Kato’s irreducibility criterion for Kac-Moody groups over local fields

Hébert, Auguste

doi:10.1090/ert/665

Kato’s irreducibility criterion for Kac-Moody groups over local fields
HTML articles powered by AMS MathViewer

by Auguste Hébert;

Represent. Theory 27 (2023), 1208-1227

DOI: https://doi.org/10.1090/ert/665

Published electronically: December 7, 2023

HTML | PDF | Request permission

Abstract:

In 2014, Braverman, Kazhdan, Patnaik and Bardy-Panse, Gaussent and Rousseau associated Iwahori-Hecke algebras to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we defined and studied their principal series representations. In 1982, Kato provided an irreducibility criterion for these representations, in the reductive case. We had obtained partially this criterion in the Kac-Moody case. In this paper, we prove this criterion in the Kac-Moody case.

References

Anders Björner and Francesco Brenti, Combinatorics of Coxeter groups, Graduate Texts in Mathematics, vol. 231, Springer, New York, 2005. MR 2133266
Nicole Bardy-Panse, Stéphane Gaussent, and Guy Rousseau, Iwahori-Hecke algebras for Kac-Moody groups over local fields, Pacific J. Math. 285 (2016), no. 1, 1–61. MR 3554242, DOI 10.2140/pjm.2016.285.1
Alexander Braverman and David Kazhdan, The spherical Hecke algebra for affine Kac-Moody groups I, Ann. of Math. (2) 174 (2011), no. 3, 1603–1642. MR 2846488, DOI 10.4007/annals.2011.174.3.5
Alexander Braverman, David Kazhdan, and Manish M. Patnaik, Iwahori-Hecke algebras for $p$-adic loop groups, Invent. Math. 204 (2016), no. 2, 347–442. MR 3489701, DOI 10.1007/s00222-015-0612-x
Vinay V. Deodhar, A note on subgroups generated by reflections in Coxeter groups, Arch. Math. (Basel) 53 (1989), no. 6, 543–546. MR 1023969, DOI 10.1007/BF01199813
Stéphane Gaussent and Guy Rousseau, Kac-Moody groups, hovels and Littelmann paths, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 7, 2605–2657 (English, with English and French summaries). MR 2498360, DOI 10.5802/aif.2423
Stéphane Gaussent and Guy Rousseau, Spherical Hecke algebras for Kac-Moody groups over local fields, Ann. of Math. (2) 180 (2014), no. 3, 1051–1087. MR 3245012, DOI 10.4007/annals.2014.180.3.5
Auguste Hébert, Decompositions of principal series representations of Iwahori-Hecke algebras for Kac-Moody groups over local fields, Pacific J. Math. 310 (2021), no. 2, 303–353. MR 4229241, DOI 10.2140/pjm.2021.310.303
Auguste Hébert, Principal series representations of Iwahori-Hecke algebras for Kac-Moody groups over local fields, Ann. Inst. Fourier (Grenoble) 72 (2022), no. 1, 187–259 (English, with English and French summaries). MR 4448595, DOI 10.5802/aif.3469
Victor G. Kac, Infinite-dimensional Lie algebras, Progress in Mathematics, vol. 44, Birkhäuser Boston, Inc., Boston, MA, 1983. An introduction. MR 739850, DOI 10.1007/978-1-4757-1382-4
Shin-ichi Kato, Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 929–943 (1982). MR 656065
Charles David Keys, On the decomposition of reducible principal series representations of $p$-adic Chevalley groups, Pacific J. Math. 101 (1982), no. 2, 351–388. MR 675406, DOI 10.2140/pjm.1982.101.351
Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, vol. 204, Birkhäuser Boston, Inc., Boston, MA, 2002. MR 1923198, DOI 10.1007/978-1-4612-0105-2
Hideya Matsumoto, Analyse harmonique dans les systèmes de Tits bornologiques de type affine, Lecture Notes in Mathematics, Vol. 590, Springer-Verlag, Berlin-New York, 1977 (French). MR 579177, DOI 10.1007/BFb0086707
R. V. Moody and A. Pianzola, On infinite root systems, Trans. Amer. Math. Soc. 315 (1989), no. 2, 661–696. MR 964901, DOI 10.1090/S0002-9947-1989-0964901-8
Mark Reeder, Nonstandard intertwining operators and the structure of unramified principal series representations, Forum Math. 9 (1997), no. 4, 457–516. MR 1457135, DOI 10.1515/form.1997.9.457
Bertrand Rémy, Groupes de Kac-Moody déployés et presque déployés, Astérisque 277 (2002), viii+348 (French, with English and French summaries). MR 1909671
Guy Rousseau, Groupes de Kac-Moody déployés sur un corps local II. Masures ordonnées, Bull. Soc. Math. France 144 (2016), no. 4, 613–692 (French, with English and French summaries). MR 3562609, DOI 10.24033/bsmf.2724
Guy Rousseau, Almost split Kac-Moody groups over ultrametric fields, Groups Geom. Dyn. 11 (2017), no. 3, 891–975. MR 3692902, DOI 10.4171/GGD/418
Allan J. Silberger, The Knapp-Stein dimension theorem for $p$-adic groups, Proc. Amer. Math. Soc. 68 (1978), no. 2, 243–246. MR 492091, DOI 10.1090/S0002-9939-1978-0492091-5
Jacques Tits, Uniqueness and presentation of Kac-Moody groups over fields, J. Algebra 105 (1987), no. 2, 542–573. MR 873684, DOI 10.1016/0021-8693(87)90214-6