Parabolic inductions for pro-\(p\)-Iwahori Hecke algebrashttps://zbmath.org/1496.220082022-11-17T18:59:28.764376Z"Abe, Noriyuki"https://zbmath.org/authors/?q=ai:abe.noriyukiThis paper belongs to a series of works by the author and collaborators on structural aspects of modulo \(p\) representations of \(p\)-adic groups.
The main results of the present paper are computations of the right and left adjoints to the parabolic induction functor. This lays the foundation for computations of \(\mathrm{Ext}\) groups for simple representations of the pro-\(p\)-Iwahori-Hecke algebra.
Let \(F\) be a non-Archimedean local field with residue characteristic \(p\), and let \(\mathbf{G}\) be a reductive algebraic group over \(F\). The pro-\(p\) Iwahori-Hecke algebra \(\mathcal{H}\) plays a major role in studying representations \(G=\mathbf{G}(F)\) over fields of characteristic \(p\).
In the previous paper [J. Reine Angew. Math. 749, 1--64 (2019; Zbl 1472.22008)], the author classified irreducible \(\mathcal{H}\)-modules in this setting. In the present paper, the author computes the left and right adjoints \(L_P\) and \(R_P\), respectively, of the parabolic induction functor \(I_P\) for \(\mathcal{H}\) and shows that they preserve simplicity and that \(L_P\) is exact. Analogous results for the group setting were obtained by the author et al. in [Trans. Am. Math. Soc. 371, No. 12, 8297--8337 (2019; Zbl 1474.22013)].
The main application given is as follows.
Let \(\pi_1\) and \(\pi_2\) be two simple \(\mathcal{H}\)-representations. By [the author, loc. cit.], \(\pi_2=I_P(\mathrm{St}(\sigma))\) is the parabolic induction of the generalized Steinberg representation attached the supersingular \(\mathcal{H}_P\)-module \(\sigma\).
Combined with the results of the present paper and suppressing some decorations, one gets
\begin{align*}
\mathrm{Ext}^i_{\mathcal{H}}(\pi_1,\pi_2) &\simeq \mathrm{Ext}^i_{\mathcal{H}}(\pi_2, I_P(\mathrm{St}(\sigma)) \\
&\simeq \mathrm{Ext}^i_{\mathcal{H}_P}(L_P(\pi_2), \mathrm{St}(\sigma)) \\
&\simeq \mathrm{Ext}^i_{\mathcal{H}_P}(I_R^P\mathrm{St}(\sigma'),\mathrm{St}(\sigma)) \\
&\simeq \mathrm{Ext}^i_{\mathcal{H}_R}(\mathrm{St}(\sigma'), R_R(\mathrm{St}(\sigma)) \\
&\simeq \mathrm{Ext}^i_{\mathcal{H}_R}(\mathrm{St}(\sigma'), \mathrm{St}(\sigma''))
\end{align*}
for some parabolic subgroup \(R\subset P\). Thus, the present paper reduces the problem to computations for Steinberg modules, a problem the author states will be taken up in a sequel.
Reviewer: Stefan Dawydiak (Toronto)Typical representations via fixed point sets in Bruhat-Tits buildingshttps://zbmath.org/1496.220092022-11-17T18:59:28.764376Z"Latham, Peter"https://zbmath.org/authors/?q=ai:latham.peter-e"Nevins, Monica"https://zbmath.org/authors/?q=ai:nevins.monicaThe paper under review aims to study the \textit{unicity of types} for tame supercuspidal representations of a \(p\)-adic reductive group \(G\). Let \(\pi\) be such a representation constructed from Yu's datum \(\Sigma\) [\textit{J.-K. Yu}, J. Am. Math. Soc. 14, No. 3, 579--622 (2001; Zbl 0971.22012)], and let \((J,\lambda)\) be the \(\mathfrak{s}_\pi\)-type associated to \(\Sigma\). Fix a maximal compact subgroup \(G_y\) of \(G\), where \(y\) is some point of the Bruhat-Tits building \(\mathscr{B}(G)\) of \(G\).
In view of the properties of types and Mackey theory, the restriction of the smooth representation \(\Pi=\text{c-Ind}^G_J\lambda\) to \(G_y\) has a decomposition of the form
\[
\Pi\vert_{G_y} = \bigoplus_{g\in G_y\backslash G/J}{}^g\tau(y, g).
\]
The authors pose a conjecture which strengthens the usual statement about unicity of types.
\textbf{Conjecture.} Suppose that \(\tau(y, g)\) contains an \(\mathfrak{s}_\pi\)-type. Then \(g^{-1}y\in \mathscr{B}(G)\) is a fixed point of the action of \(J\).
The authors prove that the condition is sufficient, and they prove two theorems towards the necessity. Recall that as part of Yu's datum, one has a twisted Levi subgroup \(G^0\), and a vertex \(x\) of \(\mathscr{B}(G^0)\) viewed as a point in \(\mathscr{B}(G)\). The first theorem asserts that if the projection of \(g^{-1}y\) onto \(\mathscr{B}(G^0)\) lies in a facet distinct from \(x\), then \(\tau(y, g)\) does not contain an \(\mathfrak{s}_\pi\)-type. The second theorem asserts that if the geodesic from \(x\) to \(g^{-1}y\) meets a point of \(\mathscr{B}(G)^{H_{t+}}\setminus \mathscr{B}(G)^{Z^0_t}\), then \(\tau(y,g)\) does not contain an \(\mathfrak{s}_\pi\)-type. Here \(H\) is a finite index subgroup of \(J\), and \(Z^0\) is the centre of \(G^0\), both equipped with natural Moy-Prasad-like decreasing filtration subgroups. Along the proof of the main results, the authors provide widely applicable tools for the study of branching rules of \(\pi\) and the unicity of types.
Reviewer: Dongwen Liu (Zhejiang)Affine Beilinson-Bernstein localization at the critical level for \(\mathrm{GL}_2\)https://zbmath.org/1496.220102022-11-17T18:59:28.764376Z"Raskin, Sam"https://zbmath.org/authors/?q=ai:raskin.samThe paper under review proves an important special case of a conjecture of Frenkel and Gaitsgory on an affine analogue of the Beilinson-Bernstein localization theorem.
To provide a context of the main theorem, let us first discuss the Beilinson--Bernstein theorem. Let \(G\) be a reductive group, \(B\) be a Borel subgroup, and \(\mathfrak{g}\) and \(\mathfrak{b}\) be their Lie algebras, respectively. One of the central problems in the subject of representation theory is to classify representations of \(\mathfrak{g}\). As the center \(Z(\mathfrak{g})\) of the universal enveloping algebra \(U(\mathfrak g)\) acts through a character, called a central character, one may as well fix the central character. Then one is led to the study of the category \(\mathfrak g\text{-mod}_0\) of representations of \(U(\mathfrak{g})\) with the central character being the same as the trivial representation.
