Note on invariance over weakly unramified characters of $p$-adic groups
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- by Kwangho Choiy;
- Proc. Amer. Math. Soc. 153 (2025), 45-53
- DOI: https://doi.org/10.1090/proc/17005
- Published electronically: November 21, 2024
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Abstract:
We shall study representation-theoretic aspects of a complex affine variety of weakly unramified characters of $p$-adic groups. This work proves the invariance of Knapp-Stein $R$-groups within finite subsets of unitary weakly unramified characters constructed over Satake parameters.References
- James Arthur, Unipotent automorphic representations: conjectures, Astรฉrisque 171-172 (1989), 13โ71. Orbites unipotentes et reprรฉsentations, II. MR 1021499
- James Arthur, On elliptic tempered characters, Acta Math. 171 (1993), no.ย 1, 73โ138. MR 1237898, DOI 10.1007/BF02392767
- A. Borel, Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, RI, 1979, pp.ย 27โ61. MR 546608
- Kuok Fai Chao and Wen-Wei Li, Dual $R$-groups of the inner forms of $\textrm {SL}(N)$, Pacific J. Math. 267 (2014), no.ย 1, 35โ90. MR 3163476, DOI 10.2140/pjm.2014.267.35
- Kwangho Choiy, Tempered non-discrete spectrum for pseudo-$z$-embedding, Proc. Amer. Math. Soc. 149 (2021), no.ย 7, 2841โ2850. MR 4257798, DOI 10.1090/proc/15426
- Kwangho Choiy, Weakly unramified representations, finite morphisms, and Knapp-Stein $R$-groups, Manuscripta Math. 172 (2023), no.ย 3-4, 871โ884. MR 4651107, DOI 10.1007/s00229-022-01434-7
- Kwangho Choiy and David Goldberg, Transfer of $R$-groups between $p$-adic inner forms of $SL_n$, Manuscripta Math. 146 (2015), no.ย 1-2, 125โ152. MR 3294420, DOI 10.1007/s00229-014-0689-3
- Kwangho Choiy and David Goldberg, Invariance of $R$-groups between $p$-adic inner forms of quasi-split classical groups, Trans. Amer. Math. Soc. 368 (2016), no.ย 2, 1387โ1410. MR 3430367, DOI 10.1090/tran/6485
- K. Choiy and D. Goldberg, Behavior of $R$-groups for $p$-adic inner forms of quasi-split special unitary groups, Bull. Iranian Math. Soc. 43 (2017), no.ย 4, 117โ141. MR 3711825
- David Goldberg, $R$-groups and elliptic representations for $\textrm {SL}_n$, Pacific J. Math. 165 (1994), no.ย 1, 77โ92. MR 1285565, DOI 10.2140/pjm.1994.165.77
- David Goldberg, Reducibility of induced representations for $\textrm {Sp}(2n)$ and $\textrm {SO}(n)$, Amer. J. Math. 116 (1994), no.ย 5, 1101โ1151. MR 1296726, DOI 10.2307/2374942
- Thomas J. Haines, The stable Bernstein center and test functions for Shimura varieties, Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp.ย 118โ186. MR 3444233
- Thomas J. Haines, On Satake parameters for representations with parahoric fixed vectors, Int. Math. Res. Not. IMRN 20 (2015), 10367โ10398. MR 3455870, DOI 10.1093/imrn/rnu254
- Thomas J. Haines, Correction to โOn Satake parameters for representations with parahoric fixed vectorsโ [ MR3455870], Int. Math. Res. Not. IMRN 13 (2017), 4160โ4170. MR 3671514, DOI 10.1093/imrn/rnx088
- Thomas J. Haines and Sean Rostami, The Satake isomorphism for special maximal parahoric Hecke algebras, Represent. Theory 14 (2010), 264โ284. MR 2602034, DOI 10.1090/S1088-4165-10-00370-5
- Harish-Chandra, Harmonic analysis on reductive $p$-adic groups, Harmonic analysis on homogeneous spaces (Proc. Sympos. Pure Math., Vol. XXVI, Williams Coll., Williamstown, Mass., 1972) Proc. Sympos. Pure Math., Vol. XXVI, Amer. Math. Soc., Providence, RI, 1973, pp.ย 167โ192. MR 340486
- Tasho Kaletha, The local Langlands conjectures for non-quasi-split groups, Families of automorphic forms and the trace formula, Simons Symp., Springer, [Cham], 2016, pp.ย 217โ257. MR 3675168
- C. David Keys, $L$-indistinguishability and $R$-groups for quasisplit groups: unitary groups in even dimension, Ann. Sci. รcole Norm. Sup. (4) 20 (1987), no.ย 1, 31โ64. MR 892141, DOI 10.24033/asens.1523
- A. W. Knapp and E. M. Stein, Irreducibility theorems for the principal series, Conference on Harmonic Analysis (Univ. Maryland, College Park, Md., 1971) Lecture Notes in Math., Vol. 266, Springer, Berlin-New York, 1972, pp.ย 197โ214. MR 422512
- Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no.ย 3, 255โ339. MR 1485921, DOI 10.1023/A:1000102604688
- R. P. Langlands, On the classification of irreducible representations of real algebraic groups, Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr., vol. 31, Amer. Math. Soc., Providence, RI, 1989, pp.ย 101โ170. MR 1011897, DOI 10.1090/surv/031/03
- Allan J. Silberger, The Knapp-Stein dimension theorem for $p$-adic groups, Proc. Amer. Math. Soc. 68 (1978), no.ย 2, 243โ246. MR 492091, DOI 10.1090/S0002-9939-1978-0492091-5
- Allan J. Silberger, The Knapp-Stein dimension theorem for $p$-adic groups, Proc. Amer. Math. Soc. 68 (1978), no.ย 2, 243โ246. MR 492091, DOI 10.1090/S0002-9939-1978-0492091-5
Bibliographic Information
- Kwangho Choiy
- Affiliation: School of Mathematical and Statistical Sciences, Southern Illinois University, Carbondale, Illinois 62901-4408
- MR Author ID: 1060574
- Email: kchoiy@siu.edu
- Received by editor(s): April 15, 2024
- Received by editor(s) in revised form: June 13, 2024, and June 28, 2024
- Published electronically: November 21, 2024
- Additional Notes: The author was supported by a gift from the Simons Foundation (#840755).
- Communicated by: David Savitt
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 153 (2025), 45-53
- MSC (2020): Primary 11F70; Secondary 11E95, 20G20, 22E50
- DOI: https://doi.org/10.1090/proc/17005
- MathSciNet review: 4840256