Split Kac-Moody Group Research Articles

DISTANCES ON A MASURE

A masure (also known as an affine ordered hovel) is a generalization of the Bruhat-Tits building that is associated with a split Kac-Moody group G over a nonarchimedean local field. This is a union of affine spaces called apartments. When G is a reductive group, is a building and there is a G-invariant distance inducing a norm on each apartment. In this paper, we study distances on inducing the affine topology on each apartment. We construct distances such that each element of G is a continuous automorphism of and we study their properties (completeness, local compactness, …).

An extension theorem for non-compact split embedded Riemannian symmetric spaces and an application to their universal property

Abstract By [5] it is known that a geodesic γ in an abstract reflection space X (in the sense of Loos, without any assumption of differential structure) canonically admits an action of a 1-parameter subgroup of the group of transvections of X. In this article, we modify these arguments in order to prove an analog of this result stating that, if X contains an embedded hyperbolic plane 𝓗 ⊂ X, then this yields a canonical action of a subgroup of the transvection group of X isomorphic to a perfect central extension of PSL2(ℝ). This result can be further extended to arbitrary Riemannian symmetric spaces of non-compact split type Y lying in X and can be used to prove that a Riemannian symmetric space and, more generally, the Kac–Moody symmetric space G/K for an algebraically simply connected two-spherical split Kac–Moody group G, as defined in [5], satisfies a universal property similar to the universal property that the group G satisfies itself.

Macdonald's formula for Kac–Moody groups over local fields

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 invariant elements in some formal series algebra C[[Y]].In this article, we prove a Macdonald's formula, i.e. an explicit formula for the image of a basis element of H. The proof involves geometric arguments in the masure associated to G and algebraic tools, including the Cherednik's representation of the Bernstein-Lusztig-Hecke algebra (introduced in a previous article) and the Cherednik's identity between some symmetrizers.

Completed Iwahori-Hecke algebras and parahoric Hecke algebras for Kac-Moody groups over local fields

Let G be a split Kac-Moody group over a non-Archimedean local field. We define a completion of the Iwahori-Hecke algebra of G, then we compute its center and prove that it is isomorphic (via the Satake isomorphism) to the spherical Hecke algebra of G. This is thus similar to the situation for reductive groups. Our main tool is the masure ℐ associated to this setting, which plays here the same role as Bruhat-Tits buildings do for reductive groups. In a second part, we associate a Hecke algebra to each spherical face F of type 0, extending a construction that was only known, in the Kac-Moody setting, for the spherical subgroup and for the Iwahori subgroup.

Gindikin–Karpelevich Finiteness for Kac–Moody Groups Over Local Fields

In this paper, we prove some finiteness results about split Kac-Moody groups over local non-archimedean fields. Our results generalize those of "An affine Gindikin-Karpelevich formula" by Alexander Braverman, Howard Garland, David Kazhdan and Manish Patnaik. We do not require our groups to be affine. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group.

Iwahori–Hecke algebras for Kac–Moody groups over local fields

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 of some chamber in the standard apartment plays the role of the Iwahori subgroup. We can define the Iwahori-Hecke algebra as the algebra of some K-bi-invariant functions on the group with support consisting of a finite union of double classes. As two chambers in the hovel are not always in a same apartment, this support has to be in some large subsemigroup of the Kac-Moody group. In the split case, we prove that the structure constants of the multiplication in this algebra are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We give a presentation of this algebra, similar to the Bernstein-Lusztig presentation in the reductive case, and embed it in a greater algebra, algebraically defined by the Bernstein-Lusztig presentation. In the affine case, this algebra contains the Cherednik's double affine Hecke algebra. Actually, our results apply to abstract "locally finite" hovels, so that we can define the Iwahori-Hecke algebra with unequal parameters.

Almost split Kac–Moody groups over ultrametric fields

For a split Kac–Moody group G over an ultrametric field K , S. Gaussent and the author defined an ordered affine hovel (for short, a masure) on which the group acts; it generalizes the Bruhat–Tits building which corresponds to the case when G is reductive. This construction was generalized by C. Charignon to the almost split case when K is a local field. We explain here these constructions with more details and prove many new properties, e.g. that the hovel of an almost split Kac–Moody group is an ordered affine hovel, as defined in a previous article.

