Semisimple Group Research Articles

Noncommutative Poisson boundaries and Furstenberg–Hamana boundaries of Drinfeld doubles

We clarify the relation between noncommutative Poisson boundaries and Furstenberg–Hamana boundaries of quantum groups. Specifically, given a compact quantum group G, we show that in many cases where the Poisson boundary of the dual discrete quantum group Gˆ has been computed, the underlying topological boundary either coincides with the Furstenberg–Hamana boundary of the Drinfeld double D(G) of G or is a quotient of it. This includes the q-deformations of compact Lie groups, free orthogonal and free unitary quantum groups, quantum automorphism groups of finite dimensional C⁎-algebras. In particular, the boundary of D(Gq) for the q-deformation of a compact connected semisimple Lie group G is Gq/T (for q≠1), in agreement with the classical results of Furstenberg and Moore on the Furstenberg boundary of GC.We show also that the construction of the Furstenberg–Hamana boundary of D(G) respects monoidal equivalence and, in fact, can be carried out entirely at the level of the representation category of G. This leads to a notion of the Furstenberg–Hamana boundary of a rigid C⁎-tensor category.

Strict quantization of coadjoint orbits

For every semisimple coadjoint orbit \hat{\mathcal{O}} of a complex connected semisimple Lie group \hat{G} , we obtain a family of \hat{G} -invariant products \hat{*}_\hbar on the space of holomorphic functions on \hat{\mathcal{O}} . For every semisimple coadjoint orbit \mathcal{O} of a real connected semisimple Lie group G , we obtain a family of G -invariant products *_\hbar on a space \mathcal{A}(\mathcal{O}) of certain analytic functions on \mathcal{O} by restriction. \mathcal{A}(\mathcal{O}) , endowed with one of the products *_\hbar , is a G -Fréchet algebra, and the formal expansion of the products around \hbar=0 determines a formal deformation quantization of \mathcal{O} , which is of Wick type if G is compact. Our construction relies on an explicit computation of the canonical element of the Shapovalov pairing between generalized Verma modules and complex analytic results on the extension of holomorphic functions.

Numerical flatness and principal bundles on Fujiki manifolds

Let M be a compact connected Fujiki manifold, G a semisimple affine algebraic group over C with one simple factor and P a fixed proper parabolic subgroup of G. For a holomorphic principal G–bundle EG over M, let EP be the holomorphic principal P–bundle EG⟶EG/P given by the quotient map. We prove that the following three statements are equivalent: (1) ad(EG) is numerically flat, (2) the holomorphic line bundle ⋀topad(EP)⁎ is nef, and (3) for every reduced irreducible compact complex analytic space Z with a Kähler form ω, holomorphic map γ:Z⟶M, and holomorphic reduction of structure group EP⊂γ⁎EG to P, the inequality degree(ad(EP))≤0 holds.

An Equivariant Atiyah–Patodi–Singer Index Theorem for Proper Actions I: The Index Formula

AbstractConsider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ on $M$, and an element $g \in G$, we define a numerical index ${\operatorname {index}}_g(D)$, in terms of a parametrix for $D$ and a trace associated to $g$. We prove an equivariant Atiyah–Patodi–Singer index theorem for this index. We first state general analytic conditions under which this theorem holds, and then show that these conditions are satisfied if $g=e$ is the identity element; if $G$ is a finitely generated, discrete group, and the conjugacy class of $g$ has polynomial growth; and if $G$ is a connected, linear, real semisimple Lie group, and $g$ is a semisimple element. In the classical case, where $M$ is compact and $G$ is trivial, our arguments reduce to a relatively short and simple proof of the original Atiyah–Patodi–Singer index theorem. In part II of this series, we prove that, under certain conditions, ${\operatorname {index}}_g(D)$ can be recovered from a $K$-theoretic index of $D$ via a trace defined by the orbital integral over the conjugacy class of $g$.

The Canonical Dimension of a Semisimple Group and the Unimodular Degree of a Root System

AbstractWe produce a short and elementary algorithm to compute an upper bound for the canonical dimension of a spit semisimple linear algebraic group. Using this algorithm we confirm previously known bounds by Karpenko [6–9] and Devyatov [5] as well as we produce new bounds (e.g., for groups of types $F_4$, adjoint $E_6$, for some semisimple groups).

Rings on Vector Abelian Groups

A multiplication on an Abelian group G is a homomorphism μ: G ⊗ G → G. An Abelian group G with a multiplication on it is called a ring on the group G. R. A. Beaumont and D. A. Lawver have formulated the problem of studying semisimple groups. An Abelian group is said to be semisimple if there exists a semisimple associative ring on it. Semisimple groups are described in the class of vector Abelian nonmeasurable groups. It is also shown that if a set I is nonmeasurable, \( G=\coprod_{i\in I}{A}_i \) is a reduced vector Abelian group, and μ is a multiplication on G, then μ is determined by its restriction on the sum \( {\displaystyle \begin{array}{c}\oplus \\ {}i\in I\end{array}}{A}_i \) this statement is incorrect if the set I is measurable or the group G is not reduced.

