Semisimple Group Research Articles

Non-Abelian symmetry-resolved entanglement entropy

We introduce a mathematical framework for symmetry-resolved entanglement entropy with a non-Abelian symmetry group. To obtain a reduced density matrix that is block-diagonal in the non-Abelian charges, we define subsystems operationally in terms of subalgebras of invariant observables. We derive exact formulas for the average and the variance of the typical entanglement entropy for the ensemble of random pure states with fixed non-Abelian charges. We focus on compact, semisimple Lie groups. We show that, compared to the Abelian case, new phenomena arise from the interplay of locality and non-Abelian symmetry, such as the asymmetry of the entanglement entropy under subsystem exchange, which we show in detail by computing the Page curve of a many-body system with SU(2)SU(2) symmetry.

Symmetries of Fano varieties

Abstract Prokhorov and Shramov proved that the BAB conjecture, which Birkar later proved, implies the uniform Jordan property for automorphism groups of complex Fano varieties of fixed dimension. This property in particular gives an upper bound on the size of finite semi-simple groups (i.e., those with no nontrivial normal abelian subgroups) acting faithfully on 𝑛-dimensional complex Fano varieties, and this bound only depends on 𝑛. We investigate the geometric consequences of an action by a certain semi-simple group: the symmetric group. We give an effective upper bound for the maximal symmetric group action on an 𝑛-dimensional Fano variety. For certain classes of varieties – toric varieties and Fano weighted complete intersections – we obtain optimal upper bounds. Finally, we draw a connection between large symmetric actions and boundedness of varieties, by showing that the maximally symmetric Fano fourfolds form a bounded family. Along the way, we also show analogues of some of our results for Calabi–Yau varieties and log terminal singularities.

A local-global principle for similarities over function fields of p-adic curves

Let p∈N be a prime with p≠2, and let K be a p-adic field. Let F be the function field of a curve over K. Let ΩF be the set of all divisorial discrete valuations of F. In this paper, we ask whether the Hasse principle holds for semisimple adjoint linear algebraic groups over F. We give a positive answer to this question for a class of adjoint classical groups.

Abelian ideals and the variety of Lagrangian subalgebras

For a semisimple algebraic group G of adjoint type with Lie algebra g over the complex numbers, we establish a bijection between the set of closed orbits of the group G⋉g⁎ acting on the variety of Lagrangian subalgebras of g⋉g⁎ and the set of abelian ideals of a fixed Borel subalgebra of g. In particular, the number of such orbits equals 2rkg by Peterson's theorem on abelian ideals.

Harmonic Riemannian submersions between Riemannian symmetric spaces of noncompact type

AbstractWe construct harmonic Riemannian submersions that are retractions from symmetric spaces of noncompact type onto their rank‐one totally geodesic subspaces. Among the consequences, we prove the existence of a nonconstant, globally defined complex‐valued harmonic morphism from the Riemannian symmetric space associated to a split real semisimple Lie group. This completes an affirmative proof of a conjecture of Gudmundsson.

Non-concentration property of Patterson–Sullivan measures for Anosov subgroups

Abstract Let G be a connected semisimple real algebraic group. For a Zariski dense Anosov subgroup $\Gamma <G$ with respect to a parabolic subgroup $P_\theta $ , we prove that any $\Gamma $ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ . More generally, we prove that for a Zariski dense $\theta $ -transverse subgroup $\Gamma <G$ , any $(\Gamma , \psi )$ -Patterson–Sullivan measure charges no mass on any proper subvariety of $G/P_\theta $ , provided the $\psi $ -Poincaré series of $\Gamma $ diverges at its abscissa of convergence. In particular, our result also applies to relatively Anosov subgroups.

A spectral theorem for compact representations and non-unitary cusp forms

We show that a compact representation of either a semisimple Lie group or a totally disconnected group has a filtration with irreducible subquotients of finite multiplicity. In the Lie group case we show the stronger assertion, that it has an orthogonal decomposition into summands of finite lengths. This generalises and simplifies a number of more special spectral theorems in Deitmar and Monheim (Math Z 284(3–4):1199–1210, 2016, https://doi.org/10.1007/s00209-016-1695-9), Müller (Int Math Res Not 9(2):2068–2109, 2011), Venkov (in: Proceedings of the Steklov Institute of Mathematics, no. 4(153), ix+163 pp. (1983), 1982). We apply it to the case of cusp forms, thus settling the spectral theory for the space of non-unitary twisted cusp forms. We finally show that the space of cusp forms is complemented by the space of Eisenstein series.

