Compact Subgroup Research Articles

Hecke eigenvalues in 𝑝-modular representations of unramified 𝑈(2,1)

Let G G be the unramified unitary group U ( 2 , 1 ) ( E / F ) U(2, 1)(E/F) defined over a non-archimedean local field F F of odd residue characteristic p p , and let K K be a maximal compact open subgroup of G G . For an irreducible smooth F ¯ p \overline {\mathbf {F}}_p -representation π \pi of G G , and a weight σ \sigma of K K contained in π \pi , we prove that π \pi admits eigenvectors for the spherical Hecke algebra H ( K , σ ) \mathcal {H}(K, \sigma ) . Our approach is close to that of Barthel–Livné [Duke Math. J. 75 (1994), pp. 261–292] for G L 2 GL_2 . For G L n ( n ≥ 3 ) GL_n (n\geq 3) , we give a sketch of an essential case where the analogue of a crucial ingredient in loc.cit fails.

Self-similarity of p-adic groups

We show that a compact open subgroup H of a simple algebraic p-adic group G is self-similar if and only if it is isotropic.

Lebesgue-Fourier algebras on coset spaces and approximate amenability

Let K be a compact subgroup of a locally compact group G. In this work, we initiate an investigation of the Lebesgue-Fourier algebra S 1 A(G/K) on the coset space G/K with pointwise multiplication; after some general results on the Banach algebra S 1 A(G/K), as our main result, we give some necessary conditions for that S 1 A(G/K) be approximately amenable in terms of algebraic and topological properties of G and K .

Black flowers and real forms of higher spin symmetries

Using Chern-Simons formulation, we investigate higher spin (HS) black holes in AdS3 with soft Heisenberg hair and establish linkage with the real forms of the underlying complexified gauge symmetries taken here as SL(N, ℂ)L × SL(N, ℂ)R. We study the various conserved currents characterizing the HS black flowers (HS-BF) and show that they can be formed of layers indexed by the elements of the centre of the gauge symmetry. This feature follows from requiring the holonomy of the asymptotic gauge connection around the thermal cycle to sit in the centre ℤN of the symmetry group. With regard to the compact subgroups of the real forms of the complexified gauge symmetry, we calculate the entropies of the HS-BF and verify that, unless we are considering trivial holonomies, there are no continuous paths joining the HS-BF to the core spin 2 black holes. As explicit illustrations, we give quantum field realisations of the soft Heisenberg hair in terms of bosonic and fermionic primary conformal fields and compute the HS-BF entropy as a function of the number of fermions occupying the ground state of the Heisenberg soft hair.

The space of locally elliptic subgroups of a locally compact group

Let G be a locally compact group. We denote by Sub ( G ) the space of closed subgroups of G equipped with the Chabauty topology. A closed subgroup H of G is called locally elliptic if every compact subset of H is contained in a compact subgroup. In this paper we establish that the subset Su b L E ( G ) of closed locally elliptic subgroups of G is Chabauty-closed if and only if the set com p n ( G ) consisting of the n-tuples of elements of G contained in a common compact subgroup, is closed in the Cartesian product G n , for every nonnegative integer n. Moreover, we prove that the subspace Su b L E ( G ) is Chabauty-closed in the case when the group G contains an open normal locally elliptic subgroup.

Some characterizations of ω-balanced topological groups with a q-point

In this paper, we study some characterizations of q-spaces, strict q-spaces and strong q-spaces under ω-balanced topological groups. First we characterize a topological group G to be ω-balanced and a q-space whenever for each open neighborhood O of the identity in G, there is a countably compact invariant subgroup H which is of countable character in G, such that H ⊆ O and the canonical quotient mapping p : G → G/H is quasi-perfect and the quotient group G/H is metrizable. Secondly, we characterize G to be ω-balanced and a strict q-space, replacing the condition of being a q-space with an appropriate condition which is designed for the so-called “strict q-spaces”. Finally, we characterize G to be ω-balanced and a strong q-space, illustrating an additional condition to be replaced to that of q-space, or strict q-space. As applications of our three main results, we found new characterizations of the ω-narrow topological groups.

Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I

Let G be a (non compact) connected, simply connected, locally compact, second countable Lie group, either abelian or unimodular of type I, and let ρ be an irreducible unitary representation of G. Then, we define the analytic torsion of G localised at the representation ρ. The idea of considering localised invariants is due to Brodzki, Niblo, Plymen and Wright [5], and was exploited in [31] to define a localised eta function. Next, let Γ be a discrete co compact subgroup of G. We use the localised analytic torsion to define the relative analytic torsion of the pair (G,Γ), and we prove that the last coincides with the Lott L2 analytic torsion of a covering space. We illustrate these constructions analysing in some details two examples: the abelian case, and the case G=H, the Heisenberg group.

Non-compact gauge groups, tensor fields and Yang-Mills-Einstein amplitudes

Scattering amplitudes in Yang-Mills-Einstein theories have been investigated mostly for compact gauge groups. While non-compact gauge groups are not physically viable in Yang-Mills theory, non-compact gaugings feature prominently in the supergravity literature, where any choice of perturbative vacuum spontaneously breaks the gauge group to a compact subgroup. In this paper, we formulate double-copy constructions for several five-dimensional N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supergravities with non-compact gauge groups. On one side of the double copy, we employ amplitudes from a super-Yang-Mills theory with a massive hypermultiplet. On the other, we use amplitudes from particular non-supersymmetric Yang-Mills-scalar theories with massive fermions, chosen to obey constraints coming from color/kinematics duality. Supergravities with massive self-dual tensors in five dimensions are also considered, showing that tensors are straightforwardly realized as double copies of gauge-theory fermions with suitable choices of signs in the corresponding solutions of the Dirac equation. We present several examples of these constructions, noting in particular the appearance of Heisenberg groups in the supergravity gauge symmetry and, in some cases, the possibility of exotic tensor-vector matter couplings.

On countably complete topological groups

In this paper, we give a characterization of countably complete topological groups and study when a countably complete subgroup of a topological group is C-embedded. We mainly show that (1) a topological group G is countably complete (the notion introduced by M. Tkachenko in 2012) iff G contains a closed r-pseudocompact subgroup H such that the quotient space G/H is completely metrizable and the canonical quotient mapping π:G→G/H satisfies that π−1(F) is r-pseudocompact in G for each r-pseudocompact set F in G/H; (2) every countably complete weakly Ψω-factorizable and ω-balanced subgroup H of a topological group G is C-embedded; (3) every countably complete subgroup H of a pointwise pseudocompact topological group G is C-embedded; (4) every uniformly strongly countably complete and weakly Ψω-factorizable subgroup H of a topological group G is C-embedded. Further, an ω-narrow locally compact subgroup H of a topological group G is C-embedded.

Maximally localized Gabor orthonormal bases on locally compact Abelian groups

A Gabor orthonormal basis, on a locally compact Abelian (LCA) group A, is an orthonormal basis of L2(A) that consists of time-frequency shifts of some template f∈L2(A). It is well known that, on Rd, the elements of such a basis cannot have a good time-frequency localization. The picture is drastically different on LCA groups containing a compact open subgroup, where one can easily construct examples of Gabor orthonormal bases with f maximally localized, in the sense that the ambiguity function of f (i.e., the correlation of f with its time-frequency shifts) has support of minimum measure, compatibly with the uncertainty principle. In this note we find all the Gabor orthonormal bases with this extremal property. To this end, we identify all the functions in L2(A) that are maximally localized in the time-frequency space in the above sense – an issue that is open even for finite Abelian groups. As a byproduct, on every LCA group containing a compact open subgroup we exhibit the complete family of optimizers for Lieb's uncertainty inequality, and we also show previously unknown optimizers on a general LCA group.