Consider the flag variety \(G/B\) and the category \(D(G/B)\) of D-modules on it. For an object \(M \in D(G/B)\), its space \(\Gamma (G/B,M)\) of global sections has an induced action of \(\mathfrak{g}\) and one can check that it has the same central character as the trivial representation. Then the celebrated Beilinson-Bernstein localization theorem says that the functor
\[
\Gamma \colon D(G/B)\to \mathfrak{g}\text{-mod}_0
\]
is in fact an equivalence of abelian and derived categories.
For an affine analogue, consider an affine Kac-Moody algebra \(\widehat{\mathfrak{g}}_\kappa \) where \(\kappa\) is a level, or a symmetric invariant bilinear form on \(\mathfrak{g}\). Feigin and Frenkel proved that the center \(Z(\widehat{\mathfrak{g}}_{\kappa }):= Z(U(\widehat{\mathfrak{g}}_\kappa ))\) of the universal enveloping algebra \(U(\widehat{\mathfrak{g}}_\kappa )\) is trivial unless \(\kappa\) is the critical level. Moreover, they identified the center at the critical level in terms of the Langlands dual group \(\check{G}\) and exhibited the following commutative diagrams: \begin{center} \begin{tikzpicture} \draw [-, thick] (0,1.5) to (0,0.5) ; \draw [-, thick] (3,1.5) to (3,0.5) ; \draw [->>, thick] (1,2) to (2,2) ; \draw [->>, thick] (1,0) to (2,0) ; \node at (-0.5,1) {\( \cong \)}; \node at (3.5,1) {\( \cong \)}; \node at (0,0) {\(\text{Fun}(\text{Op}_{\check{G}} )\)}; \node at (3,0) {\(\text{Fun}( \text{Op}_{\check{G}}^{\text{reg}})\)}; \node at (0,2) {\(Z(\widehat{\mathfrak{g}}_{\text{crit}})\)}; \node at (3,2) {\(\text{End}(\mathbb{V}_{\text{crit}})\)}; \end{tikzpicture} \end{center} where \(\mathbb{V}_{\text{crit}} := \text{Ind}^{\widehat{\mathfrak{g}}_{\text{crit}} }_{\mathfrak g[[t]]}\mathbb C\) is the vacuum module, \( \text{Op}_{\check{G}}\) is the space of \(\check{G}\)-opers on the punctured disk, and \( \text{Op}_{\check{G}}^{\text{reg}}\) is the space of \(\check{G}\)-opers on the disk. Now we are interested in understanding the category \( \widehat{\mathfrak{g}}_{\text{crit}}\text{-mod}_{\text{reg}} \) of \(\widehat{\mathfrak{g}}_{\text{crit}} \)-modules where \(Z(\widehat{\mathfrak{g}}_{\text{crit}} )\) acts through the quotient \(\text{Fun}( \text{Op}_{\check{G}}^{\text{reg}})\) of \(\text{Fun}(\text{Op}_{\check{G}} )\).
Just like the finite-dimensional case, we have a functor
\[
\Gamma \colon D_{\text{crit}}(\text{Gr}_G) \to \widehat{\mathfrak{g}}_{\text{crit}}\text{-mod}_{\text{reg}} ,
\]
where \(\text{Gr}_G:=G(K)/G(O)\) is the affine Grassmannian of \(G\) understood as a flag variety of the loop group \(G(K)\). However, this has no chance of being an equivalence; the delta function \(\delta_1\) goes to the vacuum module \(\mathbb{V}_{\text{crit}}\), but the endomorphisms of the delta function are trivial whereas the endomorphisms of the vacuum module are big, as mentioned before.
Frenkel and Gaitsgory's paper ``Local geometric Langlands correspondence and affine Kac-Moody algebras'' suggested to instead consider the following functor
\[
\Gamma^{\text{Hecke}} \colon D_{\text{crit}}(\text{Gr}_G) \otimes_{\text{Rep}(\check{G})} \text{QCoh}( \text{Op}_{\check{G}}^{\text{reg}}) \to \widehat{\mathfrak{g}}_{\text{crit}}\text{-mod}_{\text{reg}}
\]
where
\begin{itemize}
\item the geometric Satake functor \(\text{Rep}(\check{G}) \to D_{\text{crit}}(\text{Gr}_G)^{G(O)}\) is a symmetric monoidal functor and \(D_{\text{crit}}(\text{Gr}_G)^{G(O)}\) acts on \(D_{\text{crit}}(\text{Gr}_G)\);
\item as the space \( \text{Op}_{\check{G}}^{\text{reg}}\) is the moduli space of flat \(\check{G}\)-bundles on the formal disk with an extra structure, the forgetful map \( \text{Op}_{\check{G}}^{\text{reg}}\to \text{Flat}_{\check{G}}(D) = B\check{G} \) gives a symmetric monoidal functor \(\text{Rep}(\check{G})\to \text{QCoh}( \text{Op}_{\check{G}}^{\text{reg}})\).
\end{itemize}
Then their conjecture is that \(\Gamma^{\text{Hecke}}\) is an equivalence of abelian and derived categories.
The main theorem of the current paper proves the conjecture for a group \(G=GL_2\) as well as simple groups of rank 1. Let us give an idea of the proof.
\begin{itemize}
\item Frenkel and Gaitsgory proved that \(\Gamma^{\text{Hecke}}\) is fully faithful and that \(\Gamma^{\text{Hecke}}\) is an equivalence on \(I^0\)-equivariant objects, where \(I^0\) is the radical of the Iwahori subgroup \(I\);
\item Earlier results of Raskin's ``W-algebras and Whittaker categories'' and Frenkel-Gaitsgory-Vilonen's ``Whittaker patterns in the geometry of moduli spaces of bundles on curves'' showed that \(\Gamma^{\text{Hecke}}\) is an equivalence on Whittaker objects;
\item The new result of the current paper is that for \(G=PGL_2\), any \(G(K)\)-category \(C\) with a level \(\kappa\) is generated under \(G(K)\) by the \(I^0\)-equivariant objects and the Whittaker objects.
\end{itemize}
Note that the first two points work for any reductive group \(G\). Then by observing that the essential image of \(\Gamma^{\text{Hecke}}\) is a \(G(K)\)-category, one can prove the result. With some more work, this can also be used to prove t-exactness.
Later results of \textit{D. Yang} and \textit{S. Raskin} prove the conjecture for a general reductive group \(G\) [``Affine Beilinson-Bernstein localization at the critical level'', Preprint, \url{arXiv:2203.13885}], using categorical Moy-Prasad theory developed by \textit{D. Yang} [``Categorical Moy-Prasad theory Yang'', Preprint, \url{arXiv:2104.12917}] as a main new ingredient.