Commutative avatars of representations of semisimple Lie groups

Here we announce the construction and properties of a big commutative subalgebra of the Kirillov algebra attached to a finite dimensional irreducible representation of a complex semisimple Lie group. They are commutative finite flat algebras over the cohomology of the classifying space of the group. They are isomorphic with the equivariant intersection cohomology of affine Schubert varieties, endowing the latter with a new ring structure. Study of the finer aspects of the structure of the big algebras will also furnish the stalks of the intersection cohomology with ring structure, thus ringifying Lusztig's q-weight multiplicity polynomials i.e., certain affine Kazhdan-Lusztig polynomials.

Algebraic fundamental groups of fake projective planes

Fundamental groups of fake projective planes fall into fifty distinct isomorphism classes, one for each complex conjugate pair. We prove that this is not the case for their algebraic fundamental groups: there are only forty-six isomorphism classes. We show that there are four pairs of complex conjugate pairs of fake projective planes that are \operatorname{Aut}(\mathbb{C}) -equivalent and hence have mutually isomorphic algebraic fundamental groups. All other pairs of algebraic fundamental groups are shown to be distinct through explicit finite étale covers. As a by-product, this provides the first examples of commensurable but nonisomorphic lattices in a rank one semisimple Lie group that have isomorphic profinite completions.

The $J$-invariant over splitting fields of Tits algebras

We describe the $J$-invariant of a semisimple algebraic group $G$ over a generic splitting field of a Tits algebra of $G$ in terms of the $J$-invariant over the base field. As a consequence we prove a 10-year-old conjecture of Quéguiner-Mathieu, Semenov, and Zainoulline on the $J$-invariant of groups of type $\mathrm {D}_n$. In the case of type $\mathrm {D}_n$ we also provide explicit formulas for the first component and in some cases for the second component of the $J$-invariant.

Cyclic base change of cuspidal automorphic representations over function fields

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

Higher Tate traces of Chow motives

Abstract We establish the complete classification of Chow motives of projective homogeneous varieties for p-inner semi-simple algebraic groups, with coefficients in ℤ / p ⁢ ℤ {{\mathbb{Z}}/p{\mathbb{Z}}} . Our results involve a new motivic invariant, the Tate trace of a motive, defined as a pure Tate summand of maximal rank. They apply more generally to objects of the Tate subcategory generated by upper motives of irreducible, geometrically split varieties satisfying the nilpotence principle. Using Chernousov–Gille–Merkurjev decompositions and their interpretation through Bialynicki–Birula–Hesselink–Iversen filtrations due to Brosnan, we then generalize the characterization of the motivic equivalence of inner semi-simple groups through the higher Tits p-indexes. We also define the motivic splitting pattern and the motivic splitting towers of a summand of the motive of a projective homogeneous variety, which correspond for quadrics to the classical splitting pattern and Knebusch tower of the underlying quadratic form.

Invariant connections on non-irreducible symmetric spaces with simple Lie group

Consider a symmetric space G / H G/H with simple Lie group G G . We demonstrate that when G / H G/H is not irreducible, it is necessarily even dimensional and noncompact. Furthermore, the subgroup H H is also both noncompact and non-semisimple. Additionally, we establish that the only G G -invariant connection on G / H G/H is the canonical connection. On the other hand, we show that if G / H G/H has an odd dimension, it must be irreducible, and the subgroup H H must be semisimple. Finally, we present an explicit example, and we show that there exists no other torsion-free G G -invariant connection on a symmetric space G / H G/H with semisimple Lie group G G which has the same curvature as the canonical one.

A hyper-Kähler metric on the moduli spaces of monopoles with arbitrary symmetry breaking

We construct the hyper-Kähler moduli space of framed monopoles over R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^3$$\\end{document} for any connected, simply connected, compact, semisimple Lie group and arbitrary mass and charge, and hence arbitrary symmetry breaking. In order to do so, we define a configuration space of pairs with appropriate asymptotic conditions and perform an infinite-dimensional quotient construction. We make use of the b and scattering calculuses to study the relevant differential operators.

On Singularity Properties of Word Maps and Applications to Probabilistic Waring Type Problems

We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact p p -adic groups. Given a word w w in a free Lie algebra L r \mathcal {L}_{r} , it induces a word map φ w : g r → g \varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g} for every semisimple Lie algebra g \mathfrak {g} . Given two words w 1 ∈ L r 1 w_{1}\in \mathcal {L}_{r_{1}} and w 2 ∈ L r 2 w_{2}\in \mathcal {L}_{r_{2}} , we define and study the convolution of the corresponding word maps φ w 1 ∗ φ w 2 ≔ φ w 1 + φ w 2 : g r 1 + r 2 → g \varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g} . By introducing new degeneration techniques, we show that for any word w ∈ L r w\in \mathcal {L}_{r} of degree d d , and any simple Lie algebra g \mathfrak {g} with φ w ( g r ) ≠ 0 \varphi _{w}(\mathfrak {g}^{r})\neq 0 , one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking O ( d 4 ) O(d^{4}) self-convolutions of φ w \varphi _{w} . Similar results are obtained for matrix word maps. We deduce that a group word map of length ℓ \ell becomes (FRS), locally around identity, after O ( ℓ 4 ) O(\ell ^{4}) self-convolutions, for every semisimple algebraic group G _ \underline {G} . We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word w 0 = [ X , Y ] w_{0}=[X,Y] , we show that φ w 0 ∗ 4 \varphi _{w_{0}}^{*4} is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact p p -adic and arithmetic groups. The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form Z / p k Z \mathbb {Z}/p^{k}\mathbb {Z} . This allows us to relate them to properties of some natural families of random walks on finite and compact p p -adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to p p -adic probabilistic Waring type problems.

Model geometries dominated by locally finite graphs

We study model geometries of finitely generated groups. We show that a finitely generated group possesses a model geometry not dominated by a locally finite graph if and only if it contains either a commensurated finite rank free abelian subgroup, or a uniformly commensurated subgroup that is a uniform lattice in a semisimple Lie group. This characterises finitely generated groups that embed as uniform lattices in locally compact groups that are not compact-by-(totally disconnected). We also prove the only such groups of cohomological two are surface groups and generalised Baumslag–Solitar groups.

On normalizers of maximal tori in classical Lie groups

In general, the normalizer N G ( H G ) N_G(H_G) of a maximal torus H G H_G in a semisimple complex Lie group G G does not admit a presentation as a semidirect product of H G H_G and the corresponding Weyl group W G W_G . Meanwhile, such splitting exists for classical groups corresponding to the root systems A ℓ A_\ell , B ℓ B_\ell , D ℓ D_\ell . For the remaining classical groups corresponding to the root systems C ℓ C_\ell , there still exists an embedding of the Tits extension W G T W^T_G of W G W_G into the normalizer N G ( H G ) N_G(H_G) . Here an explicit unified construction is proposed for the lifts of the Weyl groups into normalizers of maximal tori for general linear and orthogonal Lie groups via embedding into the general linear groups. For the symplectic group S p 2 ℓ \mathrm {Sp}_{2\ell } , an explanation of the impossibility of embedding W S p 2 ℓ W_{\mathrm {Sp}_{2\ell }} into the normalizer N S p 2 ℓ ( H S p 2 ℓ ) N_{\mathrm {Sp}_{2\ell }}(H_{\mathrm {Sp}_{2\ell }}) is suggested. Also, explicit formulas are computed for the adjoint action of the lifts of the Weyl groups on g = L i e ( G ) \mathfrak {g}=\mathrm {Lie}(G) . Finally, examples of Weyl groups lifts are given for the groups closely associated with classical Lie groups.

Pureté de l’approximation forte sur le corps des fonctions d’une courbe algébrique complexe

Over the function field of a complex algebraic curve, strong approximation off a non-empty finite set of places holds for the complement of a codimension 2 closed subset in a homogeneous space under a semisimple algebraic group, and for the complement of a codimension 2 closed subset in an affine smooth complete intersection of low degree.

A Classification of Compact Cohomogeneity One Locally Conformal Kähler Manifolds

In this paper, we apply a result of the classification of a compact cohomogeneity one Riemannian manifold with a compact Lie group G to obtain a classification of compact cohomogeneity one locally conformal Kähler manifolds. In particular, we prove that the compact complex manifold is a complex one-dimensional torus bundle over a projective rational homogeneous, or cohomogeneity one manifold except of a class of manifolds with a generalized Hopf surface bundle over a projective rational homogeneous space. Additionally, it is a homogeneous compact complex manifold under the complexification GC of the given compact Lie group G under an extra condition that the related closed one form is cohomologous to zero on the generic G orbit. Moreover, the semi-simple part S of the Lie group action has hypersurface orbits, i.e., it is of cohomogeneity one with respect to the semi-simple Lie group S in that special case.

A remark on the Hochschild dimension of liberated quantum groups

Let A be a Hopf algebra equipped with a projection onto the coordinate Hopf algebra \mathcal{O}(G) of a semisimple algebraic group G . It is shown that if A admits a suitably non-degenerate comodule V and the induced G -module structure of V is non-trivial, then the third Hochschild homology group of A is non-trivial.