Deligne–Lusztig duality and wonderful compactification
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We use geometry of the wonderful compactification to obtain a new proof of the relation between Deligne–Lusztig (or Alvis–Curtis) duality for p-adic groups and homological duality. This provides a new way to introduce an involution on the set of irreducible representations of the group which has been defined by A. Zelevinsky for \(G=GL(n)\) and by A.-M. Aubert in general (less direct geometric approaches to this duality have been developed earlier by Schneider-Stuhler and by the second author). As a byproduct, we describe the Serre functor for representations of a p-adic group.
Mathematics Subject Classification20G05 20G25 20J05 22E35
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We thank Vladimir Drinfeld for many helpful conversations over the years. The second author is also grateful to Michael Finkelberg, Leonid Rybnikov and Jonathan Wang for motivating discussions. The impetus for writing this note came from a talk given by the second author at the Higher School for Economics (Moscow), he thanks this institution for the stimulating opportunity. Finally, we thank Victor Ginzburg for an inspiring correspondence which has motivated Sect. 3.4. J.B. acknowledges partial support by the ERC Grant 291612, R.B. was partly supported by the NSF Grant DMS-1601953 and Russian Academic Excellence Project 5-100, D.K. was supported by an EPRC Grant, their collaboration was supported by the US-Israel BSF Grant 2016363.
- 2.Aubert, A.-M.: Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie dun groupe réductif \(p\)-adique, Trans. Am. Math. Soc. 347 (1995), no. 6, 2179–2189, erratum in: Trans. Am. Math. Soc. 348 (1996), no. 11, 4687–4690Google Scholar
- 3.Bernstein, J.: Le “centre” de Bernstein. In: Deligne, P. (ed.) Travaux en Cours, Representations of Reductive Groups Over a Local Field, pp. 1–32. Hermann, Paris (1984)Google Scholar
- 5.Bernstein, J.: Second adjointness for representations of \(p\)-adic groups, preprint, available at: http://www.math.tau.ac.il/~bernstei/Unpublished_texts/Unpublished_list.html
- 8.Bezrukavnikov, R., Kaledin, D.: McKay equivalence for symplectic resolutions of singularities, Tr. Mat. Inst. Steklova 246 (2004), Algebr. Geom. Metody, Svyazi i Prilozh., 20–42; translation in Proc. Steklov Inst. Math. 2004, no. 3 (246), 13–33Google Scholar
- 9.Bondal, A.I., Kapranov, M.M.: Representable functors, Serre functors, and mutations, Math. USSR-Izv. 35 (1990), no. 3, 519–541; translated from Izv. Akad. Nauk SSSR Ser. Mat. 53 (1989), no. 6, 1183–1205Google Scholar
- 13.Gaitsgory, D.: A “strange” functional equation for Eisenstein series and miraculous duality on the moduli stack of bundles. Ann. Sci. Ec. Norm. Sup. (4) 50 (5), 1123–1162 (2017)Google Scholar
- 14.Gromov, M.: Hyperbolic Groups. In: Gersten, S.M. (ed.) “Essays in Group Theory,” Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–263. Springer, New York (1987)Google Scholar