Reviewer: Philsang Yoo (New Haven)Elements in pointed invariant cones in Lie algebras and corresponding affine pairshttps://zbmath.org/1496.220112022-11-17T18:59:28.764376Z"Neeb, Karl-Hermann"https://zbmath.org/authors/?q=ai:neeb.karl-hermann"Oeh, Daniel"https://zbmath.org/authors/?q=ai:oeh.danielThis paper is concerned with the study of a finite dimensional Lie algebra \(\mathfrak{g}\) the set of all those elements \(x\) for which the closed convex hull of the adjoint orbit contains no affine lines. And the authors also study affine pairs \((x, h)\in \mathfrak{g}\times \mathfrak{g}\) for a pointed invariant cone \(W \subseteq \mathfrak{g}\). These pairs are characterized by the relations \(x \in W\) and \([h, x] = x\).
The authors say: ``Convexity properties of adjoint orbits \(\mathcal{O}_{x} = \)Inn\((\mathfrak{g})x\) in a finite dimensional real Lie algebra, where Inn\((\mathfrak{g}) = \langle e^{ad\mathfrak{g}} \rangle\) is the group of inner automorphisms, play a role in many contexts.
The interest in affine pairs stems from their relevance in Algebraic Quantum Field Theory (AQFT), where they arise from unitary representations \((U,\mathcal{H})\) of a corresponding Lie group \(G\) and their positive cones \(W = C_{U}\).''
Let \(\mathfrak{l}\) be a Lie algebra, \(V\) an \(\mathfrak{l}\)-module, \(\mathfrak{z}\) a vector space, and \(\beta : V \times V \to \mathfrak{z}\) an \(\mathfrak{l}\)-invariant skew-symmetric bilinear map. Then \(\mathfrak{z} \times V \times \mathfrak{l}\) is a Lie algebra with respect to the bracket:
\([(z, v, x), (z^{'}, v^{'}, x^{'})] = (\beta(v, v^{'}), x.v^{'}-x^{'}.v, [x, x^{'}])\).
We write \(\mathfrak{g}(\mathfrak{l}, V, \mathfrak{z}, \beta)\) for the so-obtained Lie algebra.
The following definitions are quoted from this paper:
\(\cdot\) For \(x \in \mathfrak{g}\), co(\(x\)) := \(\overline{\mathrm{conv}(\mathcal{O}_{x})}\) for the closed convex hull of \(\mathcal{O}_{x}\),
\(C_{x} := \)cone(\(\mathcal{O}_{x}) = \overline{\mathbb{R}_{+}\mathrm{conv}(\mathcal{O}_{x})}\) for the closed convex cone generated by \(x\),
\(\mathfrak{g}_{co} := \{x \in \mathfrak{g} : \)co\((x)\) pointed\} \(\supseteq \mathfrak{g}_{c} := \{x \in\mathfrak{g} : C_{x}\) pointed\}.
\(\cdot U : G \to \)U\((\mathcal{H})\) is a unitary representation and \(\partial U(x)\) denotes the infinitesimal generator of the unitary one-parameter group \((U(\)exp\( tx))_{t\in \mathbb{R}}\), \(C_{U} := \{x \in \mathfrak{g}: -i\partial U(x) \ge 0\}\) for the positive cone of \(U\).
\(\cdot\) For \(x = x_{\mathfrak{z}} + x_{V} + x_{\mathfrak{l}} \in \mathfrak{g}\), we consider the \(\mathfrak{z}\)-valued Hamiltonian function \(H^{\mathfrak{z}}_{x}: V \to \mathfrak{z}\),
\(H^{\mathfrak{z}}_{x}(v) := P_{\mathfrak{z}}(e^{ad v}x) = x_{\mathfrak{z}} + [v, x_{V}] + \frac{1}{2}[v, [v, x_{\mathfrak{l}}]]\) \((P_{\mathfrak{z}}:\mathfrak{g} \to \mathfrak{z}\) projection),
co\(_{\mathfrak{z}}(x) := \overline{\mathrm{conv}(H^{\mathfrak{z}}_{x} (V))}\).
\(\cdot\) A finite dimensional real Lie algebra \(\mathfrak{g}\) is called \(admissible\) if it contains an Inn(\(\mathfrak{g}\))-invariant pointed generating closed convex subset \(C\).
\(\cdot\) For \(\mathfrak{g} = \mathfrak{g}(\mathfrak{l}, V , \mathfrak{z}, \beta), D_{\mathrm{can}}(z, v, x) := (z,\frac{1}{2}v, 0)\) for the canonical derivation.
\(\cdot\) \(D \in \)der\((\mathfrak{g})\) is called an Euler derivation if \(D\) is diagonalizable with Spec(\(D) \subseteq \{-1, 0, 1\}\).
\bigskip
The central result in Sect. 3 is the Characterization Theorem 3.20:
Let \(\mathfrak{g} = \mathfrak{g}(\mathfrak{l}, V, \mathfrak{z}, \beta \)) be admissible such that the representation of \(\mathfrak{l}\) on \(V\) is faithful.
Then, the convexity properties of the adjoint orbit \(\mathcal{O}_{x}\) are characterized as the properties of the Hamiltonian function \(H^{\mathfrak{z}}_{x}\), i.e., (a) co(\(x\)) is pointed if and only if co\(_{\mathfrak{z}}(x)\) is pointed, (b) \(C_{x}\) is pointed if and only if co\(_{\mathfrak{z}}(x)\) is pointed and, if \(x_{\mathfrak{l}}\) is nilpotent, then co\(_{\mathfrak{z}}(x)\) is contained in a pointed cone.
The first main result on affine pairs is the Existence Theorem 4.7:
Let \(\mathfrak{g} = \mathfrak{g}(\mathfrak{l}, V, \mathfrak{z}, \beta)\) be an admissible Lie algebra and \(x = x_{\mathfrak{z}} + x_{\mathfrak{l}} \in \mathfrak{z} + \mathfrak{l}\) be an ad-nilpotent element for which
co\((x)\) is pointed. Then there exists a derivation \(D \in D_{\mathrm{can}} + ad\mathfrak{g}\) with \(Dx = x\) and Spec\((D) \subseteq \{0,\pm \frac{1}{2} ,\pm1\}\).
Any invariant cone \(W\) generated by \(W_{\mathfrak{l}} := W \cap \mathfrak{l}\) and a central cone \(W_{\mathfrak{z}} \subseteq \mathfrak{z}\) satisfies \(e^{\mathbb{R}D}W = W\).
The second main result characterizes the existence of Euler derivations with this property, i.e., where we even have Spec\((D) \subseteq \{0,\pm1\}\) (Theorem 4.17).
