# zbMATH — the first resource for mathematics

Formulas for the dimensions of some affine Deligne-Lusztig varieties. (English) Zbl 1053.22010
Deligne-Lusztig varieties for semisimple algebraic groups $$G$$ parametrise parabolics with a given relative position with respect to their Frobenius transforms. Their cohomology realises important representations of the corresponding finite group $$G(k)$$ ($$k$$ a finite field). If one replaces $$G$$ by an algebraic loop group one can make an analogous definition, however, many important facts remain open, like non-emptiness or dimension. The paper under review studies the special cases $$\text{SL}(3)$$ and $$\text{Sp}(4)$$ ($$\text{SL}(2)$$ is also possible but too easy as a challenge), turning for these conjectures into theorems and ignorance into conjectures. The method used is a heavy dose of Bruhat-Tits theory.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 14L40 Other algebraic groups (geometric aspects)
##### Keywords:
loop groups; Deligne-Lusztig varieties
Full Text:
##### References:
 [1] F. Bruhat and J. Tits, Groupes réductifs sur un corps local, Inst. Hautes Études Sci. Publ. Math. 41 (1972), 5–251. · Zbl 0254.14017 · doi:10.1007/BF02715544 · numdam:PMIHES_1972__41__5_0 · eudml:103918 [2] ——, Groupes réductifs sur un corps local II, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376. [3] P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), 103–161. JSTOR: · Zbl 0336.20029 · doi:10.2307/1971021 · links.jstor.org [4] P. Garrett, Building and classical groups, Chapman & Hall, London, 1997. · Zbl 0933.20019 [5] D. Kazhdan and G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math 62 (1988), 129–168. · Zbl 0658.22005 · doi:10.1007/BF02787119 [6] R. Kottwitz, Orbital integrals on $$\text \mathrm GL_3,$$ Amer. J. Math. 102 (1980), 327–384. · Zbl 0437.22011 · doi:10.2307/2374243 [7] ——, Isocrystals with additional structure, Compositio Math. 56 (1985), 201–220. · Zbl 0597.20038 · numdam:CM_1985__56_2_201_0 · eudml:89735 [8] ——, Isocrystals with additional structure II, Compositio Math. 109 (1997), 255–339. · Zbl 0966.20022 · doi:10.1023/A:1000102604688 [9] R. Kottwitz and M. Rapoport, On the existence of $$F$$ -crystals, Comment. Math. Helv. 78 (2003), 153–184. · Zbl 1126.14023 · doi:10.1007/s000140300007 [10] G. Lusztig, Singularities, characters, formulas, and a q-analog of weight multiplicities, Astérisque 101/102 (1983), 208–229. · Zbl 0561.22013 [11] Y. Manin, The theory of commutative formal groups over fields of finite characteristic, Russian Math. Surveys 18 (1963), 1–81. · Zbl 0128.15603 · doi:10.1070/rm1963v018n06ABEH001142 [12] M. Rapoport, A positivity property of the Satake isomorphism, Manuscripta Math. 101 (2000), 153–166. · Zbl 0941.22006 · doi:10.1007/s002290050010 [13] ——, A guide to the reduction modulo p of Shimura varieties, preprint, · Zbl 1084.11029 · smf.emath.fr [14] D. Reuman, Determining whether certain affine Deligne–Lusztig sets are empty, preprint, [15] J.-P. Winterberger, Existence de F-cristaux avec structures supplémentaires,
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.