Geometric wave-front set may not be a singleton

Tsai, Cheng-Chiang

doi:10.1090/jams/1031

Geometric wave-front set may not be a singleton
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by Cheng-Chiang Tsai;

J. Amer. Math. Soc. 37 (2024), 281-304

DOI: https://doi.org/10.1090/jams/1031

Published electronically: August 15, 2023

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Abstract:

We show that the geometric wave-front set of specific half-integral-depth supercuspidal representations of ramified $p$-adic unitary groups is not a singleton.

References

Avraham Aizenbud, Dmitry Gourevitch, and Eitan Sayag, Irreducibility of wave-front sets for depth zero cuspidal representations, ArXiv e-prints (2022) https://arxiv.org/abs/2205.14695v1.
Dan Barbasch and Allen Moy, Local character expansions, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 5, 553–567 (English, with English and French summaries). MR 1474804, DOI 10.1016/S0012-9593(97)89931-4
Dan Ciubotaru, Lucas Mason-Brown, and Emile Okada, Wavefront sets of unipotent representations of reductive $p$-adic groups I, ArXiv e-prints (2021) https://arxiv.org/abs/2112.14354v5.
Dan Ciubotaru, Lucas Mason-Brown, and Emile Okada, The wavefront sets of unipotent supercuspidal representations, ArXiv e-prints (2022) https://arxiv.org/abs/2206.08628v2.
Tsao-Hsien Chen, Kari Vilonen, and Ting Xue, Springer correspondence for the split symmetric pair in type $A$, Compos. Math. 154 (2018), no. 11, 2403–2425. MR 3867304, DOI 10.1112/s0010437x18007443
Stephen Debacker, Homogeneity results for invariant distributions of a reductive $p$-adic group, Ann. Sci. École Norm. Sup. (4) 35 (2002), no. 3, 391–422 (English, with English and French summaries). MR 1914003, DOI 10.1016/S0012-9593(02)01094-7
Raul Gomez, Dmitry Gourevitch, and Siddhartha Sahi, Whittaker supports for representations of reductive groups, Ann. Inst. Fourier (Grenoble) 71 (2021), no. 1, 239–286 (English, with English and French summaries). MR 4275869, DOI 10.5802/aif.3372
Mark Goresky, Robert Kottwitz, and Robert MacPherson, Purity of equivalued affine Springer fibers, Represent. Theory 10 (2006), 130–146. MR 2209851, DOI 10.1090/S1088-4165-06-00200-7
Thomas C. Hales, Hyperelliptic curves and harmonic analysis (why harmonic analysis on reductive $p$-adic groups is not elementary), Representation theory and analysis on homogeneous spaces (New Brunswick, NJ, 1993) Contemp. Math., vol. 177, Amer. Math. Soc., Providence, RI, 1994, pp. 137–169. MR 1303604, DOI 10.1090/conm/177/01921
Harish-Chandra, Admissible invariant distributions on reductive $p$-adic groups, University Lecture Series, vol. 16, American Mathematical Society, Providence, RI, 1999. With a preface and notes by Stephen DeBacker and Paul J. Sally, Jr. MR 1702257, DOI 10.1090/ulect/016
Dihua Jiang, Baiying Liu, and Gordan Savin, Raising nilpotent orbits in wave-front sets, Represent. Theory 20 (2016), 419–450. MR 3564676, DOI 10.1090/ert/490
Dihua Jiang, Dongwen Liu, and Lei Zhang, Arithmetic wavefront sets and generic $l$-packets, ArXiv e-prints (2022) https://arxiv.org/abs/2207.04700v2.
Tasho Kaletha, Regular supercuspidal representations, J. Amer. Math. Soc. 32 (2019), no. 4, 1071–1170. MR 4013740, DOI 10.1090/jams/925
Ju-Lee Kim and Fiona Murnaghan, Character expansions and unrefined minimal $K$-types, Amer. J. Math. 125 (2003), no. 6, 1199–1234. MR 2018660, DOI 10.1353/ajm.2003.0043
