Book

*An introduction to Kac-Moody groups over fields*

EMS Textbooks in Mathematics (2018), European Mathematical Society (EMS), Zürich. ISBN 978-3-03719-187-3. doi | abstract**Abstract**:- The interest for Kac-Moody algebras and groups has grown exponentially in the
past decades, both in the mathematical and physics communities, and with it also the need of an introductory textbook on the topic.

The aims of this book are twofold:- to offer an accessible, reader-friendly and self-contained introduction to Kac-Moody algebras and groups;
- to clean the foundations and to provide a unified treatment of the theory.

The book starts with an outline of the classical Lie theory, used to set the scene. Part II provides a self-contained introduction to Kac-Moody algebras. The heart of the book is Part III, which develops an intuitive approach to the construction and fundamental properties of Kac-Moody groups. It is complemented by two appendices, respectively offering introductions to affine group schemes and to the theory of buildings. Many exercises are included, accompanying the readers throughout their journey.

The book assumes only a minimal background in linear algebra and basic topology, and is addressed to anyone interested in learning about Kac-Moody algebras and/or groups, from graduate (master) students to specialists.

Publications and preprints

*Cyclically reduced elements in Coxeter groups*

preprint (2018). pdf | doi | arXiv | abstract**Abstract**:- Let $W$ be a Coxeter group. We provide a precise description of the conjugacy classes in $W$, yielding an analogue of Matsumoto's theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993.

*On the structure of Kac-Moody algebras*

preprint (2018). pdf | doi | arXiv | abstract**Abstract**:- Let $A$ be a symmetrisable generalised Cartan matrix, and let $\mathfrak g(A)$ be the corresponding Kac-Moody algebra. In this paper, we address the following fundamental question on the structure of $\mathfrak g(A)$: given two homogeneous elements $x,y\in\mathfrak g(A)$, when is their bracket $[x,y]$ a nonzero element? As an application of our results, we give a description of the solvable and nilpotent graded subalgebras of $\mathfrak g(A)$.

*Closed sets of real roots in Kac-Moody root systems*(with Pierre-Emmanuel Caprace)

preprint (2018). pdf | doi | arXiv | abstract**Abstract**:- In this note, we provide a complete description of the closed sets of real roots in a Kac-Moody root system.

*On geodesic ray bundles in buildings*

Geometriae Dedicata (2018). 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.

*Half-Lie groups*(with Karl-Hermann Neeb)

Transform. Groups 23 (2018), no. 3, 801-840. pdf | doi | arXiv | abstract**Abstract**:- 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.

*Isomorphisms of twisted Hilbert loop algebras*(with Karl-Hermann Neeb)

Canad. J. Math. 69 (2017), 453-480. pdf | doi | arXiv | abstract**Abstract**:- 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.

*Open subgroups of locally compact Kac-Moody groups*(with Pierre-Emmanuel Caprace)

Math. Z. 274 Nr. 1-2 (2013), pp. 291–313. pdf | doi | arXiv | abstract**Abstract**:- 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 $R$.