Cocenters and representations of pro-𝑝 Hecke algebras

He, Xuhua; Nie, Sian

doi:10.1090/ert/498

Cocenters and representations of pro-$p$ Hecke algebras
HTML articles powered by AMS MathViewer

by Xuhua He and Sian Nie

Represent. Theory 21 (2017), 82-105

DOI: https://doi.org/10.1090/ert/498

Published electronically: June 23, 2017

PDF | Request permission

Abstract:

In this paper, we study the relation between the cocenter $\overline {{\tilde {\mathcal H}}}$ and the representations of an affine pro-$p$ Hecke algebra ${\tilde {\mathcal H}}={\tilde {\mathcal H}}(0, -)$. As a consequence, we obtain a new criterion on supersingular representations: a (virtual) representation of ${\tilde {\mathcal H}}$ is supersingular if and only if its character vanishes on the non-supersingular part of the cocenter $\overline {\tilde {\mathcal H}}$.

References

N. Abe, Mod $p$ parabolic induction for Pro-$p$-Iwahori Hecke algebra, arXiv:1406.1003.
Colin J. Bushnell and Philip C. Kutzko, Smooth representations of reductive $p$-adic groups: structure theory via types, Proc. London Math. Soc. (3) 77 (1998), no. 3, 582–634. MR 1643417, DOI 10.1112/S0024611598000574
W. Casselman, Characters and Jacquet modules, Math. Ann. 230 (1977), no. 2, 101–105. MR 492083, DOI 10.1007/BF01370657
D. Ciubotaru and X. He, Cocenters and representations of affine Hecke algebra, arXiv:1409.0902, to appear in JEMS.
P. Deligne and G. Lusztig, Representations of reductive groups over finite fields, Ann. of Math. (2) 103 (1976), no. 1, 103–161. MR 393266, DOI 10.2307/1971021
Meinolf Geck, Sungsoon Kim, and Götz Pfeiffer, Minimal length elements in twisted conjugacy classes of finite Coxeter groups, J. Algebra 229 (2000), no. 2, 570–600. MR 1769289, DOI 10.1006/jabr.1999.8253
M. Geck and G. Pfeiffer, On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), no. 1, 79–94.
Xuhua He, Geometric and homological properties of affine Deligne-Lusztig varieties, Ann. of Math. (2) 179 (2014), no. 1, 367–404. MR 3126571, DOI 10.4007/annals.2014.179.1.6
Xuhua He and Sian Nie, Minimal length elements of extended affine Weyl groups, Compos. Math. 150 (2014), no. 11, 1903–1927. MR 3279261, DOI 10.1112/S0010437X14007349
Xuhua He, Centers and cocenters of 0-Hecke algebras, Representations of reductive groups, Progr. Math., vol. 312, Birkhäuser/Springer, Cham, 2015, pp. 227–240. MR 3495798, DOI 10.1007/978-3-319-23443-4_{8}
Xuhua He and Sian Nie, Minimal length elements of finite Coxeter groups, Duke Math. J. 161 (2012), no. 15, 2945–2967. MR 2999317, DOI 10.1215/00127094-1902382
Xuhua He and Sian Nie, Minimal length elements of extended affine Weyl groups, Compos. Math. 150 (2014), no. 11, 1903–1927. MR 3279261, DOI 10.1112/S0010437X14007349
X. He and S. Nie, Cocenters and representations of affine $0$-Hecke algebras, arXiv:1502.02184.
N. Iwahori and H. Matsumoto, On some Bruhat decomposition and the structure of the Hecke rings of ${\mathfrak {p}}$-adic Chevalley groups, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 5–48. MR 185016, DOI 10.1007/BF02684396
George Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), no. 3, 599–635. MR 991016, DOI 10.1090/S0894-0347-1989-0991016-9
Rachel Ollivier, Parabolic induction and Hecke modules in characteristic $p$ for $p$-adic $\textrm {GL}_n$, Algebra Number Theory 4 (2010), no. 6, 701–742. MR 2728487, DOI 10.2140/ant.2010.4.701
Rachel Ollivier, Compatibility between Satake and Bernstein isomorphisms in characteristic $p$, Algebra Number Theory 8 (2014), no. 5, 1071–1111. MR 3263136, DOI 10.2140/ant.2014.8.1071
Timo Richarz, On the Iwahori Weyl group, Bull. Soc. Math. France 144 (2016), no. 1, 117–124. MR 3481263, DOI 10.24033/bsmf.2708
Marie-France Vignéras, Induced $R$-representations of $p$-adic reductive groups, Selecta Math. (N.S.) 4 (1998), no. 4, 549–623. MR 1668044, DOI 10.1007/s000290050040
Marie-France Vignéras, Erratum: “Pro-$p$-Iwahori Hecke ring and supersingular $\overline {\mathbf F}_p$-representations” [Math. Ann. 331 (2005), no. 3, 523–556; MR2122539], Math. Ann. 333 (2005), no. 3, 699–701. MR 2198804, DOI 10.1007/s00208-005-0679-6
Marie-France Vigneras, The pro-$p$-Iwahori Hecke algebra of a reductive $p$-adic group I, Compos. Math. 152 (2016), no. 4, 693–753. MR 3484112, DOI 10.1112/S0010437X15007666
M.-F. Vignéras, The pro-$p$-Iwahori-Hecke algebra of a reductive $p$-adic group III, J. Inst. Math. Jussieu (2015), 1–38.
M.-F. Vignéras, The pro-$p$-Iwahori-Hecke algebra of a reductive $p$-adic group IV (Levi subgroups and central extensions), preprint.
Marie-France Vignéras, The pro-$p$ Iwahori Hecke algebra of a reductive $p$-adic group, V (parabolic induction), Pacific J. Math. 279 (2015), no. 1-2, 499–529. MR 3437789, DOI 10.2140/pjm.2015.279.499