Spherical Hecke algebras for Kac-Moody groups over local fields

We define the spherical Hecke algebra H for an almost split Kac-Moody group G over a local non-archimedean field. We use the hovel I associated to this situation, which is the analogue of the Bruhat-Tits building for a reductive group. The stabilizer K of a special point on the standard apartment plays the role of a maximal open compact subgroup. We can define H as the algebra of K-bi-invariant functions on G with almost finite support. As two points in the hovel are not always in a same apartment, this support has to be in some large subsemigroup G+ of G. We prove that the structure constants of H are polynomials in the cardinality of the residue field, with integer coefficients depending on the geometry of the standard apartment. We also prove the Satake isomorphism between H and the algebra of Weyl invariant elements in some completion of a Laurent polynomial algebra. In particular, H is always commutative. Actually, our results apply to abstract ''locally finite'' hovels, so that we can define the spherical algebra with unequal parameters.

On topological split Kac-Moody groups and their twin buildings

We prove that a two-spherical split Kac–Moody group over a local field naturally provides a topological twin building in the sense of Kramer. This existence result and the local-to-global principle for twin building topologies combined with the theory of Moufang foundations as introduced and studied by Muhlherr, Ronan, and Tits allows one to immediately obtain a classification of two-spherical split Moufang topological twin buildings whose underlying Coxeter diagram contains no loop and no isolated vertices. we obtain a similar classification for split Moufang topological twin buildings.

Mini-Workshop: Symmetric Varieties and Involutions of Algebraic Groups

The topics of this conference all in some way evolved from the classical theory of real and complex Lie groups. Indeed, one of the important mathematical goals during the 1950's was to find analogs of the semisimple Lie groups of exceptional type over arbitrary fields. Chevalley completed the first crucial step by producing his famous basis theorem for simple complex Lie algebras, and later Steinberg succeeded in describing these analogs group-theoretically. An important development due to Tits was the theory of groups with a BN -pair and invented buildings; these buildings belong to arbitrary Chevalley groups as naturally as the projective spaces belong to the special linear groups. Since then the theories of algebraic groups and of buildings developed into various directions. However, due to their common origin both theories often lead naturally to similar questions which were attacked by completely different means. In the context of this conference, the PhD thesis by Bernhard Mühlherr and the work by Aloysius Helminck and coauthors on involutions of algebraic groups illustrate this in a quite remarkable way. Both projects contributed strongly to the understanding of the geometry of involutions of algebraic groups, but surprisingly each one has gone unnoticed by the researchers of the other until recently. One of the main objectives of this conference was to bring these two theories closer to each other. The first two lectures on Monday morning familiarised all participants with the concept of buildings; Pierre-Emmanuel Caprace explained the foundations of buildings from the simplicial point of view, while Bernhard Mühlherr introduced the chamber system approach to buildings and explained the power of filtrations when studying sub-geometries of (twin) buildings that arise from the action of certain subgroups of the isometry group of the (twin) building. As these two lectures were of an introductory nature and since their content is already well documented (we refer to the recently published book by Abramenko and Brown for the theory of buildings and to the contributions of Alice Devillers and of Hendrik Van Maldeghem to the Oberwolfach report 20/2008 for an account on filtrations and their powerful applications), we do not include acstracts of these lectures. On Monday afternoon and on Friday morning Aloysius Helminck presented the theory of involutions of algebraic groups, while in Monday's final lecture Max Horn showed how to combine Helminck's theory with M"uhlherr's PhD thesis in order to obtain general and powerful results on the geometry of involutions of groups with a root group datum, a class of groups that contains the semisimple algebraic groups, the split Kac–Moody groups, and the split finite groups of Lie type. Most of Tuesday and part of Wednesday were focussed on the Tits centre conjecture. In a series of two lectures Gerhard Röhrle and Michael Bate presented an algebraic-group approach towards proving the conjecture, while on Tuesday afternoon Katrin Tent presented a combinatorial approach and on Wednesday morning Linus Kramer reported on metric considerations in the context of the Tits centre conjecture. It is our impression that these four lectures have triggered additional activity towards proving the centre conjecture, and that one or more of these approaches will be successful in the near future. The fourth talk on Tuesday afternoon was given by Yiannis Sakellaridis on spherical varieties and automorphic forms, while the second talk on Wednesday by Lizhen Ji presented compactifications of locally symmetric spaces. Thursday's talks by Sergey Shpectorov and by Paul Levy concentrated on involutions of affine buildings, respectively automorphisms of finite order of semisimple Lie algebras. The remaining three talks were more topologically in nature. Guy Rousseau presented his theory of microaffine buildings, hovels, and Kac–Moody groups over ultrametric fields on Thursday. Thursday's fourth talk was by Bertrand Rémy on Satake compactifications of buildings via Berkovich theory. The conference was concluded by Pierre-Emmanuel Caprace's report on aspects of the structure of locally compact groups. We are particularly pleased by the lively interaction between the participants during the long afternoon breaks (each morning's lectures finished at 11.30 a.m. while the afternoon sessions only started at 4.20 p.m.) and during the evenings.