Ju-Lee Kim and Fiona Murnaghan, K-types and $\Gamma$-asymptotic expansions, J. Reine Angew. Math. 592 (2006), 189–236. MR 2222734, DOI 10.1515/CRELLE.2006.027
G. Lusztig, A class of irreducible representations of a Weyl group, Nederl. Akad. Wetensch. Indag. Math. 41 (1979), no. 3, 323–335. MR 546372, DOI 10.1016/1385-7258(79)90036-2
G. Lusztig, Intersection cohomology complexes on a reductive group, Invent. Math. 75 (1984), no. 2, 205–272. MR 732546, DOI 10.1007/BF01388564
George Lusztig, Character sheaves. I-V, Adv. Math. (1985).
George Lusztig, A unipotent support for irreducible representations, Adv. Math. 94 (1992), no. 2, 139–179. MR 1174392, DOI 10.1016/0001-8708(92)90035-J
George Lusztig and Zhiwei Yun, $\mathbf {Z}/m$-graded Lie algebras and perverse sheaves, I, Represent. Theory 21 (2017), 277–321. MR 3697026, DOI 10.1090/ert/500
C. Mœglin, Front d’onde des représentations des groupes classiques $p$-adiques, Amer. J. Math. 118 (1996), no. 6, 1313–1346 (French, with French summary). MR 1420926, DOI 10.1353/ajm.1996.0051
C. Mœglin and J.-L. Waldspurger, Modèles de Whittaker dégénérés pour des groupes $p$-adiques, Math. Z. 196 (1987), no. 3, 427–452 (French). MR 913667, DOI 10.1007/BF01200363
Emile Takahiro Okada, The wavefront set over a maximal unramified field extension, ArXiv e-prints (2021) https://arxiv.org/abs/2107.10591v3.
Alexei Oblomkov and Zhiwei Yun, Geometric representations of graded and rational Cherednik algebras, Adv. Math. 292 (2016), 601–706. MR 3464031, DOI 10.1016/j.aim.2016.01.015
Modèle de Whittaker et caractères de représentations, Non-commutative harmonic analysis (Actes Colloq., Marseille-Luminy, 1974) Lecture Notes in Math., Vol. 466, Springer, Berlin-New York, 1975, pp. 151–171 (French). MR 393355
Mark Reeder and Jiu-Kang Yu, Epipelagic representations and invariant theory, J. Amer. Math. Soc. 27 (2014), no. 2, 437–477. MR 3164986, DOI 10.1090/S0894-0347-2013-00780-8
Toshiaki Shoji, Green functions of reductive groups over a finite field, The Arcata Conference on Representations of Finite Groups (Arcata, Calif., 1986) Proc. Sympos. Pure Math., vol. 47, Amer. Math. Soc., Providence, RI, 1987, pp. 289–301. MR 933366, DOI 10.1090/pspum/047.1/933366
T. A. Springer, Trigonometric sums, Green functions of finite groups and representations of Weyl groups, Invent. Math. 36 (1976), 173–207. MR 442103, DOI 10.1007/BF01390009
Cheng-Chiang Tsai, Inductive structure of Shalika germs and affine Springer fibers, ArXiv e-prints (2015) https://arxiv.org/abs/1512.00445.
Cheng-Chiang Tsai, A formula for certain Shalika germs of ramified unitary groups, Compos. Math. 153 (2017), no. 1, 175–213. MR 3622875, DOI 10.1112/S0010437X16007843
Kari Vilonen and Ting Xue, Character sheaves for classical symmetric pairs, Represent. Theory 26 (2022), 1097–1144. With an appendix by Dennis Stanton. MR 4497390, DOI 10.1090/ert/622
Jean-Loup Waldspurger, Représentations de réduction unipotente pour $\textrm {SO}(2n+1)$, III: exemples de fronts d’onde, Algebra Number Theory 12 (2018), no. 5, 1107–1171 (French, with English and French summaries). MR 3840872, DOI 10.2140/ant.2018.12.1107
Jean-Loup Waldspurger, Fronts d’onde des représentations tempérées et de réduction unipotente pour $\textrm {SO}(2n+1)$, Tunis. J. Math. 2 (2020), no. 1, 43–95 (French, with English and French summaries). MR 3933392, DOI 10.2140/tunis.2020.2.43