Infinite-dimensional Lie groups and algebras, and their representations
Groups with a BN-pair and buildings
Locally compact groups
Publications and Preprints
On geodesic ray bundles in buildings
preprint (2017). pdf |
doi |
arXiv |
abstract
Abstract:
Let $X$ be a building, identified with its Davis realisation. In this paper, we provide for each $x\in X$ and each $\eta$ in the visual boundary $\partial X$ of $X$ a description of the geodesic ray bundle $\mathrm{Geo}(x,\eta)$, namely, of the reunion of all combinatorial geodesic rays (corresponding to infinite minimal galleries in the chamber graph of $X$) starting from $x$ and pointing towards $\eta$. When $X$ is locally finite and hyperbolic, we show that the symmetric difference between $\mathrm{Geo}(x,\eta)$ and $\mathrm{Geo}(y,\eta)$ is always finite, for $x,y\in X$ and $\eta\in\partial X$. This gives a positive answer to a question of Huang, Sabok and Shinko in the setting of buildings. Combined with their results and a construction of Bourdon, our results then yield examples of hyperbolic groups $G$ with Kazhdan's property (T) such that the $G$-action on its Gromov boundary is hyperfinite.
In this paper we study the Lie theoretic properties of a class of topological groups which carry a Banach manifold structure but whose multiplication is not smooth. If $G$ and $N$ are Banach-Lie groups and $\pi : G \to \mathrm{Aut}(N)$ is a homomorphism defining a continuous action of $G$ on $N$, then $H := N \rtimes_\pi G$ is a Banach manifold with a topological group structure for which the left multiplication maps are smooth, but the right multiplication maps need not to be. We show that these groups share surprisingly many properties with Banach-Lie groups: (a) for every regulated function $\xi : [0,1] \to T_1H$ the initial value problem $\dot\gamma(t) = \gamma(t)\xi(t)$, $\gamma(0)= 1_H$, has a solution and the corresponding evolution map from curves in $T_1H$ to curves in $H$ is continuous; (b) every $C^1$-curve $\gamma$ with $\gamma(0) = 1$ and $\gamma'(0) = x$ satisfies $\lim_{n \to \infty} \gamma(t/n)^n = \exp(tx)$; (c) the Trotter formula holds for $C^1$ one-parameter groups in $H$; (d) the subgroup $N^\infty$ of elements with smooth $G$-orbit maps in $N$ carries a natural Fréchet-Lie group structure for which the $G$-action is smooth; (e) the resulting Fréchet-Lie group $H^\infty := N^\infty \rtimes G$ is also regular in the sense of (a).
Positive energy representations of double extensions of Hilbert loop algebras (with Karl-Hermann Neeb)
J. Math. Soc. Japan 69 Nr. 4 (2017), 1485-1518. pdf |
doi |
arXiv |
abstract
Abstract:
A real Lie algebra with a compatible Hilbert space structure (in the sense that the scalar product is invariant) is called a Hilbert-Lie algebra.
Such Lie algebras are natural infinite-dimensional analogues of the compact Lie algebras; in particular, any infinite-dimensional simple Hilbert-Lie algebra $\mathfrak{k}$ is of one of the four classical types $A_J$, $B_J$, $C_J$ or $D_J$ for some infinite set $J$. Imitating the construction of affine Kac-Moody algebras, one can then consider affinisations of $\mathfrak{k}$, that is, double extensions of (twisted) loop algebras over $\mathfrak{k}$. Such an affinisation $\mathfrak{g}$ of $\mathfrak{k}$ possesses a root space decomposition with respect to some Cartan subalgebra $\mathfrak{h}$, whose corresponding root system yields one of the seven locally affine root systems (LARS) of type $A_J^{(1)}$, $B^{(1)}_J$, $C^{(1)}_J$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ or $BC_J^{(2)}$.
Let $D\in\mathrm{der}(\mathfrak{g})$ with $\mathfrak{h}\subseteq\mathrm{ker}D$ (a diagonal derivation of $\mathfrak{g}$). Then every highest weight representation $(\rho_{\lambda},L(\lambda))$ of $\mathfrak{g}$ with highest weight $\lambda$ can be extended to a representation $\widetilde{\rho}_{\lambda}$ of the semi-direct product $\mathfrak{g}\rtimes \mathbb{R} D$.
In this paper, we characterise all pairs $(\lambda,D)$ for which the representation $\widetilde{\rho}_{\lambda}$ is of positive energy, namely, for which the spectrum of the operator $-i\widetilde{\rho}_{\lambda}(D)$ is bounded from below.