Reviewer: Takao Imai (Chiba)Lie groups of \(C^k\)-maps on non-compact manifolds and the fundamental theorem for Lie group-valued mappingshttps://zbmath.org/1496.220122022-11-17T18:59:28.764376Z"Alzaareer, Hamza"https://zbmath.org/authors/?q=ai:alzaareer.hamzaThe article studies the existence of Lie group structures on groups of the form \(C^k(M,K), k \in \mathbb{N}\), where \(M\) is a non-compact smooth manifold (possibly with boundary) and \(K\) is a (possibly infinite-dimensional) Lie group. For \(k=\infty\) these groups are known in the literature as current groups. Here Lie group means locally convex Lie group (a la [\textit{K.-H. Neeb}, Jpn. J. Math. (3) 1, No. 2, 291--468 (2006; Zbl 1161.22012)]), where differentiability is understood in the Bastiani calculus. As a tool, a version of the fundamental theorem for Lie algebra-valued functions is provided. Under some technical conditions (involving the regularity of the target Lie group), the author establishes the existence of Lie group structures on \(C^k(M,K)\). In particular, it is shown that \(C^k (\mathbb{R},K)\) admits a Lie group structure under some conditions on \(K\). These results were a stepping stone for the generalised versions on Lie group structures constructed later in [\textit{H. Glöckner} and \textit{A. Schmeding}, Ann. Global Anal. Geom. 61, No. 2, 359--398 (2022; Zbl 1484.58005)].
Reviewer: Alexander Schmeding (Bodø)Riemannian submersions of \(\mathrm{SO}_0(2,1)\)https://zbmath.org/1496.220132022-11-17T18:59:28.764376Z"Byun, Taechang"https://zbmath.org/authors/?q=ai:byun.taechang\textit{U. Pinkall} [Invent. Math. 81, 379--386 (1985; Zbl 0585.53051)] proved that, for the Hopf bundle \(S^1\to S^3\to S^2\), the parallel displacement along a simple closed curve in the base space depends only on the area surrounded by the curve. The paper under review studies similar question for the Riemannian submersions \(G \to N\backslash G\), \(G \to A\backslash G\), \(G \to K\backslash G\), and \(G \to NA\backslash G\), where \(G=NAK\) is the Iwasawa decomposition of the Lie group \(G=\mathrm{SO}^0(2,1)\).
Reviewer: Anton Galaev (Hradec Králové)\(G\)-connectedness in topological groups with operationshttps://zbmath.org/1496.400202022-11-17T18:59:28.764376Z"Mucuk, Osman"https://zbmath.org/authors/?q=ai:mucuk.osman"Çakallı, Hüseyin"https://zbmath.org/authors/?q=ai:cakalli.huseyinSummary: It is a well-known fact that for a Hausdorff topological group \(X\), the limits of convergent sequences in \(X\) define a function denoted by \(\lim\) from the set of all convergent sequences in \(X\) to \(X\). This notion has been modified by \textit{J. Connor} and \textit{K. G. Grosse-Erdmann} [Rocky Mt. J. Math. 33, No. 1, 93--121 (2003; Zbl 1040.26001)] for real functions by replacing \(\lim\) with an arbitrary linear functional \(G\) defined on a linear subspace of the vector space of all real sequences. Recently, some authors have extended the concept to the topological group setting and introduced the concepts of \(G\)-continuity, \(G\)-compactness and \(G\)-connectedness. In this paper, we present some results about \(G\)-hulls, \(G\)-connectedness and \(G\)-fundamental systems of \(G\)-open neighbourhoods for a wide class of topological algebraic structures called groups with operations, which include topological groups, topological rings without identity, R-modules, Lie algebras, Jordan algebras, and many others.Samples of homogeneous functionshttps://zbmath.org/1496.410022022-11-17T18:59:28.764376Z"Dăianu, Dan M."https://zbmath.org/authors/?q=ai:daianu.dan-mSummary: We present an algorithm for extracting samples of homogeneity from functions between sets endowed with actions. In this way, we extend Taylor's formula with the remainder of Peano type in a very wide framework. We illustrate the versatility of this procedure by giving approximations with polynomials of homogeneous functions for some non-differentiable functions.The circular maximal operator on Heisenberg radial functionshttps://zbmath.org/1496.420222022-11-17T18:59:28.764376Z"Beltran, David"https://zbmath.org/authors/?q=ai:beltran.david"Guo, Shaoming"https://zbmath.org/authors/?q=ai:guo.shaoming"Hickman, Jonathan"https://zbmath.org/authors/?q=ai:hickman.jonathan"Seeger, Andreas"https://zbmath.org/authors/?q=ai:seeger.andreasDenote by \(\mathbb{H}^n\) the Heisenberg group, the set \(\mathbb{R}\times\mathbb{R}^{2n}\) equipped with the following non-commutative group operation: for all \((u,x), (v,y)\in\mathbb{H}^n\),
\[
(u,x)\cdot (v,y):=(u+v+x^TBy,x+y).
\]
Here, \(B:=\begin{pmatrix}0&-bI_n\\
bI_n&0\end{pmatrix}\) for some \(b\neq 0\) (one typically choose \(b=1/2\)). For \(\mu_1\equiv\mu\), the normalized surface measure on \(\{0\}\times\mathbb{S}^{2n-1}\), let \(\mu_t\) denote its dilation supported on \(t\mathbb{S}^{2n-1}\). For a function \(f:\mathbb{H}^n\rightarrow\mathbb{C}\), one may formally define its spherical means as \[ f\ast\mu_t(u,x):=\int_{\mathbb{S}^{2n-1}}\!f(u-tx^TBy,x-ty)\,d\mu(y) \] and its spherical maximal function as
\[
Mf(u,x):=\sup_{t>0}|f\ast\mu_t(u,x)|.
\]
In this paper, the authors complement known \(L^p\)-boundedness results for \(M\) on \(\mathbb{H}^n\), \(n\geq{2}\), by initiating the study of the case \(n=1\), where currently nothing is known for any \(p<\infty\). For \(2<p\leq\infty\), they show the existence of a constant \(C_p\), depending only on \(p\), such that
\[
\|Mf\|_{L^p(\mathbb{H}^1)}\leq C_p\|f\|_{L^p(\mathbb{H}^1)}
\]
for all \(\mathbb{H}\)-radial functions \(f\) on \(\mathbb{H}^1\). A function \(f:\mathbb{H}^1\rightarrow\mathbb{C}\) is said to be \(\mathbb{H}\)-radial if \(f(u,Rx)=f(u,x)\) for all \((u,x)\in\mathbb{H}^1\) and all \(R\) belonging to the special orthogonal group, \(SO(2)\). Equivalently, \(f\) is \(\mathbb{H}\)-radial if and only if there exists some function \(f_0:\mathbb{R}\times[0,\infty)\rightarrow\mathbb{C}\) such that \(f(u,x)=f_0(u,|x|)\) for all \((u,x)\in\mathbb{H}^1\).