Projective Anosov representations, convex cocompact actions, and rigidity

In this paper we show that many projective Anosov representations act convex cocompactly on some properly convex domain in real projective space. In particular, if a non-elementary word hyperbolic group is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface, then any projective Anosov representation of that group acts convex cocompactly on some properly convex domain in real projective space. We also show that if a projective Anosov representation preserves a properly convex domain, then it acts convex cocompactly on some (possibly different) properly convex domain. We then give three applications. First, we show that Anosov representations into general semisimple Lie groups can be defined in terms of the existence of a convex cocompact action on a properly convex domain in some real projective space (which depends on the semisimple Lie group and parabolic subgroup). Next, we prove a rigidity result involving the Hilbert entropy of a projective Anosov representation. Finally, we prove a rigidity result which shows that the image of the boundary map associated to a projective Anosov representation is rarely a $C^2$ submanifold of projective space. This final rigidity result also applies to Hitchin representations.

Arithmetic groups and the Lehmer conjecture

We generalize a result of Sury [Sur92] and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis [Mar91]) is equivalent to a weak form of Lehmer’s conjecture. We include a short survey of related results and conjectures.

The moduli space of non-abelian vortices in Yang–Mills–Chern–Simons–Higgs theory

We determine the dimension of the moduli space of non-abelian vortices in Yang–Mills–Chern–Simons–Higgs theory in 2 + 1 dimensions for gauge groups with G′ being an arbitrary semi-simple group and n 0 the greatest common divisor of the abelian charges of the G′ invariants. The calculation is carried out using a Callias-type index theorem, the moduli matrix approach and a D-brane setup in type IIB string theory. We prove that the index theorem gives the number of zeromodes or moduli of the non-abelian vortices, extend the moduli matrix approach to the Yang–Mills–Chern–Simons–Higgs theory and finally derive the effective Lagrangian of Collie and Tong using string theory.

Strong and uniform boundedness of groups

A group [Formula: see text] is called bounded if every conjugation-invariant norm on [Formula: see text] has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

Representation theoretic embedding of twisted Dirac operators

Let G G be a non-compact connected semisimple real Lie group with finite center. Suppose L L is a non-compact connected closed subgroup of G G acting transitively on a symmetric space G / H G/H such that L ∩ H L\cap H is compact. We study the action on L / L ∩ H L/L\cap H of a Dirac operator D G / H ( E ) D_{G/H}(E) acting on sections of an E E -twist of the spin bundle over G / H G/H . As a byproduct, in the case of ( G , H , L ) = ( S L ( 2 , R ) × S L ( 2 , R ) , Δ ( S L ( 2 , R ) × S L ( 2 , R ) ) , S L ( 2 , R ) × S O ( 2 ) ) (G,H,L)=(SL(2,{\mathbb R})\times SL(2,{\mathbb R}),\Delta (SL(2,{\mathbb R})\times SL(2,{\mathbb R})),SL(2,{\mathbb R})\times SO(2)) , we identify certain representations of L L which lie in the kernel of D G / H ( E ) D_{G/H}(E) .

On two geometric means and sum of adjoint orbits

In this paper, we study the metric geometric mean introduced by Pusz and Woronowicz and the spectral geometric mean introduced by Fiedler and Pták, originally for positive definite matrices. The relation between t-metric geometric mean and t-spectral geometric mean is established via log majorization. The result is then extended in the context of symmetric space associated with a noncompact semisimple Lie group. For any Hermitian matrices X and Y, So's matrix exponential formula asserts that there are unitary matrices U and V such thateX/2eYeX/2=eUXU⁎+VYV⁎. In other words, the Hermitian matrix log⁡(eX/2eYeX/2) lies in the sum of the unitary orbits of X and Y. So's result is also extended to a formula for adjoint orbits associated with a noncompact semisimple Lie group.

Equivariant maps for measurable cocycles with values into higher rank Lie groups

Let $G$ a semisimple Lie group of non-compact type and let $\mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $\mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $\mathbb{H}^2$ or $\text{SL}(3,\mathbb{R})/\text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $\Gamma=\pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $\rho:\Gamma \rightarrow G$ with a measurable $\rho$-equivariant map $\psi:Y \rightarrow \mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Y\rightarrow \mathcal{X}_G$ which is smooth, $\rho$-equivariant and with uniformly bounded Jacobian. In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if $(\Omega,\mu_\Omega)$ is a standard Borel probability $\Gamma$-space, let $\sigma:\Gamma \times \Omega \rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y \times \Omega \rightarrow \mathcal{X}_G$ which is $\sigma$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.

Explicit generators of some pro-$p$ groups via Bruhat-Tits theory

Given a semisimple group over a local field of residual characteristic p, its topological group of rational points admits maximal pro-p-subgroups. Quasi-split simply-connected semisimple groups can be described in the combinatorial terms of valued root groups, thanks to Bruhat-Tits theory. In this context, it becomes possible to compute explicitly a minimal generating set of the (all conjugated) maximal pro-p-subgroups thanks to parametrizations of a suitable maximal torus and of corresponding root groups. We show that the minimal number of generators is then linear with respect to the rank of a suitable root system.