Around the Lie correspondence for complete Kac-Moody groups and Gabber-Kac simplicity
preprint (2015). pdf |
doi |
arXiv |
abstract
Abstract:
Let $k$ be a field and $A$ be a generalised Cartan matrix, and let ${\mathfrak G}_A(k)$ be the corresponding minimal Kac-Moody group of simply connected type over $k$. Consider the completion ${\mathfrak G}_A^{pma}(k)$ of ${\mathfrak G}_A(k)$ introduced by O. Mathieu and G. Rousseau, and let ${\mathfrak U}_A^{ma+}(k)$ denote the unipotent radical of the positive Borel subgroup of ${\mathfrak G}_A^{pma}(k)$. In this paper, we exhibit some functoriality dependence of the groups ${\mathfrak U}_A^{ma+}(k)$ and ${\mathfrak G}_A^{pma}(k)$ on their Lie algebra.
We also produce a large class of examples of minimal Kac-Moody groups ${\mathfrak G}_A(k)$ that are not dense in their Mathieu-Rousseau completion ${\mathfrak G}_A^{pma}(k)$.
Finally, we explain how the problematic of providing a unified theory of complete Kac-Moody groups is related to the conjecture of Gabber-Kac simplicity of ${\mathfrak G}_A^{pma}(k)$, stating that every normal subgroup of ${\mathfrak G}_A^{pma}(k)$ that is contained in ${\mathfrak U}_A^{ma+}(k)$ must be trivial.
We present several motivations for the study of this conjecture, as well as several applications of our functoriality theorem, with contributions to the question of (non-)linearity of ${\mathfrak U}_A^{ma+}(k)$, and to the isomorphism problem for complete Kac-Moody groups over finite fields. For $k$ finite, we also make some observations on the structure of ${\mathfrak U}_A^{ma+}(k)$ in the light of some important concepts from the theory of pro-$p$ groups.
The closest infinite dimensional relatives of compact Lie algebras
are Hilbert-Lie algebras, i.e. real Hilbert spaces with a Lie algebra
structure for which the scalar product is invariant.
Locally affine Lie algebras (LALAs)
correspond to double extensions of (twisted) loop algebras
over simple Hilbert-Lie algebras $\mathfrak{k}$, also called affinisations of $\mathfrak{k}$.
They possess a root space decomposition
whose corresponding root system is a locally affine root system
of one of the $7$ families $A_J^{(1)}$, $B_J^{(1)}$, $C_J^{(1)}$, $D_J^{(1)}$, $B_J^{(2)}$, $C_J^{(2)}$ and $BC_J^{(2)}$ for some infinite set $J$. To each of these types corresponds a "minimal" affinisation of some simple Hilbert-Lie algebra $\mathfrak{k}$, which we call standard.
In this paper, we give for each affinisation $\mathfrak{g}$ of a simple Hilbert-Lie algebra $\mathfrak{k}$ an explicit isomorphism from $\mathfrak{g}$ to one of the standard affinisations of $\mathfrak{k}$.
The existence of such an isomorphism could also be derived from the classification
of locally affine root systems, but
for representation theoretic purposes it is crucial to obtain it explicitely
as a deformation between two twists which is compatible
with the root decompositions.
We illustrate this by applying our isomorphism theorem to the study of positive energy highest weight representations of $\mathfrak{g}$.
In subsequent work, the present paper will be used to obtain a complete classification
of the positive energy highest weight representations of affinisations of $\mathfrak{k}$.
Positive energy representations for locally finite split Lie algebras (with Karl-Hermann Neeb)
Int. Math. Res. Notices (2016); 2016 (21): 6689-6712. pdf |
doi |
arXiv |
abstract
Abstract:
Let $\mathfrak g$ be a locally finite split simple complex Lie algebra of type $A_J$, $B_J$, $C_J$ or $D_J$ and $\mathfrak h \subseteq \mathfrak g$ be a splitting Cartan subalgebra. Fix $D \in \mathrm{der}(\mathfrak g)$ with $\mathfrak h \subseteq \ker D$ (a diagonal derivation). Then every unitary highest weight representation $(\rho_\lambda, V^\lambda)$ of $\mathfrak g$ extends to a representation $\tilde\rho_\lambda$ of the semidirect product $\mathfrak g \rtimes \mathbb C D$ and we say that $\tilde\rho_\lambda$ is a positive energy representation if the spectrum of $-i\tilde\rho_\lambda(D)$ is bounded from below. In the present note we characterise all pairs $(\lambda,D)$ with $\lambda$ bounded for which this is the case.
