Geometric structure in smooth dual and local Langlands conjecture
Tóm tắt
This expository paper first reviews some basic facts about p-adic fields, reductive p-adic groups, and the local Langlands conjecture. If G is a reductive p-adic group, then the smooth dual of G is the set of equivalence classes of smooth irreducible representations of G. The representations are on vector spaces over the complex numbers. In a canonical way, the smooth dual is the disjoint union of subsets known as the Bernstein components. According to a conjecture due to ABPS (Aubert–Baum–Plymen–Solleveld), each Bernstein component has a geometric structure given by an appropriate extended quotient. The paper states this ABPS conjecture and then indicates evidence for the conjecture, and its connection to the local Langlands conjecture.
Tài liệu tham khảo
Arthur J.: A note on L-packets. Pure Appl. Math. Q. 2, 199–217 (2006)
J. Arthur, The Endoscopic Classification of Representations: Orthogonal and Symplectic Groups, Amer. Math. Soc. Colloq. Publ., 61, Amer. Math. Soc., Providence, RI, 2013.
A.-M. Aubert, P. Baum and R.J. Plymen, The Hecke algebra of a reductive p-adic group: a geometric conjecture, In: Noncommutative Geometry and Number Theory, (eds. C. Consani and M. Marcolli), Aspects Math., 37, Vieweg Verlag, Wiesbaden, 2006, pp. 1–34.
Aubert A.-M, Baum P, Plymen R.J: Geometric structure in the representation theory of p-adic groups. C. R. Math. Acad. Sci. Paris 345, 573–578 (2007)
Aubert A.-M, Baum P, Plymen R.J: Geometric structure in the principal series of the p-adic group G 2. Represent. Theory 15, 126–169 (2011)
A.-M. Aubert, P. Baum and R.J. Plymen, Geometric structure in the representation theory of reductive p-adic groups. II, In: Harmonic Analysis on Reductive p-adic Groups, (eds. R.S. Doran, P.J. Sally, Jr. and L. Spice), Contemp. Math., 543, Amer. Math. Soc., Providence, RI, 2011, pp. 71–90.
A.-M. Aubert, P. Baum, R.J. Plymen and M. Solleveld, Geometric structure and the local Langlands conjecture, preprint, arXiv:1211.0180v2.
A.-M. Aubert, P. Baum, R.J. Plymen and M. Solleveld, The local Langlands correspondence for inner forms of SL n , preprint, arXiv:1305.2638.
A.-M. Aubert, P. Baum, R.J. Plymen and M. Solleveld, On the local Langlands correspondence for non-tempered representations, Münster J. Math., to appear.
Baum P, Nistor V: Periodic cyclic homology of Iwahori–Hecke algebras. C. R. Acad. Sci. Paris Sér. I Math. 332, 783–788 (2001)
Baum P, Nistor V: Periodic cyclic homology of Iwahori–Hecke algebras.. K-Theory 27, 329–357 (2002)
J.N. Bernstein, Le centre de Bernstein. Edited by P. Deligne, In: Representations of Reductive Groups Over a Local Field, Travaux en Cours, Hermann, Paris, 1984, pp. 1–32.
J.N. Bernstein and S. Gelbart, An Introduction to the Langlands Program, Birkhäuser, 2003.
J.N. Bernstein and K.E. Rumelhart, Representations of p-adic Groups, Harvard Mathematics Department Lecture Notes, 1993.
A. Borel, Automorphic L-functions, In: Automorphic Forms, Representations and L-Functions, Proc. Sympos. Pure Math., 33, Amer. Math. Soc., Providence, RI, 1979, pp. 27–61.
J. Dixmier, C*-Algebras, North-Holland, 1977.
M. Harris and R. Taylor, The Geometry and Cohomology of Some Simple Shimura Varieties, Ann. of Math. Stud., 151, Princeton Univ. Press, Princeton, NJ, 2001.
Henniart G.: Une preuve simple de conjectures de Langlands pour GL(n) sur un corps p-adique. Invent. Math. 139, 439–455 (2000)
K. Hiraga and H. Saito, On L-packets for Inner Forms of SL n , Mem. Amer. Math. Soc., 215, Amer. Math. Soc., Providence, RI, 2012.
K. Iwasawa, Local Class Field Theory, Oxford Sci. Publ., Oxford Math. Monogr., Clarendon Press, Oxford Univ. Press, New York, 1986.
Kato S.-I.: A realization of irreducible representations of affine Weyl groups. Nederl. Akad. Wetensch. Indag. Math. 45, 193–201 (1983)
Kazhdan D, Lusztig G: Proof of the Deligne–Langlands conjecture for Hecke algebras. Invent. Math. 87, 153–215 (1987)
Kazhdan D, Nistor V, Schneider P: Hochschild and cyclic homology of finite type algebras. Selecta Math. (N.S.) 4, 321–359 (1998)
R.P. Langlands, Problems in the theory of automorphic forms, In: Lectures in Modern Analysis and Applications. III, Lecture Notes in Math., 170, Springer-Verlag, 1970, pp. 18–61.
Lusztig G.: Some examples of square integrable representations of semisimple p-adic groups. Trans. Amer. Math. Soc. 277, 623–653 (1983)
Lusztig G.: Cells in affine Weyl groups. III. J. Fac. Sci. Univ. Tokyo Sect IA Math. 34, 223–243 (1987)
Lusztig G.: Cells in affine Weyl groups. IV. J. Fac. Sci. Univ. Tokyo Sect IA Math. 36, 297–328 (1989)
Lusztig G.: Representations of affine Hecke algebras. Astérisque. 171-172, 73–84 (1989)
C.P. Mok, Endoscopic classification of representations of quasi-split unitary groups, preprint, arXiv:1206.0882; Mem. Amer. Math. Soc., to appear.
Reeder M.: Isogenies of Hecke algebras and a Langlands correspondence for ramified principal series representations. Represent. Theory 6, 101–126 (2002)
D. Renard, Représentations des groupes réductifs p-adiques, Cours Spec., 17, Soc. Math. France, Paris, 2010.
Scholze P.: The local Langlands correspondence for GL n over p-adic fields. Invent. Math. 192, 663–715 (2013)
Solleveld M.: On the classification of irreducible representations of affine Hecke algebras with unequal parameters. Represent. Theory 16, 1–87 (2012)
D.A. Vogan, Jr., The local Langlands conjecture, In: Representation Theory of Groups and Algebras, Contemp. Math., 145, Amer. Math. Soc., Providence, RI, 1993, pp. 305–379.
Zelevinsky A.V.: Induced representations of reductive p-adic groups. II. On irreducible representations of GL(n). Ann. Sci. École Norm. Sup. 13(4), 165–210 (1980)