Proper affine actions: a sufficient criterion

For a semisimple real Lie group G with a representation \(\rho \) on a finite-dimensional real vector space V, we give a sufficient criterion on \(\rho \) for existence of a group of affine transformations of V whose linear part is Zariski-dense in \(\rho (G)\) and that is free, nonabelian and acts properly discontinuously on V. This new criterion is more general than the one given in Smilga (Groups Geom Dyn 12(2):449–528, 2018), insofar as it also deals with “swinging” representations. When G is split, almost all the irreducible representations of G that have 0 as a weight satisfy this criterion. We conjecture that it is actually a necessary and sufficient criterion.

Langlands duality and Poisson–Lie duality via cluster theory and tropicalization

Let G be a connected semisimple Lie group. There are two natural duality constructions that assign to G: its Langlands dual group \(G^\vee \), and its Poisson–Lie dual group \(G^*\), respectively. The main result of this paper is the following relation between these two objects: the integral cone defined by the cluster structure and the Berenstein–Kazhdan potential on the double Bruhat cell \(G^{\vee ; w_0, e} \subset G^\vee \) is isomorphic to the integral Bohr–Sommerfeld cone defined by the Poisson structure on the partial tropicalization of \(K^* \subset G^*\) (the Poisson–Lie dual of the compact form \(K \subset G\)). By Berenstein and Kazhdan (in: Contemporary mathematics, vol. 433. American Mathematical Society, Providence, pp 13–88, 2007), the first cone parametrizes the canonical bases of irreducible G-modules. The corresponding points in the second cone belong to integral symplectic leaves of the partial tropicalization labeled by the highest weight of the representation. As a by-product of our construction, we show that symplectic volumes of generic symplectic leaves in the partial tropicalization of \(K^*\) are equal to symplectic volumes of the corresponding coadjoint orbits in \({{\,\mathrm{Lie}\,}}(K)^*\). To achieve these goals, we make use of (Langlands dual) double cluster varieties defined by Fock and Goncharov (Ann Sci Ec Norm Supér (4) 42(6):865–930, 2009). These are pairs of cluster varieties whose seed matrices are transpose to each other. There is a naturally defined isomorphism between their tropicalizations. The isomorphism between the cones described above is a particular instance of such an isomorphism associated to the double Bruhat cells \(G^{w_0, e} \subset G\) and \(G^{\vee ; w_0, e} \subset G^\vee \).

Folded and contracted solutions of coupled classical dynamical Yang–Baxter and reflection equations

In this paper we give a concrete recipe how to construct triples of algebra-valued meromorphic functions on a complex vector space a satisfying three coupled classical dynamical Yang–Baxter equations and an associated classical dynamical reflection equation. Such triples provide the local factors of a consistent system of first order differential operators on a, generalising asymptotic boundary Knizhnik–Zamolodchikov–Bernard (KZB) equations.The recipe involves folding and contracting a-invariant and θ-twisted symmetric classical dynamical r-matrices along an involutive automorphism θ. In case of the universal enveloping algebra of a simple Lie algebra g we determine the subclass of Schiffmann’s classical dynamical r-matrices which are a-invariant and θ-twisted.The paper starts with a section highlighting the connections between asymptotic (boundary) KZB equations, representation theory of semisimple Lie groups, and integrable quantum field theories.

Alcove Paths and Gelfand–Tsetlin Patterns

In their study of the equivariant K-theory of the generalized flag varieties G/P, where G is a complex semisimple Lie group, and P is a parabolic subgroup of G, Lenart and Postnikov introduced a combinatorial tool, called the alcove path model. It provides a model for the highest weight crystals with dominant integral highest weights, generalizing the model by semistandard Young tableaux. In this paper, we prove a simple and explicit formula describing the crystal isomorphism between the alcove path model and the Gelfand–Tsetlin pattern model for type A.

Positive Scalar Curvature on Spin Pseudomanifolds: the Fundamental Group and Secondary Invariants

In this paper we continue the study of positive scalar curvature (psc) metrics on a depth-1 Thom-Mather stratified space $M_\Sigma$ with singular stratum $\beta M$ (a closed manifold of positive codimension) and associated link equal to $L$, a smooth compact manifold. We briefly call such spaces manifolds with $L$-fibered singularities. Under suitable spin assumptions we give necessary index-theoretic conditions for the existence of wedge metrics of positive scalar curvature. Assuming in addition that $L$ is a simply connected homogeneous space of positive scalar curvature, $L=G/H$, with the semisimple compact Lie group $G$ acting transitively on $L$ by isometries, we investigate when these necessary conditions are also sufficient. Our main result is that our conditions are indeed sufficient for large classes of examples, even when $M_\Sigma$ and $\beta M$ are not simply connected. We also investigate the space of such psc metrics and show that it often splits into many cobordism classes.

On the rate of equidistribution of expanding translates of horospheres in $\Gamma\backslash G$

Let G be a semisimple Lie group and \Gamma a lattice in G . We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space \Gamma\backslash G .