Representations of PGL ( 2 ) of a local field and harmonic cochains on graph

Paul Broussous[1]

  • [1] Université de Poitiers, UMR6086 CNRS, SP2MI – Téléport, Bd M. et P. Curie BP 30179, 86962 Futuroscope Chasseneuil Cedex

Annales de la faculté des sciences de Toulouse Mathématiques (2009)

  • Volume: 18, Issue: 3, page 541-559
  • ISSN: 0240-2963

Abstract

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We give combinatorial models for non-spherical, generic, smooth, complex representations of the group G = PGL ( 2 , F ) , where F is a non-Archimedean locally compact field. More precisely we carry on studying the graphs ( X ˜ k ) k ≥ 0 defined in a previous work. We show that such representations may be obtained as quotients of the cohomology of a graph X ˜ k , for a suitable integer k , or equivalently as subspaces of the space of discrete harmonic cochains on such a graph. Moreover, for supercuspidal representations, these models are unique.

How to cite

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Broussous, Paul. "Representations of ${\rm PGL}(2)$ of a local field and harmonic cochains on graph." Annales de la faculté des sciences de Toulouse Mathématiques 18.3 (2009): 541-559. <http://eudml.org/doc/10116>.

@article{Broussous2009,
abstract = {We give combinatorial models for non-spherical, generic, smooth, complex representations of the group $G=\{\rm PGL\}(2,F)$, where $F$ is a non-Archimedean locally compact field. More precisely we carry on studying the graphs $(\{\tilde\{X\}\}_k )_\{k\ge 0\}$ defined in a previous work. We show that such representations may be obtained as quotients of the cohomology of a graph $\{\tilde\{X\}\}_k$, for a suitable integer $k$, or equivalently as subspaces of the space of discrete harmonic cochains on such a graph. Moreover, for supercuspidal representations, these models are unique.},
affiliation = {Université de Poitiers, UMR6086 CNRS, SP2MI – Téléport, Bd M. et P. Curie BP 30179, 86962 Futuroscope Chasseneuil Cedex},
author = {Broussous, Paul},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
language = {eng},
month = {7},
number = {3},
pages = {541-559},
publisher = {Université Paul Sabatier, Toulouse},
title = {Representations of $\{\rm PGL\}(2)$ of a local field and harmonic cochains on graph},
url = {http://eudml.org/doc/10116},
volume = {18},
year = {2009},
}

TY - JOUR
AU - Broussous, Paul
TI - Representations of ${\rm PGL}(2)$ of a local field and harmonic cochains on graph
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2009/7//
PB - Université Paul Sabatier, Toulouse
VL - 18
IS - 3
SP - 541
EP - 559
AB - We give combinatorial models for non-spherical, generic, smooth, complex representations of the group $G={\rm PGL}(2,F)$, where $F$ is a non-Archimedean locally compact field. More precisely we carry on studying the graphs $({\tilde{X}}_k )_{k\ge 0}$ defined in a previous work. We show that such representations may be obtained as quotients of the cohomology of a graph ${\tilde{X}}_k$, for a suitable integer $k$, or equivalently as subspaces of the space of discrete harmonic cochains on such a graph. Moreover, for supercuspidal representations, these models are unique.
LA - eng
UR - http://eudml.org/doc/10116
ER -

References

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  5. Casselman (W.).— On some results of Atkin and Lehner, Math. Ann. 201, p. 301-314 (1873). Zbl0239.10015MR337789
  6. Jacquet (H.), Langlands (R.P.).— Automorphic forms on GL ( 2 ) , Lecture Notes in Math., 114, Springer Verlag (1970). Zbl0236.12010MR401654
  7. Jacquet (H.), Piateski-Shapiro (I.), Shalika (J.).— Conducteur des représentations du groupe linéaire, Math. Ann., 256 no. 2, p. 199-214 (1981). Zbl0443.22013MR620708
  8. Serre (J.-P.).— Trees, Springer, 2nd ed.2002. Zbl1013.20001MR607504
  9. Waldspurger (J.-L.).— Correspondance de Shimura, J. Math. Pures et Appl., 59, p. 1-133 (1980). Zbl0412.10019MR577010

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