Startseite Generic cuspidal representations of 𝑈(2, 1)
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

Generic cuspidal representations of 𝑈(2, 1)

  • Santosh Nadimpalli ORCID logo EMAIL logo
VerÜffentlicht/Copyright: 21. März 2021
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 33 Heft 3

Abstract

Let 𝐹 be a non-Archimedean local field, and let 𝜎 be a non-trivial Galois involution with fixed field F0. When the residue characteristic of F0 is odd, using the construction of cuspidal representations of classical groups by Stevens, we classify generic cuspidal representations of U⁢(2,1)⁢(F/F0).

Keywords: Generic representations; cuspidal representations; unitary group in 3 variables; Bushnell–Kutzko types
MSC 2010: 22E50; 11F70

Funding source: Nederlandse Organisatie voor Wetenschappelijk Onderzoek

Award Identifier / Grant number: 639.032.528

Funding statement: The author was supported by the NWO Vidi grant “A Hecke algebra approach to the local Langlands correspondence” (nr. 639.032.528).

Appendix A Appendix: Filtration of Uder induced by lattice sequences

In this section, we fix some representatives for 𝐺-conjugacy classes of self-dual lattice sequences on 𝑉 and describe an⁢(Λ) for n∈Z. Then we use them to determine Uder∩an⁢(Λ) for n∈Z. These calculations are used in showing certain representations are non-generic.

A.1 The unramified case

We begin with the case where F/F0 is unramified. Let Λ1 be the lattice sequence of periodicity 2 and

Λ1⁢(-1)=Λ1⁢(0)=oF⁢e1⊕oF⁢e0⊕oF⁢e-1.

The filtration {an⁢(Λ)∣n∈Z} of EndF⁡(V) is given by

(A.1)a2⁢m-1⁢(Λ1)=ϖm⁢(oFoFoFoFoFoFoFoFoF)∩g and a2⁢m⁢(Λ1)=ϖm⁢(oFoFoFoFoFoFoFoFoF)∩g

for all m∈Z. Let Λ2 be a period 2 lattice sequence given by

Λ2⁢(0)=oF⁢e1⊕oF⁢e0⊕pF⁢e-1 and Λ2⁢(1)=oF⁢e1⊕pF⁢e0⊕pF⁢e-1.

The filtration {an⁢(Λ2)∣n∈Z} is given by

(A.2)a2⁢m⁢(Λ2)=ϖm⁢(oFoFpF-1pFoFoFpFpFoF)∩g and a2⁢m+1⁢(Λ2)=ϖm⁢(pFoFoFpFpFoFpF2pFpF)∩g

for all m∈Z. Let Λ3 be the lattice sequence of period 4 given by

Λ3⁢(-1)=oF⁢e1⊕oF⁢e0⊕oF⁢e-1,Λ3⁢(0)=oF⁢e1⊕oF⁢e0⊕pF⁢e-1,
Λ3⁢(1)=oF⁢e1⊕pF⁢e0⊕pF⁢e-1,Λ3⁢(2)=pF⁢e1⊕pF⁢e0⊕pF⁢e-1.
The filtration {an⁢(Λ3)∣n∈Z} on 𝔤 is given by

(A.3)a4⁢m+r⁢(Λ3)={ϖm⁢(oFoFoFpFoFoFpFpFoF)∩gif⁢r=0,ϖm⁢(pFoFoFpFpFoFpFpFpF)∩gif⁢r=1,ϖm⁢(pFpFoFpFpFpFpFpFpF)∩gif⁢r=2,ϖm⁢(pFpFpFpFpFpFpF2pFpF)∩gif⁢r=3.

Although, there is a lattice sequence, say Λ4, with period 6, we do not need to write it down explicitly. This corresponds to type (A) strata, and in this case, all representations are generic. The filtration {Uder∩an⁢(Λ1)∣n∈Z} is given by

Uder∩a2⁢m-1⁢(Λ1)=Uder∩a2⁢m⁢(Λ1)=Uder⁢(m)

for m∈Z. The filtration {Uder∩an⁢(Λ2)∣n∈Z} is given by

Uder∩a2⁢m⁢(Λ2)=Uder⁢(m-1) and Uder∩a2⁢m+1⁢(Λ2)=Uder⁢(m).

The filtration Λ3 is given by

Uder∩a4⁢m+r⁢(Λ2)={Uder⁢(m)if⁢r=0,Uder⁢(m)if⁢r=1,Uder⁢(m)if⁢r=2,Uder⁢(m+1)if⁢r=3.