Isomorphisms of unitary forms of Kac–Moody groups over finite fields

We use the methods developed in [Pierre-Emmanuel Caprace, “Abstract” homomorphisms of split Kac–Moody groups, Mem. Amer. Math. Soc. 198 (2009); Pierre-Emmanuel Caprace, Bernhard Mühlherr, Isomorphisms of Kac–Moody groups, Invent. Math. 161 (2005) 361–388; Pierre-Emmanuel Caprace, Bernhard Mühlherr, Isomorphisms of Kac–Moody groups which preserve bounded subgroups, Adv. Math. 206 (2006) 250–278] to solve the isomorphism problem of unitary forms of infinite split Kac–Moody groups over finite fields of square order.

“Abstract” homomorphisms of split Kac-Moody groups

Cette these est consacree a une classe de groupes, appeles groupes de Kac-Moody, qui generalise de facon naturelle les groupes de Lie semi-simples, ou plus precisement, les groupes algebriques reductifs, dans un contexte infini-dimensionnel. On s'interesse plus particulierement au probleme d'isomorphismes pour ces groupes, en vue d'obtenir un analogue infini-dimensionnel de la celebre theorie des homomorphismes 'abstraits' de groupes algebriques simples, due a Armand Borel et Jacques Tits.Le probleme d'isomorphismes qu'on etudie s'avere etre un cas particulier d'un probleme plus general, qui consiste a caracteriser les homomorphismes de groupes algebriques vers les groupes de Kac-Moody, dont l'image est bornee. Ce probleme peut a son tour s'enoncer comme un probleme de rigidite pour les actions de groupes algebriques sur les immeubles, via l'action naturelle d'un groupe de Kac-Moody sur une paire d'immeubles jumeles. Les resultats partiels, relatifs a ce probleme de rigidite, que nous obtenons, nous permettent d'apporter une solution complete au probleme d'isomorphismes pour les groupes de Kac-Moody deployes.En particulier, on obtient un resultat de devissage pour les automorphismes de ces objets. Celui-ci fournit a son tour une description complete de la structure du groupe d'automorphismes d'un groupe de Kac-Moody deploye sur un corps de caracteristique~$0$.Nos arguments permettent egalement de traiter de facon analogue certaines formes anisotropes de groupes de Kac-Moody complexes, appelees formes unitaires. On montre en particulier que la topologie Hausdorff naturelle que portent ces formes est un invariant de leur structure de groupe abstrait. Ceci generalise un resultat bien connu de H. Freudenthal pour les groupes de Lie compacts.Enfin, l'on s'interesse aux homomorphismes de groupes de Kac-Moody a image fini-dimensionnelle, et l'on demontre la non-existence de tels homomorphismes a noyau central, lorsque le domaine est un groupe de Kac-Moody de type indefini sur un corps infini. Ceci reduit un probleme ouvert, dit probleme de linearite pour les groupes de Kac-Moody, au cas de corps de base finis.

Loop Groups and Twin Buildings

In these notes we describe some buildings related to complex Kac–Moody groups. First we describe the spherical building of SLn(ℂ) (i.e. the projective geometry PG(ℂn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field ℂ(z). Then we describe the same building in terms of complex Laurent polynomials, and introduce the Veronese representation, which is an equivariant embedding of the building into an affine Kac–Moody algebra. Next, we introduce topological twin buildings. These buildings can be used for a proof which is a variant of the proof by Quillen and Mitchell, of Bott periodicity which uses only topological geometry. At the end we indicate very briefly that the whole process works also for affine real almost split Kac–Moody groups.