Spinorial description for Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT

We introduce a novel spinorial description for the higher-spin gauge theory induced by the IKKT matrix model on an FLRW spacetime with Lorentzian signature, called Lorentzian hs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathfrak{hs} $$\\end{document}-IKKT theory. The new description is based on Weyl spinors transforming under the space-like isometry subgroup SL(2, ℂ) of the structure group SO(2, 4) ≃ SU(2, 2). It allows us to exploit the full power of the spinor formalism in Lorentzian signature, in contrast to a previous formalism based on the compact subgroup SU(2)L × SU(2)R of SU(2, 2). Some cubic vertices of the Yang-Mills sector and the corresponding scattering amplitudes are computed. We observe that the n-point (for n ≥ 4) tree-level amplitudes are typically non-trivial on-shell, but exponentially suppressed in the late-time regime. While Lorentz invariance of the higher-spin amplitudes is not manifest, it is expected to be restored by higher-spin gauge invariance.

The Weyl law for congruence subgroups and arbitrary K∞-types

Let G be a reductive algebraic group over Q and Γ⊂G(Q) an arithmetic subgroup. Let K∞⊂G(R) be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of G(R) in L2(Γ﹨G(R)) of a fixed K∞-type σ. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type σ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about L-functions occurring in the constant terms of Eisenstein series. If G satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.

Paley Wiener Theorem on a Reductive Lie Group

Let G be a locally compact group, K a maximal compact subgroup of G and δ on arbitrary class of irreducible unitary representations of K. The spherical Grassmannian Gp,δ is an equivalence class of spherical functions of type δ−positive of height p. In this work, we give an extension of orbital integral with respect to δ, when G is reductive Lie group. Moreover, if the discret serie is not empty, we give an extension of Paley-Wiener theorem using a compact Cartan subgroup of G.

Simple approaches for evaluation of OTU quality based on dissimilarity arrays

An accurate and complete taxonomic description of the diversity present in an environmental sample is out of reach at this time. Instead, metabarcoding is used today and it is expected that OTUs represent a category relevant for biodiversity inventories on a molecular basis. However, artefacts in the production of OTUs can occur at different stages and may impact ecological conclusions. We propose to evaluate the quality of OTUs in a sample by characterising the deviation of each OTU’s dissimilarity array from that of an ideal OTU where all sequences are at distances smaller than the barcoding gap. We consider two deviations: the creation of composed OTUs, corresponding to the artificial merging of several OTUs and the creation of noisy OTUs that contain some sequences that are loosely associated with the core sequence of the OTUs and that do not form a compact subgroup. We propose a simple and automatic 2-step method that successively categorises the OTUs of a sample as composed or single and then identifies OTUs with noise amongst the single ones. The associated code is available at https://forgemia.inra.fr/alain.franc/otu_shape. We applied the method on 32 samples of diatoms from Arcachon Bay (France) that represent contrasted environmental conditions and we obtained good agreement with expert categorisation of OTUs. We suggest that single OTUs without noise can be used as such for further ecological studies. Composed OTUs should be post-treated with classical clustering or community detection tools. The quality of single OTUs with noise remains to be further tested via supplementary studies on a diversity of organisms.

Geodesics on adjoint orbits of SL(n, ℝ)

In this paper, we examine the geodesics on adjoint orbits of [Formula: see text] that are equipped with [Formula: see text]-invariant metrics, where [Formula: see text] is the maximal compact subgroup. Our primary technique involves translating this problem into a geometric problem in the tangent bundle of certain [Formula: see text]-flag manifolds and describing the geodesic equations with respect to the Sasaki metric on the tangent bundle. Additionally, we utilize tools from Lie Theory to obtain explicit descriptions of families of geodesics. We provide a detailed analysis of the case for [Formula: see text].

