Related papers: Split Kac-Moody groups over a local field, II. Ord…

The purpose of this paper is to describe some conjectures and results on the existence and uniqueness of invariant measures on formal completions of Kac-Moody groups and associated homogeneous spaces. Existence is rigorously established in…

Let $G$ be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix $A$, as constructed by Tits. It is known that $G$ admits the structure of a BN-pair, and acts on its corresponding building. We study the complete…

Automorphisms of finite order and real forms of "smooth" affine Kac-Moody algebras are studied, i.e. of 2-dimensional extensions of the algebra of smooth loops in a simple Lie algebra. It is shown that they can be parametrized by certain…

The paper is devoted to model-theoretic properties of Kac-Moody groups with the focus on elementary equivalence of Kac-Moody groups. We show that elementary equivalence of (untwisted) affine Kac-Moody groups implies coincidence of their…

Masures (previously also known as hovels) are a generalization of the theory of affine buildings for arbitrary $p$-adic Kac-Moody groups. Gaussent and Rousseau invented masures to compute the Satake transform for $p$-adic Kac-Moody groups.…

The Kac-Moody affine Hecke algebra $\mathcal{H}$ was first constructed as the Iwahori-Hecke algebra of a $p$-adic Kac-Moody group by work of Braverman, Kazhdan, and Patnaik, and by work of Bardy-Panse, Gaussent, and Rousseau. Since…

This survey paper presents the discrete group viewpoint on Kac-Moody groups. Over finite fields, the latter groups are finitely generated; they act on new buildings enjoying remarkable negative curvature properties. The study of these…

We define the spherical Hecke algebra for an (untwisted) affine Kac-Moody group over a local non-archimedian field. We prove a generalization of the Satake isomorphism for these algebras, relating it to integrable representations of the…

In this paper, we establish that complete Kac-Moody groups over finite fields are abstractly simple. The proof makes an essential use of Mathieu-Rousseau's construction of complete Kac-Moody groups over fields. This construction has the…

In the context of affine complex Kac-Moody algebras, we define the meaning of nilpotent orbits under the adjoint action of the maximal Kac-Moody group. We also give a parameterization of nilpotent orbits of $\mathfrak{sl}_n^{(1)}(\mathbb…

For an almost split Kac-Moody group G over a local non-archimedean field, the last two authors constructed a spherical Hecke algebra H (over the complex numbers C, say) and its Satake isomorphism with the commutative algebra of Weyl…

We define the Iwahori-Hecke algebra for an almost split Kac-Moody group over a local non-archimedean field. We use the hovel associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The fixer K…

Let $G$ be a minimal split Kac-Moody group over a valued field {\mathcal{K}. Motivated by the representation theory of $G$, we define two topologies of topological group on $G$, which take into account the topology on {\mathcal{K}.

Kac-Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most…

We define the formal affine Demazure algebra and formal affine Hecke algebra associated to a Kac-Moody root system. We prove the structure theorems of these algebras, hence, extending several result and construction (presentation in terms…

Let $\mathfrak{G}$ be a split reductive group, $\mathbb{k}$ be a field and $\varpi$ be an indeterminate. In order to study $\mathfrak{G}(\mathbb{k}[\varpi,\varpi^{-1}])$ and $\mathfrak{G}(\mathbb{k}(\varpi))$, one can make them act on their…

The goal of the present paper is to obtain new free field realizations of affine Kac-Moody algebras motivated by geometric representation theory for generalized flag manifolds of finite-dimensional semisimple Lie groups. We provide an…

For any Kac-Moody root data $\mathcal D$, D. Muthiah and D. Orr have defined a partial order on the semi-direct product $W^+$ of the integral Tits cone with the vectorial Weyl group of $\mathcal D$, and a strictly compatible $\mathbb…

We review the categorical representation of a Kac-Moody algebra on unipotent representations of finite unitary groups in non-defining characteristic given by the authors. Then, we extend this construction to finite reductive groups of types…

We compute the mod $p$ cohomology algebra of a family of infinite discrete Kac-Moody groups of rank two defined over finite fields of characteristic different from $p$.