A.2 The ramified case

Now, assume that F/F0 is a ramified extension, and let Λ1 and Λ2 be the lattice sequences of period 2 given by

Λ1⁢(-1)=Λ⁢(0)=oF⁢e1⊕oF⁢e0⊕oF⁢e-1,
Λ2⁢(0)=oF⁢e1⊕oF⁢e0⊕pF⁢e-1 and Λ2⁢(1)=oF⁢e1⊕pF⁢e0⊕pF⁢e-1.
The filtration {an⁢(Λ1)∣n∈Z} is similar to the filtration in (A.1). The filtration {an⁢(Λ2)∣n∈Z}, in this case, is similar to the filtration in (A.2). We will not require to write the filtrations {an⁢(Λ′)∣n∈Z} for which P0⁢(Λ′) is an Iwahori subgroup of 𝐺. The filtration {Uder∩an⁢(Λ1)∣n∈Z} is given by

Uder∩a2⁢m-1⁢(Λ1)=Uder∩a2⁢m⁢(Λ1)=Uder⁢([m/2])

for all m∈Z. The filtration {Uder∩an⁢(Λ2)∣n∈Z} is given by

Uder∩a2⁢m-1⁢(Λ2)=Uder∩a2⁢m⁢(Λ2)=Uder⁢([(m-1)/2])

for any m∈Z.

Acknowledgements

I want to thank Maarten Solleveld for many useful discussions and clarifications during the course of this work. I want to thank Shaun Stevens for answering some questions on his paper, for his interest, and other clarifications. I want to thank Peter Badea for suggesting some useful references. I want to thank Kam-Fai Geo Tam for helpful discussions.

  1. Communicated by: Freydoon Shahidi

References

[1] I. N. Bernšteĭn and A. V. Zelevinskiĭ, Representations of the group G⁢L⁢(n,F), where 𝐹 is a local non-Archimedean field, Uspehi Mat. Nauk 31 (1976), no. 3(189), 5–70. 10.1070/RM1976v031n03ABEH001532Suche in Google Scholar

[2] L. Blasco, Description du dual admissible de U⁢(2,1)⁢(F) par la théorie des types de C. Bushnell et P. Kutzko, Manuscripta Math. 107 (2002), no. 2, 151–186. 10.1007/s002290100231Suche in Google Scholar

[3] L. Blasco, Types, paquets et changement de base: l’exemple de U⁢(2,1)⁢(F0). I. Types simples maximaux et paquets singletons, Canad. J. Math. 60 (2008), no. 4, 790–821. 10.4153/CJM-2008-035-6Suche in Google Scholar

[4] C. Blondel and S. Stevens, Genericity of supercuspidal representations of 𝑝-adic Sp4, Compos. Math. 145 (2009), no. 1, 213–246. 10.1112/S0010437X08003849Suche in Google Scholar

[5] P. Broussous and S. Stevens, Buildings of classical groups and centralizers of Lie algebra elements, J. Lie Theory 19 (2009), no. 1, 55–78. Suche in Google Scholar

[6] C. J. Bushnell and P. C. Kutzko, The Admissible Dual of GL⁢(N) via Compact Open Subgroups, Ann. of Math. Stud. 129, Princeton University, Princeton, 1993. 10.1515/9781400882496Suche in Google Scholar

[7] S. DeBacker and M. Reeder, Depth-zero supercuspidal 𝐿-packets and their stability, Ann. of Math. (2) 169 (2009), no. 3, 795–901. 10.4007/annals.2009.169.795Suche in Google Scholar

[8] S. DeBacker and M. Reeder, On some generic very cuspidal representations, Compos. Math. 146 (2010), no. 4, 1029–1055. 10.1112/S0010437X10004653Suche in Google Scholar

[9] S. Gelbart, H. Jacquet and J. Rogawski, Generic representations for the unitary group in three variables, Israel J. Math. 126 (2001), 173–237. 10.1007/BF02784154Suche in Google Scholar

[10] S. Gelbart and F. Shahidi, Analytic Properties of Automorphic 𝐿-Functions, Perspect. Math. 6, Academic Press, Boston, 1988. Suche in Google Scholar

[11] D. C. Jabon, The Supercuspidal Representations of U(2,1) and GSP(4) via Hecke Algebra Isomorphisms, ProQuest LLC, Ann Arbor, 1989; Thesis (Ph.D.)–The University of Chicago. Suche in Google Scholar

[12] H. Jacquet and R. P. Langlands, Automorphic Forms on GL⁢(2), Lecture Notes in Math. 114, Springer, Berlin, 1970. 10.1007/BFb0058988Suche in Google Scholar

[13] M. Miyauchi and S. Stevens, Semisimple types for 𝑝-adic classical groups, Math. Ann. 358 (2014), no. 1–2, 257–288. 10.1007/s00208-013-0953-ySuche in Google Scholar

