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Kac-Moody groups, hovels and Littelmann paths. (English) Zbl 1161.22007
The authors give the definition of a kind of building \(\mathcal{I}\) for a symmetrizable Kac-Moody group over a field \(K\) endowed with a discrete valuation and with a residue field containing \(\mathbb{C}\), which they call a hovel, due to its pathological behaviour. This concept allows them to generalize some previous results obtained by themselves, separately, and in a joint paper of St. Gaussent and P. Littelmann [Duke Math. J. 127, 35–88 (2005; Zbl 1078.22007)], by introducing a new set \({\mathcal I}\) associated with the Kac-Moody group \(G\) over \(K = {\mathbb C}((t))\), or, even more generally, over any discretely valuated field \(K\) with residue field containing \({\mathbb C}\). By using that concept they get good generalizations of Gaussent-Littelmann’s results and obtain a new result which is an analogue of some previous ones in the semisimple case. In the two last sections they give the definitions of LS paths, Hecke paths, dual dimension and codimension, and they prove some characterizations of them.

MSC:
22E46 Semisimple Lie groups and their representations
20G05 Representation theory for linear algebraic groups
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
20E42 Groups with a \(BN\)-pair; buildings
51E24 Buildings and the geometry of diagrams
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References:
[1] Bardy, Nicole, Systèmes de racines infinis, Mém. Soc. Math. Fr. (N.S.), 65, vi+188 pp., (1996) · Zbl 0880.17019
[2] Brown, Kenneth S., Buildings, (1989), Springer · Zbl 0715.20017
[3] Bruhat, F.; Tits, J., Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math., 41, 5-251, (1972) · Zbl 0254.14017
[4] Garland, Howard, A Cartan decomposition for p-adic loop groups, Math. Ann., 302, 151-175, (1995) · Zbl 0837.22013
[5] Gaussent, Stéphane; Littelmann, Peter, LS galleries, the path model and MV cycles, Duke Math. J., 127, 35-88, (2005) · Zbl 1078.22007
[6] Kac, Victor G., Infinite dimensional Lie algebras, (1990), Cambridge University Press · Zbl 0716.17022
[7] Kac, Victor G.; Peterson, Dale H., Élie Cartan et les mathématiques d’aujourd’hui, Lyon (1984), Defining relations of certain infinite dimensional groups, 165-208, (1985), Astérisque n o hors série · Zbl 0625.22014
[8] Kapovich, Misha; Millson, John J., A path model for geodesics in Euclidean buildings and its applications to representation theory, Geometry, Groups and Dynamics, 2, 3, 405-480, (2008) · Zbl 1147.22011
[9] Kumar, Shrawan, Kac-Moody groups, their flag varieties and representation theory, (2002), Progress in Math. 204 Birkhäuser · Zbl 1026.17030
[10] Littelmann, Peter, A Littlewood-Richardson rule for symmetrizable Kac-Moody algebras, Invent. Math., 116, 1-3, 329-346, (1994) · Zbl 0805.17019
[11] Littelmann, Peter, Paths and root operators in representation theory, Annals of Math., 142, 499-525, (1995) · Zbl 0858.17023
[12] Littelmann, Peter, Algebraic groups and their representations (Cambridge, 1997), 517, The path model, the quantum Frobenius map and standard monomial theory, 175-212, (1998), Kluwer Acad. Publ., Dordrecht · Zbl 0938.14031
[13] Mirković, Ivan; Vilonen, Kari, Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett., 7, 13-24, (2000) · Zbl 0987.14015
[14] Moody, Robert; Pianzola, Arturo, Lie algebras with triangular decompositions, (1995), Wiley-Interscience, New York · Zbl 0874.17026
[15] Rémy, Bertrand, Groupes de Kac-Moody déployés et presque déployés, Astérisque, 277, viii+348 pp., (2002) · Zbl 1001.22018
[16] Ronan, Mark A., Lectures on buildings, (1989), Perspectives in Math. 7 Academic Press · Zbl 0694.51001
[17] Rousseau, Guy, Groupes de Kac-Moody déployés sur un corps local, immeubles microaffines, Compositio Mathematica, 142, 501-528, (2006) · Zbl 1094.22003
[18] Rousseau, Guy; Bessières, L.; Parreau, A.; Rémy, B., Géométries à courbure négative ou nulle, groupes discrets et rigidité, Grenoble 2004, 18, Euclidean buildings, 77-116, (2008), Séminaires et congrès, Soc. Math. France
[19] Rousseau, Guy, Groupes de Kac-Moody déployés sur un corps local 2, masures ordonnées, (2008) · Zbl 1401.20055
[20] Rousseau, Guy, Masures affines, (2008)
[21] Tits, Jacques; Rosati, L. A., Buildings and the geometry of diagrams, Como (1984), 1181, Immeubles de type affine, 159-190, (1986), Lecture Notes in Math. · Zbl 0611.20026
[22] Tits, Jacques, Uniqueness and presentation of Kac-Moody groups over fields, J. of Algebra, 105, 542-573, (1987) · Zbl 0626.22013
[23] Tits, Jacques; Liebeck, M.; Saxl, J., Groups combinatorics and geometry, Durham (1990), 165, Twin buildings and groups of Kac-Moody type, 249-286, (1992), London Math. Soc. lecture note · Zbl 0851.22023
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