×

Ideal triangles in Euclidean buildings and branching to Levi subgroups. (English) Zbl 1272.20033

Let \(\underline G\) be a connected reductive group. The authors study connections between the (spherical) Hecke ring of \(\underline G\) and the representation ring of the Langlands dual group for \(\underline G\).

MSC:

20E42 Groups with a \(BN\)-pair; buildings
20G15 Linear algebraic groups over arbitrary fields
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Beilinson, A.; Drinfeld, V., Quantization of Hitchinʼs integrable system and Hecke eigensheaves, preprint, available at:
[2] Belkale, P.; Kumar, S., Eigencone, saturation and Horn problems for symplectic and odd orthogonal groups, J. Algebraic Geom., 19, 199-242 (2010) · Zbl 1233.20040
[3] Beauville, A.; Laszlo, Y., Conformal blocks and generalized theta function, Comm. Math. Phys., 164, 385-419 (1994) · Zbl 0815.14015
[4] Berenstein, A.; Sjamaar, R., Coadjoint orbits, moment polytopes, and the Hilbert-Mumford criterion, J. Amer. Math. Soc., 13, 2, 433-466 (2000) · Zbl 0979.53092
[5] Bruhat, F.; Tits, J., Groupes réductifs sur un corps local. I, Inst. Hautes Études Sci. Publ. Math., 41, 5-251 (1972) · Zbl 0254.14017
[6] Foth, P., Generalized Kostant convexity theorems, Proc. Amer. Math. Soc., 137, 1, 297-301 (2009) · Zbl 1163.53054
[7] Haines, T. J., Structure constants for Hecke and representations rings, Int. Math. Res. Not. IMRN, 39, 2103-2119 (2003) · Zbl 1071.22020
[8] Haines, T. J., Equidimensionality of convolution morphisms and applications to saturation problems, Adv. Math., 207, 1, 297-327 (2006) · Zbl 1161.20043
[9] Görtz, U.; Haines, T.; Kottwitz, R.; Reuman, D., Dimension of some affine Deligne-Lusztig varieties, Ann. Sci. École Norm. Sup. (4), 39, 467-511 (2006) · Zbl 1108.14035
[10] Görtz, U.; Haines, T.; Kottwitz, R.; Reuman, D., Affine Deligne-Lusztig varieties in affine flag varieties, Compos. Math., 146, 1339-1382 (2010) · Zbl 1229.14036
[11] Haines, T.; Ngô, B. C., Alcoves associated to special fibers of local models, Amer. J. Math., 124, 6, 1125-1152 (2002) · Zbl 1047.20037
[12] Kapovich, M.; Kumar, S.; Millson, J., The eigencone and saturation for Spin(8), Pure Appl. Math. Q., 5, 2, 755-780 (2009), (Hirzebruch special issue, Part 1) · Zbl 1188.20045
[13] Kapovich, M.; Leeb, B.; Millson, J., Convex functions on symmetric spaces, side lengths of polygons and the stability inequalities for weighted configurations at infinity, J. Differential Geom., 81, 297-354 (2009) · Zbl 1167.53044
[14] Kapovich, M.; Leeb, B.; Millson, J., Polygons in buildings and their refined side lengths, Geom. Funct. Anal., 19, 1081-1100 (2009) · Zbl 1205.53037
[15] Kapovich, M.; Leeb, B.; Millson, J., The generalized triangle inequalities in symmetric spaces and buildings with applications to algebra, Mem. Amer. Math. Soc., 192 (2008) · Zbl 1140.22009
[16] Kapovich, M.; Millson, J., Structure of the tensor product semigroup, Asian J. Math., 10, 3, 493-540 (2006), (S.S. Chern memorial issue) · Zbl 1108.22010
[17] Kapovich, M.; Millson, J., A path model for geodesics in Euclidean buildings and applications to representation theory, Groups Geom. Dyn., 2, 405-480 (2008) · Zbl 1147.22011
[18] Kottwitz, R.; Rapoport, M., Minuscule alcoves for \(G L_n\) and \(G S p_{2 n}\), Manuscripta Math., 102, 4, 403-428 (2000) · Zbl 0981.17003
[19] Littelmann, P., Characters of representations and paths in \(h_R^\ast \), (Representation Theory and Automorphic Forms. Representation Theory and Automorphic Forms, Proc. Sympos. Pure Math., vol. 61 (1997), American Mathematical Society: American Mathematical Society Rhode Island), 29-49 · Zbl 0892.17010
[20] Mirkovic, I.; Vilonen, K., Perverse sheaves on affine Grassmannians and Langlands duality, Math. Res. Lett., 7, 1, 13-24 (2000) · Zbl 0987.14015
[21] Mirkovic, I.; Vilonen, K., Geometric Langlands duality and representations of algebraic groups over commutative rings (2004), preprint · Zbl 1138.22013
[22] Ngô, B. C.; Polo, P., Résolutions de Demazure affines et formule de Casselman-Shalika géométrique, J. Algebraic Geom., 10, 3, 515-547 (2001) · Zbl 1041.14002
[23] Sam, S., Symmetric quivers, invariant theory, and saturation theorems for the classical groups (2010)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.