The authors accomplish this by reducing the problem to studying the boundedness of a maximal function given by \(\sup_{t>0} |A_tf|\), where \(\{A_t\}\) are non-convolution averaging operators on \(\mathbb{R}^2\). While the reduction is not difficult, the associated curve distribution has vanishing rotational and cinematic curvatures, precluding the straightforward application of the standard techniques used to study the Euclidean spherical maximal function. A significant portion of this paper is spent overcoming these challenges, along the way performing an \(L^2\) analysis of two-parameter oscillatory integrals with two-sided fold singularities.
The appendices contain, among other things, a discussion of the use of repeated integration by parts often seen when studying oscillatory integrals.
Reviewer: Ryan Gibara (Cincinnati)An uncertainty principle for spectral projections on rank one symmetric spaces of noncompact typehttps://zbmath.org/1496.430052022-11-17T18:59:28.764376Z"Ganguly, Pritam"https://zbmath.org/authors/?q=ai:ganguly.pritam"Thangavelu, Sundaram"https://zbmath.org/authors/?q=ai:thangavelu.sundaramThe authors present a weaker version of Chernoff's theorem for Bessel and Jacobi operators. This result is used to prove a refined version of Ingham's theorem for the Helgason Fourier transform on rank one Riemannian symmetric spaces of noncompact type. The authors also prove an Ingham type uncertainty principle for the generalized spectral projections associated to the Laplace-Beltrami operator. Similar Ingham type results for the generalized spectral projections associated to Dunkl Laplacian are also discussed.
Reviewer: Ashish Bansal (Delhi)The Haagerup property for twisted groupoid dynamical systemshttps://zbmath.org/1496.460512022-11-17T18:59:28.764376Z"Kwaśniewski, Bartosz K."https://zbmath.org/authors/?q=ai:kwasniewski.bartosz-kosma"Li, Kang"https://zbmath.org/authors/?q=ai:li.kang"Skalski, Adam"https://zbmath.org/authors/?q=ai:skalski.adam-gSummary: We introduce the Haagerup property for twisted groupoid \(C^\ast\)-dynamical systems in terms of naturally defined positive-definite operator-valued multipliers. By developing a version of `the Haagerup trick' we prove that this property is equivalent to the Haagerup property of the reduced crossed product \(C^\ast\)-algebra with respect to the canonical conditional expectation \(E\). This extends a theorem of Dong and Ruan [\textit{Z.~Dong} and \textit{Z.-J. Ruan}, Integral Equations Oper. Theory 73, No.~3, 431--454 (2012; Zbl 1263.46043)]
for discrete group actions, and implies that a given Cartan inclusion of separable \(C^\ast\)-algebras has the Haagerup property if and only if the associated Weyl groupoid has the Haagerup property in the sense of Tu [\textit{J.-L. Tu}, \(K\)-Theory 17, No. 3, 215--264 (1999; Zbl 0939.19001)]. We use the latter statement to prove that every separable \(C^\ast\)-algebra which has the Haagerup property with respect to some Cartan subalgebra satisfies the Universal Coefficient Theorem. This generalises a recent result of Barlak and Li [\textit{S.~Barlak} and \textit{X.~Li}, Adv. Math. 316, 748--769 (2017; Zbl 1382.46048)] on the UCT for nuclear Cartan pairs.Gaffney-Friedrichs inequality for differential forms on Heisenberg groupshttps://zbmath.org/1496.490222022-11-17T18:59:28.764376Z"Franchi, Bruno"https://zbmath.org/authors/?q=ai:franchi.bruno"Montefalcone, Francescopaolo"https://zbmath.org/authors/?q=ai:montefalcone.francescopaolo"Serra, Elena"https://zbmath.org/authors/?q=ai:serra.elenaSummary: In this paper, we will prove several generalized versions, dependent on different boundary conditions, of the classical Gaffney-Friedrichs inequality for differential forms on Heisenberg groups. In the first part of the paper, we will consider horizontal differential forms and the horizontal differential. In the second part, we shall prove the counterpart of these results in the context of Rumin's complex.Left invariant special Kähler structureshttps://zbmath.org/1496.530402022-11-17T18:59:28.764376Z"Valencia, Fabricio"https://zbmath.org/authors/?q=ai:valencia.fabricioAuthors' abstract: We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.
Reviewer: Mohammed El Aïdi (Bogotá)Area of intrinsic graphs and coarea formula in Carnot groupshttps://zbmath.org/1496.530432022-11-17T18:59:28.764376Z"Julia, Antoine"https://zbmath.org/authors/?q=ai:julia.antoine"Nicolussi Golo, Sebastiano"https://zbmath.org/authors/?q=ai:nicolussi-golo.sebastiano"Vittone, Davide"https://zbmath.org/authors/?q=ai:vittone.davideThe authors consider submanifolds of sub-Riemannian Carnot groups with intrinsic \(C^1\) regularity \((C^1_H )\). The first main result in the present paper is an area formula for \(C^1_H\) intrinsic graphs; as an application, the authors deduce density properties for Hausdorff measures on rectifiable sets. The second main result is a coarea formula for slicing \(C^1_H\) submanifolds into level sets of a \(C^1_H\) function
Reviewer: Peibiao Zhao (Nanjing)An operator related to the sub-Laplacian on the quaternionic Heisenberg grouphttps://zbmath.org/1496.530442022-11-17T18:59:28.764376Z"Wang, Haimeng"https://zbmath.org/authors/?q=ai:wang.haimeng"Wang, Bei"https://zbmath.org/authors/?q=ai:wang.beiThe authors in the present paper study an operator related to the sub-Laplacian on the nonisotropic quaternionic Heisenberg group and construct the fundamental solution for this operator. For the isotropic case, the authors derive the closed form of this solution. The techniques the authors used can be applied to the standard Heisenberg group. The authors also give the connection between this operator and the Heisenberg sub-Laplacian.
Reviewer: Peibiao Zhao (Nanjing)A note on the nonexistence of a complex threefold as a conjugate orbit of \(G_2\)https://zbmath.org/1496.530782022-11-17T18:59:28.764376Z"Guan, Daniel"https://zbmath.org/authors/?q=ai:guan.daniel-zhuang-dan"Wang, Zhonghua"https://zbmath.org/authors/?q=ai:wang.zhonghuaSummary: The existence or nonexistence of a complex structure on \(\mathrm{S}^6\) was a long standing unsolved problem. There is a well-known orbit \(O(\varLambda)\) in \(G_2\) which is diffeomorphic to \(\mathrm{S}^6\) and used by Gábor Etesi in an effort to find a complex structure. Etesi suggested to give a complex structure in \(\mathrm{S}^6\) through this orbit. In Daniel Guan's earlier paper, he proved that the orbit can not be a complex submanifold. Since there was not a clear description of the map from \(O(\varLambda)\) to \(\mathrm{S}^6\) in that paper, we give another clearer, simpler and explicit proof of that result in this paper.On symplectic transformationshttps://zbmath.org/1496.530902022-11-17T18:59:28.764376Z"Springer, T. A."https://zbmath.org/authors/?q=ai:springer.tonny-albert.1Summary: This is an English translation of the Ph.D. thesis `Over symplectische transformaties' that Tonny Albert Springer, `born in's-Gravenhage in 1926', submitted as thesis for -- as is stated on the original frontispiece - \textit{the degree of doctor in mathematics and physics at Leiden University on the authority of the rector magnificus Dr. J.H. Boeke, professor in the faculty of law, to be defended against the objections of the Faculty of Mathematics and Physics on Wednesday October 17 1951 at 4 p.m.}, with promotor Prof. dr. H. D. Kloosterman.Chains in partially ordered spaceshttps://zbmath.org/1496.540172022-11-17T18:59:28.764376Z"Lawson, Jimmie"https://zbmath.org/authors/?q=ai:lawson.jimmie-dAfter a careful introduction and survey of existing results, the author provides some elegant new results, some of which generalize results of \textit{L. W. Anderson} and \textit{L. E. Ward jun.} [Proc. Glasg. Math. Assoc. 5, 1--3 (1961; Zbl 0098.25801)]. For example, the author shows that in a locally compact, semiclosed (= \(T_1\)-ordered) partially ordered topological space \(X\), for any \(a, b\) in a closed connected chain \(C\) with \(a < b\), the interval \([a,b]\) is compact, connected, and has the order topology. If \(S\) is a locally compact connected semilattice with \(0\) and either (a) \(S\) is locally order dense, has a closed order, and is semitopological or (b) \(S\) has small semilattices and every \(x, y \in S\) can be connected with a subcontinuum of \(S\), then for each \(x \in S\), there exists a compact connected chain \(C\) with \(0 = \inf C \in C\) and \(x = \sup C \in C\).
Reviewer: Thomas Richmond (Bowling Green)Some pseudocompact-like properties in certain topological groupshttps://zbmath.org/1496.540202022-11-17T18:59:28.764376Z"Tomita, Artur Hideyuki"https://zbmath.org/authors/?q=ai:tomita.artur-hideyuki"Trianon-Fraga, Juliane"https://zbmath.org/authors/?q=ai:trianon-fraga.julianeLet \(\omega^{\ast}\) be the set of non-principal (free) ultrafilters on \(\omega\). Let \(X\) be a topological space, for each \(x\in X\), \(p\in\omega^{\ast}\) and a sequence \((x_n)_{n\in\omega}\) in \(X\), recall that \(x\) is a \textit{\(p\)-limit point} of \((x_n)_{n\in\omega}\) if, for every neighborhood \(U\) of \(x\), \(\{n\in\omega: x_n\in U\}\in p\). It is easy to show that for a space \(X\), \(X\) is countably compact if and only if every sequence in \(X\) has a \(p\)-limit, for some \(p\in\omega^{\ast}\), \(X\) is countably pracompact if and only if there exists a dense subset \(D\) of \(X\) such that every sequence in \(D\) has a \(p\)-limit in \(X\), for some \(p\in\omega^{\ast}\), and \(X\) is pseudocompact if and only if for every countable family \(\{U_n:n\in\omega\}\) of nonempty open subsets of \(X\), there exist \(x\in X\) and \(p\in\omega^{\ast}\) such that, for each neighborhood \(V\) of \(x\), \(\{n\in\omega: V\cap U_n\neq\emptyset\}\in p\). A topological space \(X\) is called \textit{selectively pseudocompact}, defined in [\textit{S. García-Ferreira} and \textit{Y. F. Ortiz-Castillo}, Commentat. Math. Univ. Carol. 55, No. 1, 101--109 (2014; Zbl 1313.54054)] under the name \textit{strong pseudompactness}, if for each sequence \((U_n)_{n\in\omega}\) of nonempty open subsets of \(X\) there are a sequence \((x_n)_{n\in\omega}\), \(x\in X\) and \(p\in\omega^{\ast}\) such that \(x\) is a \(p\)-limit of \((x_n)_{n\in\omega}\) and, for each \(n\in\omega\), \(x_n\in U_n\). These notions are related as follows:
\[
\text{countable compactness} \Rightarrow \text{countable pracompactness}
\]
\[ \Rightarrow \text{selective pseudocompactness} \Rightarrow \text{pseudocompactness.}
\]
\textit{S. Garcia-Ferreira} and \textit{A. H. Tomita} [Topology Appl. 192, 138--144 (2015; Zbl 1330.54045)] gave examples of a selectively pseudocompact group which is not countably compact, and of a pseudocompact group which is not selectively pseudocompact. In this paper, the authors prove that there exists a topological group which is selectively pseudocompact but is not countably pracompact. They also prove that assuming the existence of a single selective ultrafilter, there exists a topological group which is not countably pracompact and has all powers selectively pseudocompact.
The question whether there exists a countably compact group without non-trivial convergent sequences in ZFC has been left open, however, \textit{M. Hrušák} et al. [Trans. Am. Math. Soc. 374, No. 2, 1277--1296 (2021; Zbl 1482.22002)] finally proved that in ZFC, there exists a Hausdorff countably compact topological Boolean group (of size \(\mathfrak c\)) without non-trivial convergent sequences. In this paper, the authors construct, in ZFC, a Hausdorff countably compact topological Boolean group of size \(2^{\mathfrak c}\) without non-trivial convergent sequences answering a question posed in [\textit{M. K. Bellini} et al., Topology Appl. 296, Article ID 107684, 14 p. (2021; Zbl 1481.54029)].
Reviewer: Kohzo Yamada (Shizuoka)Group topologies making every continuous homomorphic image to a compact group connectedhttps://zbmath.org/1496.540212022-11-17T18:59:28.764376Z"Yañez, Víctor Hugo"https://zbmath.org/authors/?q=ai:yanez.victor-hugoIn this paper, the author proves that if an abelian group \(G\) can be equipped with a group topology making all of its continuous homomorphic images to a compact group connected, then it admits a MinAP (minimally almost periodic) group topology, hence for every positive natural number \(m\) the subgroup \(mG\) of \(G\) is either the trivial group or has infinite cardinality by a result of \textit{D. Dikranjan} and \textit{D. Shakhmatov} [``Final solution of Protasov-Comfort's problem on minimally almost periodic group topologies'', Preprint, \url{arXiv:1410.3313}].