Vascular compactness of unruptured brain arteriovenous malformation predicts risk of hemorrhage after stereotactic radiosurgery

The aim of the study was to investigate whether morphology (i.e. compact/diffuse) of brain arteriovenous malformations (bAVMs) correlates with the incidence of hemorrhagic events in patients receiving Stereotactic Radiosurgery (SRS) for unruptured bAVMs. This retrospective study included 262 adult patients with unruptured bAVMs who underwent upfront SRS. Hemorrhagic events were defined as evidence of blood on CT or MRI. The morphology of bAVMs was evaluated using automated segmentation which calculated the proportion of vessel, brain tissue, and cerebrospinal fluid in bAVMs on T2-weighted MRI. Compactness index, defined as the ratio of vessel to brain tissue, categorized bAVMs into compact and diffuse types based on the optimal cutoff. Cox proportional hazard model was used to identify the independent factors for post-SRS hemorrhage. The median clinical follow-ups was 62.1 months. Post-SRS hemorrhage occurred in 13 (5.0%) patients and one of them had two bleeds, resulting in an annual bleeding rate of 0.8%. Multivariable analysis revealed bAVM morphology (compact versus diffuse), bAVM volume, and prescribed margin dose were significant predictors. The post-SRS hemorrhage rate increased with larger bAVM volume only among the diffuse nidi (1.7 versus 14.9 versus 30.6 hemorrhage per 1000 person-years in bAVM volume < 20 cm3 versus 20–40 cm3 versus > 40 cm3; p = 0.022). The significantly higher post-SRS hemorrhage rate of Spetzler-Martin grade IV–V compared with grade I–III bAVMs (20.0 versus 3.3 hemorrhages per 1000 person-years; p = 0.001) mainly originated from the diffuse bAVMs rather than the compact subgroup (30.9 versus 4.8 hemorrhages per 1000 person-years; p = 0.035). Compact and smaller bAVMs, with higher prescribed margin dose harbor lower risks of post-SRS hemorrhage. The post-SRS hemorrhage rate exceeded 2.2% annually within the diffuse and large (> 40 cm3) bAVMs and the diffuse Spetzler-Martin IV–V bAVMs. These findings may help guide patient selection of SRS for the unruptured bAVMs.

Growing trees from compact subgroups

We establish a new connection between local and large-scale structure in compactly generated totally disconnected locally compact (t.d.l.c.) groups G , finding a sufficient condition for G to have more than one end in terms of its compact subgroups. The condition actually results in an action of a quotient group G/N on a tree with faithful micro-supported action on the boundary, where N is compact, and is closely related to the Boolean algebra formed by the centralisers of the subgroups of G/N with open normaliser. As an application, we find a sufficient condition, given a one-ended t.d.l.c. group G , for all direct factors of open subgroups of G to be trivial or open.

Time optimal problems on Lie groups and applications to quantum control

&lt;abstract&gt;&lt;p&gt;In this paper we introduce a natural compactification of a left (right) invariant affine control system on a semi-simple Lie group $ G $ in which the control functions belong to the Lie algebra of a compact Lie subgroup $ K $ of $ G $ and we investigate conditions under which the time optimal solutions of this compactified system are "approximately" time optimal for the original system. The basic ideas go back to the papers of R.W. Brockett and his collaborators in their studies of time optimal transfer in quantum control (&lt;sup&gt;[&lt;xref ref-type="bibr" rid="b1"&gt;1&lt;/xref&gt;]&lt;/sup&gt;, &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b2"&gt;2&lt;/xref&gt;]&lt;/sup&gt;). We showed that every affine system can be decomposed into two natural systems that we call horizontal and vertical. The horizontal system admits a convex extension whose reachable sets are compact and hence posess time-optimal solutions. We then obtained an explicit formula for the time-optimal solutions of this convexified system defined by the symmetric Riemannian pair $ (G, K) $ under the assumption that the Lie algebra generated by the control vector fields is equal to the Lie algebra of $ K $.&lt;/p&gt; &lt;p&gt;In the second part of the paper we applied our results to the quantum systems known as Icing $ n $-chains (introduced in &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b2"&gt;2&lt;/xref&gt;]&lt;/sup&gt;). We showed that the two-spin chains conform to the theory in the first part of the paper but that the three-spin chains show new phenomena that take it outside of the above theory. In particular, we showed that the solutions for the symmetric three-spin chains studied by (&lt;sup&gt;[&lt;xref ref-type="bibr" rid="b3"&gt;3&lt;/xref&gt;]&lt;/sup&gt;, &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b4"&gt;4&lt;/xref&gt;]&lt;/sup&gt;) are solvable in terms of elliptic functions with the solutions completely different from the ones encountered in the two-spin chains.&lt;/p&gt;&lt;/abstract&gt;