If $U_1(\mathcal H)$ is the unitary group of Schatten class $1$ on an infinite dimensional real, complex or quaternionic Hilbert space and $\lambda$ is bounded, then we accordingly obtain a characterisation of those highest weight representations $\pi_\lambda$ satisfying the positive energy condition with respect to the continuous $\mathbb R$-action induced by $D$. In this context the representation $\pi_\lambda$ is norm continuous and our results imply the remarkable result that, for positive energy representations, adding a suitable inner derivation to $D$, we can achieve that the minimal eigenvalue of $\tilde\rho_\lambda(D)$ is $0$ (minimal energy condition). The corresponding pairs $(\lambda,D)$ satisfying the minimal energy condition are rather easy to describe explicitly.
Conjugacy classes and straight elements in Coxeter groups
Journal of Algebra 407 (2014), pp. 68-80. pdf |
doi |
arXiv |
abstract
Abstract:
Let $W$ be a Coxeter group. In this paper, we establish that, up to going to
some finite index normal subgroup $W_0$ of $W$, any two cyclically reduced expressions of conjugate elements of $W_0$ only differ by a sequence of braid relations and cyclic shifts. This thus provides a simple description of conjugacy classes in $W_0$. As a byproduct of our methods, we also obtain a characterisation of straight elements of $W$, namely of those elements $w\in W$ for which $\ell(w^n)=n\ell(w)$ for any integer $n$. In particular, we generalise previous characterisations of straight elements within the class of so-called cyclically fully commutative (CFC) elements, and we give a shorter and more transparent proof that Coxeter elements are straight.
Topological Kac-Moody groups and their subgroups
Ph.D. thesis (April 2013), Université Catholique de Louvain. pdf |
abstract
Abstract:
In this thesis, we establish several structure results for topological Kac-Moody groups (minimal or maximal). This thesis may also serve as an introduction to Kac-Moody theory.
Abstract simplicity of locally compact Kac-Moody groups
Compositio Math. 150 (2014), pp. 713-728. pdf |
doi | arXiv |
abstract
Abstract:
In this paper, we establish that complete Kac–Moody groups over
finite fields are abstractly simple. The proof makes an essential use of O. Mathieu’s construction of complete Kac–Moody groups over fields. This construction
has the advantage that both real and imaginary root spaces of the Lie algebra lift to root subgroups over arbitrary fields. A key point in our proof is the
fact, of independent interest, that both real and imaginary root subgroups are contracted by conjugation of positive powers of suitable Weyl group elements.
A fixed point theorem for Lie groups acting on buildings and applications to Kac-Moody theory
Forum Mathematicum 27 Nr. 1 (2015), pp. 449–466. pdf |
doi | arXiv |
abstract
Abstract:
We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac-Moody group are obtained by exponentiating ad-locally finite elements of the corresponding Lie algebra.
Let G be a complete Kac-Moody group over a finite field. It is known that G possesses a BN-pair structure, all of whose parabolic subgroups are open in G. We show that, conversely, every open subgroup of G has finite index in some parabolic subgroup. The proof uses some new results on parabolic closures in Coxeter groups. In particular, we give conditions ensuring that the parabolic closure of the product of two elements in a Coxeter group contains the respective parabolic closures of those elements.
Can an anisotropic reductive group admit a Tits system? (with Pierre-Emmanuel Caprace)
Pure and Appl. Math. Quart. 7 Nr. 3 (2011), pp. 539–558. pdf |
doi | arXiv |
abstract
Abstract:
Seeking for a converse to a well-known theorem by Borel-Tits,
we address the question whether the group of rational points $G(k)$ of an anisotropic reductive $k$-group may admit a split spherical BN-pair.
We show that if $k$ is a perfect field or a local field, then such a BN-pair must be virtually trivial. We also consider arbitrary compact groups and show
that the only abstract BN-pairs they can admit are spherical, and even virtually trivial provided they are split.
Angle-deformations in Coxeter groups (with Bernhard Mühlherr)
Algebraic & Geometric Topology 8 (2008), pp. 2175–2208 pdf |
doi | arXiv |
abstract
Abstract:
The isomorphism problem for Coxeter groups has been reduced
to its 'reflection preserving version' by B. Howlett and the second author. Thus, in order to solve it, it suffices to determine for a given Coxeter
system $(W,R)$ all Coxeter generating sets S of W which are contained in $R^W$, the set of reflections of $(W,R)$. In this paper, we provide a further reduction:
it suffices to determine all Coxeter generating sets $S$ in $R^W$ which are sharp-angled with respect to $R$.