[14] A. Moy, Representations of U⁢(2,1) over a 𝑝-adic field, J. Reine Angew. Math. 372 (1986), 178–208. 10.1515/crll.1986.372.178Suche in Google Scholar

[15] F. Murnaghan, Local character expansions for supercuspidal representations of U⁢(3), Canad. J. Math. 47 (1995), no. 3, 606–640. 10.4153/CJM-1995-032-xSuche in Google Scholar

[16] V. Paskunas and S. Stevens, On the realization of maximal simple types and epsilon factors of pairs, Amer. J. Math. 130 (2008), no. 5, 1211–1261. 10.1353/ajm.0.0022Suche in Google Scholar

[17] F. Rodier, Whittaker models for admissible representations of reductive 𝑝-adic split groups, Harmonic Analysis on Homogeneous Spaces, Proc. Sympos. Pure Math. 26 American Mathematical Society, Providence (1973), 425–430. 10.1090/pspum/026/0354942Suche in Google Scholar

[18] J. D. Rogawski, Automorphic Representations of Unitary Groups in Three Variables, Annals of Mathematics Studies 123, Princeton University Press, Princeton, NJ, 1990. 10.1515/9781400882441Suche in Google Scholar

[19] F. Shahidi, A proof of Langlands’ conjecture on Plancherel measures; complementary series for 𝑝-adic groups, Ann. of Math. (2) 132 (1990), no. 2, 273–330. 10.2307/1971524Suche in Google Scholar

[20] J. A. Shalika, The multiplicity one theorem for GLn, Ann. of Math. (2) 100 (1974), 171–193. 10.2307/1971071Suche in Google Scholar

[21] S. Stevens, Semisimple characters for 𝑝-adic classical groups, Duke Math. J. 127 (2005), no. 1, 123–173. 10.1215/S0012-7094-04-12714-9Suche in Google Scholar

[22] S. Stevens, The supercuspidal representations of 𝑝-adic classical groups, Invent. Math. 172 (2008), no. 2, 289–352. 10.1007/s00222-007-0099-1Suche in Google Scholar

Received: 2020-04-24
Revised: 2021-01-25
Published Online: 2021-03-21
Published in Print: 2021-05-01

Š 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Weighted value distributions of the Riemann zeta function on the critical line
  3. Equivariant prequantization and the moment map
  4. Covering classes and 1-tilting cotorsion pairs over commutative rings
  5. Higher pullbacks of modular forms on orthogonal groups
  6. Distribution of root numbers of Hecke characters attached to some elliptic curves
  7. Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions
  8. On the number of zeros of diagonal cubic forms over finite fields
  9. Generic cuspidal representations of 𝑈(2, 1)
  10. A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas
  11. Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces
  12. Integral foliated simplicial volume and S1-actions
  13. Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature
  14. Vector bundles 𝐸 on ℙ3 with homological dimension 2 and 𝜒(End 𝐸) = 1
  15. On counting cuspidal automorphic representations for GSp(4)
  16. Algebraic cycles and intersections of a quadric and a cubic
https://doi.org/10.1515/forum-2020-0099
SchlagwĂśrter fĂźr diesen Artikel
Generic representations; cuspidal representations; unitary group in 3 variables; Bushnell–Kutzko types

Artikel in diesem Heft

  1. Frontmatter
  2. Weighted value distributions of the Riemann zeta function on the critical line
  3. Equivariant prequantization and the moment map
  4. Covering classes and 1-tilting cotorsion pairs over commutative rings
  5. Higher pullbacks of modular forms on orthogonal groups
  6. Distribution of root numbers of Hecke characters attached to some elliptic curves
  7. Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions
  8. On the number of zeros of diagonal cubic forms over finite fields
  9. Generic cuspidal representations of 𝑈(2, 1)
  10. A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas
  11. Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces
  12. Integral foliated simplicial volume and S1-actions
  13. Improved generalized periods estimates over curves on Riemannian surfaces with nonpositive curvature
  14. Vector bundles 𝐸 on ℙ3 with homological dimension 2 and 𝜒(End 𝐸) = 1
  15. On counting cuspidal automorphic representations for GSp(4)
  16. Algebraic cycles and intersections of a quadric and a cubic
Jetzt anmelden und 20% sparen
Institutioneller Zugang
Wie funktioniert Authentifizierung?
Haben Sie eine Idee, wie wir unsere Website verbessern kĂśnnen?
Bitte schreiben Sie uns.
© 2025 De Gruyter Brill
Heruntergeladen am 1.10.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2020-0099/html?lang=de
Button zum nach oben scrollen