Dual Blobs and Plancherel Formulas

Ju-Lee Kim

Bulletin de la Société Mathématique de France (2004)

  • Volume: 132, Issue: 1, page 55-80
  • ISSN: 0037-9484

Abstract

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Let k be a p -adic field. Let G be the group of k -rational points of a connected reductive group 𝖦 defined over k , and let 𝔤 be its Lie algebra. Under certain hypotheses on 𝖦 and k , wequantifythe tempered dual G ^ of G via the Plancherel formula on 𝔤 , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on 𝔤 and G . As a consequence, we prove that any tempered representation contains a good minimal 𝖪 -type; we extend this result to irreducible admissible representations.

How to cite

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Kim, Ju-Lee. "Dual Blobs and Plancherel Formulas." Bulletin de la Société Mathématique de France 132.1 (2004): 55-80. <http://eudml.org/doc/272457>.

@article{Kim2004,
abstract = {Let $k$ be a $p$-adic field. Let $G$ be the group of $k$-rational points of a connected reductive group $\mathsf \{G\}$ defined over $k$, and let $\mathfrak \{g\}$ be its Lie algebra. Under certain hypotheses on $\mathsf \{G\}$ and $k$, wequantifythe tempered dual $\{\widehat\{G\}\}$ of $G$ via the Plancherel formula on $\mathfrak \{g\}$, using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $\mathfrak \{g\}$ and $G$. As a consequence, we prove that any tempered representation contains a good minimal $\mathsf \{K\}$-type; we extend this result to irreducible admissible representations.},
author = {Kim, Ju-Lee},
journal = {Bulletin de la Société Mathématique de France},
keywords = {representations; $p$-adic groups; Plancherel formula; character expansions},
language = {eng},
number = {1},
pages = {55-80},
publisher = {Société mathématique de France},
title = {Dual Blobs and Plancherel Formulas},
url = {http://eudml.org/doc/272457},
volume = {132},
year = {2004},
}

TY - JOUR
AU - Kim, Ju-Lee
TI - Dual Blobs and Plancherel Formulas
JO - Bulletin de la Société Mathématique de France
PY - 2004
PB - Société mathématique de France
VL - 132
IS - 1
SP - 55
EP - 80
AB - Let $k$ be a $p$-adic field. Let $G$ be the group of $k$-rational points of a connected reductive group $\mathsf {G}$ defined over $k$, and let $\mathfrak {g}$ be its Lie algebra. Under certain hypotheses on $\mathsf {G}$ and $k$, wequantifythe tempered dual ${\widehat{G}}$ of $G$ via the Plancherel formula on $\mathfrak {g}$, using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $\mathfrak {g}$ and $G$. As a consequence, we prove that any tempered representation contains a good minimal $\mathsf {K}$-type; we extend this result to irreducible admissible representations.
LA - eng
KW - representations; $p$-adic groups; Plancherel formula; character expansions
UR - http://eudml.org/doc/272457
ER -

References

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  1. [1] J. Adler – « Refined anisotropic k -types and supercuspidal representations », Pacific J. Math.185 (1998), p. 1–32. Zbl0924.22015MR1653184
  2. [2] J. Adler & S. DeBacker – « Some applications of Bruhat-Tits theory to harmonic analysis on the Lie algebra of a reductive p -adic group », Michigan J. Math. 50 (2002), no. 2, p. 263–286. Zbl1018.22013MR1914065
  3. [3] J. Adler & A. Roche – « An intertwining result for p -adic groups », Canad. J. Math. 52 (2000), no. 3, p. 449–467. Zbl1160.22304MR1758228
  4. [4] D. Barbasch & A. Moy – « Local character expansions », Ann. Sci. École Norm. Sup. 30 (1997), no. 5, p. 553–567. Zbl0885.22021MR1474804
  5. [5] S. DeBacker – « Homogeneity results for invariant distributions of a reductive p -adic group », Ann. Sci. École Norm. Sup.35 (2002), p. 391–422. Zbl0999.22013MR1914003
  6. [6] —, « Parameterizing nilpotent orbits via Bruhat-Tits theory », Ann. of Math.156 (2002), p. 295–331. Zbl1015.20033MR1935848
  7. [7] J. Dixmier – Les C * -algèbres et leurs représentations, Éditions Jacques Gabay, Paris, 1996. Zbl0174.18601
  8. [8] Harish-Chandra –« The Plancherel formula for reductive p -adic groups », Collected Papers, vol. 4, Springer-Verlag, Berlin, 1976. 
  9. [9] —, Admissible invariant distributions on reductive p -adic groups, University Lecture Series, vol. 16, Amer. Math. Soc., Providence, RI, 1999, Preface and notes by Stephen DeBacker and Paul J.Sally, Jr. Zbl0928.22017MR1702257
  10. [10] R. Howe – « Kirillov theory for compact p -adic groups », Pacific J. Math.73 (1977), p. 365–381. Zbl0385.22007MR579176
  11. [11] —, « Some qualitative results on the representation theory or Gl n over a p -adic field », Pacific J. Math.73 (1977), p. 479–538. Zbl0385.22009MR492088
  12. [12] J. Kim – « Hecke algebras of classical groups over p -adic fields and supercuspidal representations », Amer. J. Math.121 (1999), p. 967–1029. Zbl0933.22024MR1713299
  13. [13] J. Kim & F. Murnaghan – « Character expansions and unrefined minimal 𝖪 -types », Preprint, 2002. Zbl1037.22035
  14. [14] A. Moy & G. Prasad – « Unrefined minimal K -types for p -adic groups », Invent. Math.116 (1994), p. 393–408. Zbl0804.22008MR1253198
  15. [15] —, « Jacquet functors and unrefined minimal K -types », Comment. Math. Helv.71 (1996), p. 98–121. Zbl0860.22006MR1371680
  16. [16] G. Prasad – « Galois fixed points in the Bruhat-Tits buildings of a reductive group », Bull. Soc. Math. France129 (2001), p. 169–174. Zbl0992.20032MR1871292
  17. [17] G. Rousseau – « Immeubles des groupes réductifs sur les corps locaux », Thèse, Paris XI, 1977. Zbl0412.22006
  18. [18] J.-L. Waldspurger – « Homogénéité de certaines distributions sur les groupes p -adiques », Inst. Hautes Études Sci. Publ. Math.81 (1995), p. 25–72. Zbl0841.22009MR1361755
  19. [19] —, « La formule de Plancherel pour les groupes p -adiques d’après Harish-Chandra », J. Inst. Math. Jussieu 2 (2003), no. 2, p. 235–333. Zbl1029.22016MR1989693
  20. [20] J. Yu – « Construction of tame supercuspidal representations », J. Amer. Math. Soc. 14 (2001), no. 3, p. 579–622. Zbl0971.22012MR1824988

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