Reviewer: Fucai Lin (Zhangzhou)Normalizers of chains of discrete \(p\)-toral subgroups in compact Lie groupshttps://zbmath.org/1496.550152022-11-17T18:59:28.764376Z"Belmont, Eva"https://zbmath.org/authors/?q=ai:belmont.eva"Castellana, Natàlia"https://zbmath.org/authors/?q=ai:castellana.natalia"Grbić, Jelena"https://zbmath.org/authors/?q=ai:grbic.jelena"Lesh, Kathryn"https://zbmath.org/authors/?q=ai:lesh.kathryn"Strumila, Michelle"https://zbmath.org/authors/?q=ai:strumila.michelleLet \(G\) be a compact Lie group and \(BG\) denote its classifying space. Decompositions of \(BG^{\wedge}_{p}\) have been studied by many authors. This work finds a new decomposition, namely the \textit{normalizer} decomposition.
The setup is as follows. Let \(p\) be a prime number. A group is called \(p\)-toral of rank \(r\) if it is an extension of a torus of rank \(r\) by a finite \(p\)-group. A group is called a discrete \(p\)-torus of rank \(r\) if it isomorphic to a product \((\mathbb{Z}/p^{\infty})^{r}\), a discrete \(p\)-toral group of rank \(r\) is an extension of a discrete torus by a finite \(p\)-group. A discrete \(p\)-toral subgroup \( P\subseteq G\) is snugly embedded if \(P\) is a maximal discrete \(p\)-toral subgroup in its closure \(\overline{P}\). Let \(\mathbf{P}\subseteq G\) be a \(p\)-toral subgroup and \(P\subseteq\mathbf{P}\) be a snugly embedded discrete \(p\)-toral subgroup with \(\overline{P}=\mathbf{P}\), then \(P\) is called a discretization of \(\mathbf{P}\). Let \(\mathcal{F}_{\mathcal{S}}(G)\) be a saturated fusion system in \(G\), where \(\mathcal{S}\) is a maximal discrete \(p\)-toral subgroup of \(G\). A subgroup \(P\subseteq \mathcal{S}\) is called \(\mathcal{F}\)-centric if given a conjugate \(Q\) of \(P\) in \( \mathcal{S}\), then \(C_{\mathcal{S}}(Q)=Z(P)\); a subgroup \(P\subseteq\mathcal{S}\) is \(\mathcal{F}\)-radical if \(Out_{G}(P)=N_{G}(P)/(C_{G}(P)P)\) has no nontrivial normal subgroups. For \(p\)-toral subgroups the corresponding definitions are as follows, a \(p\)-toral subgroup \(\mathbf{P}\subseteq G\) is \(p\)-centric if \(Z(P)\) is a maximal \(p\)-toral subgroup of \(C_{G}(\mathbf{P})\), a \(p\)-toral subgroup \(\mathbf{P}\subseteq G\) is \(p\)-stubborn in \(G\) if \(N_{G}(\mathbf{P})/\mathbf{P} \) is finite and has no nontrivial normal \(p\)-subgroups.
We may now state one of the two main results.
\textbf{Theorem 4.3.} Let \(\mathbf{S}\) be a maximal \(p\)-toral subgroup of a compact Lie group \(G\), let \(S\subset \mathbf{S}\) be a \(p\) discretization. The closure \(P\mapsto \overline{P}\) defines an injective map
\begin{align*}
\{P_{0}\subseteq \cdots \subseteq P_{k}\mid \text{all } P_{i} \text{ are }&\mathcal{F}\text{-centric and }\mathcal{F}\text{-radical}\}/G\\
&\downarrow\\
\{ \mathbf{P}_{0}\subseteq\cdots \subseteq\mathbf{P}_{k}\mid \text{all } \mathbf{P}_{i}\text{ are } p \text{-toral, } & p \text{-centric, and }p \text{-stubborn } \}/G.
\end{align*}
Moreover, the map is a bijection if \(\pi_{0}(G)\) is a \(p\)-group.
The normalizing decomposition is the following
\textbf{Theorem 5.1} Let \( \mathbf{P}_{0}\subseteq \cdots \subseteq\mathbf{P}_{k}\) be a chain of \(p\)-toral subgroups of a compact Lie group \(G\). Let \(P_{0}\subseteq \cdots \subseteq P_{k}\) be a chain of \(p\)-toral subgroups such that \(P_{i}\) is a discretization of \(\mathbf{P}_{i}\). Then, the map
\[
N_{G}(P_{0}\subseteq \cdots \subseteq P_{k})\to N_{G}( \mathbf{P}_{0}\subseteq \cdots \subseteq \mathbf{P}_{k} )
\]
induces a mod-\(p\) equivalence of classifying spaces.