Local parameters of supercuspidal representations

Abstract For a connected reductive group G over a nonarchimedean local field F of positive characteristic, Genestier-Lafforgue and Fargues-Scholze have attached a semisimple parameter ${\mathcal {L}}^{ss}(\pi )$ to each irreducible representation $\pi $ . Our first result shows that the Genestier-Lafforgue parameter of a tempered $\pi $ can be uniquely refined to a tempered L-parameter ${\mathcal {L}}(\pi )$ , thus giving the unique local Langlands correspondence which is compatible with the Genestier-Lafforgue construction. Our second result establishes ramification properties of ${\mathcal {L}}^{ss}(\pi )$ for unramified G and supercuspidal $\pi $ constructed by induction from an open compact (modulo center) subgroup. If ${\mathcal {L}}^{ss}(\pi )$ is pure in an appropriate sense, we show that ${\mathcal {L}}^{ss}(\pi )$ is ramified (unless G is a torus). If the inducing subgroup is sufficiently small in a precise sense, we show $\mathcal {L}^{ss}(\pi )$ is wildly ramified. The proofs are via global arguments, involving the construction of Poincaré series with strict control on ramification when the base curve is ${\mathbb {P}}^1$ and a simple application of Deligne’s Weil II.

Satake-Furstenberg compactifications and gradient map

Let $G$ be a real semisimple Lie group with finite center and let $\mathfrak g=\mathfrak k \oplus \mathfrak p$ be a Cartan decomposition of its Lie algebra. Let $K$ be a maximal compact subgroup of $G$ with Lie algebra $\mathfrak k$ and let $\tau$ be an irreducible representation of $G$ on a complex vector space $V$. Let $h$ be a Hermitian scalar product on $V$ such that $\tau(G)$ is compatible with respect to $\mathrm{U}(V,h)^{\mathbb C}$. We denote by $\mu_{\mathfrak p}:\mathbb P(V) \longrightarrow \mathfrak p$ the $G$-gradient map and by $\mathcal O$ the unique closed orbit of $G$ in $\mathbb P(V)$, which is a $K$-orbit, contained in the unique closed orbit of the Zariski closure of $\tau(G)$ in $\mathrm{U}(V,h)^{\mathbb C}$. We prove that up to equivalence the set of irreducible representations of parabolic subgroups of $G$ induced by $\tau$ are completely determined by the facial structure of the polar orbitope $\mathcal E=\mathrm{conv}(\mu_{\mathfrak p} (\mathcal O))$. Moreover, any parabolic subgroup of $G$ admits a unique closed orbit which is well-adapted to $\mathcal O$ and $\mu_{\mathfrak p}$ respectively. These results are new also in the complex reductive case. The connection between $\mathcal E$ and $\tau$ provides a geometrical description of the Satake compactifications without root data. In this context the properties of the Bourguignon-Li-Yau map are also investigated. Given a measure $\gamma$ on $\mathcal O$, we construct a map $\Psi_\gamma$ from the Satake compactification of $G/K$ associated to $\tau$ and $\mathcal E$. If $\gamma$ is a $K$-invariant measure then $\Psi_\gamma$ is an homeomorphism of the Satake compactification and $\mathcal E$. Finally, we prove that for a large class of measures the map $\Psi_\gamma$ is surjective.