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D. Luna Institut Fourier Grenoble, France d.luna@wanadoo.fr

The mo del wonderful variety
A homogeneous space (under a reductive group G) is called a "model homogeneous space", if it is quasi-affine, and if its algebra of regular functions contains every irreducible representation of G exactly once. In [G-Z 84], [G-Z 85] and [A-H-V 98] are given examples of model homogeneous spaces for classical groups and for the simple group of type E 8. We will explain the complete classification of model homogeneous for all semi-simple connected groups, obtained in [Lu 07]. For every such group G, there exists a unique wonderful variety MG (called the "model wonderful variety" of G) whose orbits parametrize the isomorphisms classes of model homogeneous spaces, in the following way : for every x MG , the homogeneous space G/(Gx )# is a model homogeneous space (where (Gx )# denotes the kernel of all characters of Gx , the isotropy group of G at x), and every model homogeneous spaces of G is obtained in this way, up to isomorphism. We will give details of the (surprisingly varied) structure of MG for the different simple groups G, and will explain the relation with the results of Gel'fand­Zelevisky and Adams­ Huang­Vogan mentioned above. We will also mention the relation between our results and recent papers of Brion [Br 06] and [B-CF 06]. Finally, we will explain the ideas of the proof, which uses the combinatorial theory of "spherical systems" and their "augmentations", and talk about the role wonderful varieties play in problems concerning spherical varieties. References [A-H-V 98] Adams, J., Huang J.-S., Vogan, D. : Functions on the model orbit in E8 , Represent. Theory 2 (1998), 224-263 [B-CF 06] Bravi, P., Cupit-Foutou S. : Equivariant deformations of the affine multicone over a flag variety, 30 pages, math.AG/0603690 [Br 07] Brion, M. : The total coordinate ring of a wonderful variety, J. of Algebra, Vol. 313, Special Issue in Honor of Ernest Vinberg (2007), 61-99 [G-Z 84] Gelfand, I. M., Zelevinski A. V. : Models of representations of classical groups and their hidden symmetries, Funct. Anal. and Appl., t.18, n-3, 14-31 (1984) [G-Z 85] Gelfand, I. M., Zelevinski A. V. : Representation models for classical groups and their higher symmetries, SMF, Astґ erisque, hors sґ erie, 117-128 (1985) [Lu 07] Luna, D. : La variґ ґ magnifique mod` J. of Algebra, Vol. 313, Special Issue ete ele, in Honor of Ernest Vinberg (2007), 292-319

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