Reviewer: Daniel Juan Pineda (Michoacán)Lie symmetry analysis, conservation laws and separation variable type solutions of the time-fractional porous medium equationhttps://zbmath.org/1496.740512022-11-17T18:59:28.764376Z"Yang, Ying"https://zbmath.org/authors/?q=ai:yang.ying"Wang, Lizhen"https://zbmath.org/authors/?q=ai:wang.lizhen(no abstract)Simultaneous vs. non-simultaneous measurements in quantum and classical mechanicshttps://zbmath.org/1496.810262022-11-17T18:59:28.764376Z"Chudak, N. O."https://zbmath.org/authors/?q=ai:chudak.n-o"Potiienko, O. S."https://zbmath.org/authors/?q=ai:potiienko.o-s"Sharph, I. V."https://zbmath.org/authors/?q=ai:sharph.i-vSummary: In a traditional implementation of the relativity principles, different observers consider \textit{the same} events and relate their space-time coordinates through the Lorentz transformation. In this paper we consider the problems where it is impossible to use the non-simultaneous measurements in any inertial reference frame. In such case different observers have to use \textit{different} sets of events, the space-time coordinates of which are impossible to relate through Lorentz transform. Therefore we suggest another way of implementing the relativity principles, and discuss some of its consequences and prospects.Optical effects of domain wallshttps://zbmath.org/1496.810682022-11-17T18:59:28.764376Z"Khoze, Valentin V."https://zbmath.org/authors/?q=ai:khoze.valentin-v"Milne, Daniel L."https://zbmath.org/authors/?q=ai:milne.daniel-lSummary: Domain walls arise in theories where there is spontaneous symmetry breaking of a discrete symmetry such as \(\mathbb{Z}_N\) and are a feature of many BSM models. In this work we consider the possibility of detecting domain walls through their optical effects and specify three different methods of coupling domain walls to the photon. We consider the effects of these couplings in the context of gravitational wave detectors, such as LIGO, and examine the sensitivity of these experiments to domain wall effects. In cases where gravitational wave detectors are not sensitive we examine our results in the context of axion experiments and show how effects of passing domain walls can be detected at interferometers searching for an axion signal.The noncommutative space of light-like worldlineshttps://zbmath.org/1496.810692022-11-17T18:59:28.764376Z"Ballesteros, Angel"https://zbmath.org/authors/?q=ai:ballesteros.angel"Gutierrez-Sagredo, Ivan"https://zbmath.org/authors/?q=ai:gutierrez-sagredo.ivan"Herranz, Francisco J."https://zbmath.org/authors/?q=ai:herranz.francisco-joseSummary: The noncommutative space of light-like worldlines that is covariant under the light-like (or null-plane) \(\kappa\)-deformation of the (3+1) Poincaré group is fully constructed as the quantization of the corresponding Poisson homogeneous space of null geodesics. This new noncommutative space of geodesics is five-dimensional, and turns out to be defined as a quadratic algebra that can be mapped to a non-central extension of the direct sum of two Heisenberg-Weyl algebras whose noncommutative parameter is just the Planck scale parameter \(\kappa^{-1}\). Moreover, it is shown that the usual time-like \(\kappa\)-deformation of the Poincaré group does not allow the construction of the Poisson homogeneous space of light-like worldlines. Therefore, the most natural choice in order to model the propagation of massless particles on a quantum Minkowski spacetime seems to be provided by the light-like \(\kappa\)-deformation.Constructing the bulk at the critical point of three-dimensional large \(N\) vector theorieshttps://zbmath.org/1496.810862022-11-17T18:59:28.764376Z"Johnson, Celeste"https://zbmath.org/authors/?q=ai:johnson.celeste"Mulokwe, Mbavhalelo"https://zbmath.org/authors/?q=ai:mulokwe.mbavhalelo"Rodrigues, João P."https://zbmath.org/authors/?q=ai:rodrigues.joao-pSummary: In the context of the \(AdS_4/CFT_3\) correspondence between higher spin fields and vector theories, we use the constructive bilocal fields based approach to this correspondence, to demonstrate, at the \textit{IR} critical point of the interacting vector theory and directly in the bulk, the removal of the \(\Delta = 1\) (\(s = 0\)) state from the higher spins field spectrum, and to exhibit simple Klein-Gordon higher spin Hamiltonians. The bulk variables and higher spin fields are obtained in a simple manner from boundary bilocals, by the change of variables previously derived for the \textit{UV} critical point (in momentum space), together with a field redefinition.Effective electromagnetic actions for Lorentz violating theories exhibiting the axial anomalyhttps://zbmath.org/1496.810892022-11-17T18:59:28.764376Z"Gómez, Andrés"https://zbmath.org/authors/?q=ai:gomez.andres"Martín-Ruiz, A."https://zbmath.org/authors/?q=ai:martin-ruiz.alberto"Urrutia, Luis F."https://zbmath.org/authors/?q=ai:urrutia.luis-fSummary: The CPT odd contribution to the effective electromagnetic action deriving from the vacuum polarization tensor in a large class of fermionic systems exhibiting Lorentz invariance violation (LIV) is calculated using thermal field theory methods, focusing upon corrections depending on the chemical potential. The systems considered exhibit the axial anomaly and their effective actions are described by axion electrodynamics whereby all the LIV parameters enter in the coupling \(\operatorname{\Theta}(x)\) to the unmodified Pontryagin density. A preliminary application to type-I tilted Weyl semimetals is briefly presented.On the off-shell superfield Lagrangian formulation of \(4D\), \(\mathcal{N} = 1\) supersymmetric infinite spin theoryhttps://zbmath.org/1496.810902022-11-17T18:59:28.764376Z"Buchbinder, I. L."https://zbmath.org/authors/?q=ai:buchbinder.ioseph-l"Fedoruk, S. A."https://zbmath.org/authors/?q=ai:fedoruk.s-a"Isaev, A. P."https://zbmath.org/authors/?q=ai:isaev.aleksei-petrovich"Krykhtin, V. A."https://zbmath.org/authors/?q=ai:krykhtin.v-aSummary: We develop a complete off-shell Lagrangian description of the free \(4D\), \(\mathcal{N} = 1\) supersymmetric theory of infinite spin. Bosonic and fermionic fields are formulated in terms of spin-tensor fields with dotted and undotted indices. The corresponding Lagrangians for bosonic and fermionic infinite spin fields entering into the on-shell supersymmetric model are derived within the BRST method. Lagrangian for this supersymmetric model is written in terms of the complex infinite spin bosonic field and infinite spin fermionic Weyl field subject to supersymmetry transformations. The fields involved into the on-shell supersymmetric Lagrangian can be considered as components of six infinite spin chiral and antichiral multiplets. These multiplets are extended to the corresponding infinite spin chiral and antichiral superfields so that two chiral and antichiral superfields contain among the components the basic fields of an infinite spin supermultiplet and extra four chiral and antichiral superfields containing only the auxiliary fields needed for the Lagrangian formulation. The superfield Lagrangian is constructed in terms of these six chiral and antichiral supefields, and we show that the component form of this superfield Lagrangian exactly coincides with the previously found component supersymmetric Lagrangian after eliminating the component fields added to construct (anti)chiral superfields.Super Hirota bilinear equations for the super modified BKP hierarchyhttps://zbmath.org/1496.811122022-11-17T18:59:28.764376Z"Chen, Huizhan"https://zbmath.org/authors/?q=ai:chen.huizhanSummary: In this paper, the super modified BKP (SmBKP) hierarchy is constructed from the perspective of the neutral free superfermions by using highest weight representations of the infinite-dimensional Lie superalgebra \(\mathfrak{b}_{\infty|\infty}(\mathfrak{g})\). Based upon this, the corresponding super Hirota bilinear identity of the SmBKP hierarchy is obtained by using the super Boson-Fermion correspondence of type B, and some specific examples of super Hirota bilinear equations are given. The super bilinear identity with respect to super wave and adjoint wave functions is also constructed. At last, we also give a class of solutions other than group orbit by the neutral free superfermions.Topological confinement in Skyrme holographyhttps://zbmath.org/1496.830262022-11-17T18:59:28.764376Z"Cartwright, Casey"https://zbmath.org/authors/?q=ai:cartwright.casey"Harms, Benjamin"https://zbmath.org/authors/?q=ai:harms.benjamin-c"Kaminski, Matthias"https://zbmath.org/authors/?q=ai:kaminski.matthias"Thomale, Ronny"https://zbmath.org/authors/?q=ai:thomale.ronny