Note on invariance over weakly unramified characters of 𝑝-adic groups

Choiy, Kwangho

doi:10.1090/proc/17005

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, Astérisque 171-172 (1989), 13–71. Orbites unipotentes et représentations, II. MR 1021499
James Arthur, On elliptic tempered characters, Acta Math. 171 (1993), no. 1, 73–138. 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. 27–61. 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. 1, 35–90. 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. 7, 2841–2850. MR 4257798, DOI 10.1090/proc/15426
Kwangho Choiy, Weakly unramified representations, finite morphisms, and Knapp-Stein $R$-groups, Manuscripta Math. 172 (2023), no. 3-4, 871–884. 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. 1-2, 125–152. 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. 2, 1387–1410. 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. 4, 117–141. MR 3711825
David Goldberg, $R$-groups and elliptic representations for $\textrm {SL}_n$, Pacific J. Math. 165 (1994), no. 1, 77–92. 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. 5, 1101–1151. 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. 118–186. MR 3444233
Thomas J. Haines, On Satake parameters for representations with parahoric fixed vectors, Int. Math. Res. Not. IMRN 20 (2015), 10367–10398. MR 3455870, DOI 10.1093/imrn/rnu254
Thomas J. Haines, Correction to “On Satake parameters for representations with parahoric fixed vectors” [ MR3455870], Int. Math. Res. Not. IMRN 13 (2017), 4160–4170. 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), 264–284. 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. 167–192. 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. 217–257. MR 3675168
C. David Keys, $L$-indistinguishability and $R$-groups for quasisplit groups: unitary groups in even dimension, Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 1, 31–64. 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. 197–214. MR 422512
Robert E. Kottwitz, Isocrystals with additional structure. II, Compositio Math. 109 (1997), no. 3, 255–339. 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. 101–170. 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. 2, 243–246. 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. 2, 243–246. MR 492091, DOI 10.1090/S0002-9939-1978-0492091-5

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Note on invariance over weakly unramified characters of $p$-adic groups
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Abstract: