Distinction of the Steinberg representation III: the tamely ramified case
Annales de l'Institut Fourier, Volume 67 (2017) no. 4, p. 1521-1607
Let F be a nonarchimedean local field, let E be a Galois quadratic extension of F and let G be a quasisplit group defined over F; a conjecture by Dipendra Prasad states that the Steinberg representation St E of G(E) is then χ-distinguished for a given unique character χ of G(F), and that χ occurs with multiplicity 1 in the restriction of St E to G(F). In the first two papers of the series, Broussous and the author have proved the Prasad conjecture when G is F-split and E/F is unramified; this paper deals with the tamely ramified case, still with G F-split.
Soit F un corps local non archimédien, soit E une extension galoisienne quadratique de F et soit G un groupe quasi-déployé défini sur F ; d’après une conjecture de Dipendra Prasad, la représentation de Steinberg St E de G(E) est alors χ-distinguée (relativement à G(E)/G(F)) pour un unique caractére χ de G(F), et χ apparaît avec multiplicité 1 dans la restriction de St E à G(F). Dans les deux premiers articles de la série, Broussous et l’auteur ont démontré la conjecture de Prasad pour G F-déployé et E/F non ramifiée ; cet article traite le cas modérément ramifié, toujours avec G F-déployé.
Received : 2014-08-28
Revised : 2016-04-20
Accepted : 2016-06-14
Published online : 2017-09-26
DOI : https://doi.org/10.5802/aif.3116
Classification:  20G25,  22E50
Keywords: p-adic algebraic groups, Steinberg representation, distinguished representations, tame ramification
@article{AIF_2017__67_4_1521_0,
     author = {Court\`es, Fran\c cois},
     title = {Distinction of the Steinberg representation III: the tamely ramified case},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {4},
     year = {2017},
     pages = {1521-1607},
     doi = {10.5802/aif.3116},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_4_1521_0}
}
Courtès, François. Distinction of the Steinberg representation III: the tamely ramified case. Annales de l'Institut Fourier, Volume 67 (2017) no. 4, pp. 1521-1607. doi : 10.5802/aif.3116. https://aif.centre-mersenne.org/item/AIF_2017__67_4_1521_0/

[1] Anandavardhanan, U. N.; Rajan, Conjeeveram S. Distinguished representations, base change and reducibility for unitary groups., Int. Math. Res. Not., Tome 14 (2005), pp. 841-854 | Article

[2] Borel, Armand; Tits, Jacques Groupes réductifs, Publications Mathêmatiques de l’IHES, Tome 27 (1965), pp. 659-755

[3] Bourbaki, Nicolas Groupes et algèbres de Lie, chapitre 5: Groupes engendrés par des réflexions, Hermann (1968)

[4] Bourbaki, Nicolas Groupes et algèbres de Lie, chapitre 6: Systèmes de racines, Hermann (1968)

[5] Broussous, Paul; Courtès, François Distinction of the Steinberg representation, Int. Math. Res. Not., Tome 11 (2014), pp. 3140-3157 | Article

[6] Brown, Kenneth S. Buildings, Springer (1996), viii+215 pages

[7] Bruhat, François; Tits, Jacques Groupes réductifs sur un corps local. I. Données radicielles valuées, Publ. Math. Inst. Hautes Etudes Sci., Tome 41 (1972), pp. 5-251 | Article

[8] Carter, Roger W. Finite groups of Lie type, John Wiley & Sons (1985), xii+544 pages

[9] Courtès, François Parametrization of tamely ramified maximal tori using bounded subgroups, Ann. Mat. Pura Appl., Tome 188 (2009) no. 1, pp. 1-33 | Article

[10] Courtès, François Distinction of the Steinberg representation II: an equality of characters, Forum Math., Tome 27 (2015) no. 6, pp. 3461-3475 | Article

[11] Debacker, Stephen Parameterizing conjugacy classes of maximal unramified tori with Bruhat–Tits theory, Mich. Math. J., Tome 54 (2006) no. 1, pp. 157-178 | Article

[12] Delorme, Patrick; Sécherre, Vincent An analogue of the Cartan decomposition for p-adic symmetric spaces of split p-adic reductive groups, Pac. J. Math, Tome 251 (2011) no. 1, pp. 1-21 | Article

[13] Dynkin, Evgeniĭ Borisovich Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc., Transl., Tome 6 (1957) no. 2, pp. 111-243

[14] Macdonald, Ian Grant The Poincaré series of a Coxeter group, Mathematische Annalen, Tome 199 (1972), pp. 161-174 | Article

[15] Matringe, Nadir Distinction of the Steinberg representation for inner forms of GL n (2016) (http://arxiv.org/abs/1602.05101 )

[16] Prasad, Dipendra Invariant forms for representations of GL 2 over a local field, Amer. J. Math, Tome 114 (1992) no. 6, pp. 1317-1363 | Article

[17] Prasad, Dipendra On a conjecture of Jacquet about distinguished representations of GL n , Duke Math J., Tome 109 (2001) no. 1, pp. 67-78 | Article

[18] Prasad, Dipendra A “relative” local Langlands correspondence (2015) (http://arxiv.org/abs/1512.04347 )

[19] Shalika, Joseph A. On the space of cusp forms of a p-adic Chevalley group, Ann. Math., Tome 92 (1970) no. 2, pp. 262-278 | Article

[20] Springer, Tonny A. Linear algebraic groups, Birkhäuser, Progress in Mathematics, Tome 9 (1998), x+334 pages

[21] Tits, Jacques Classification of algebraic semisimple groups, Proc. Symp. Pure Math., Tome 9 (1966), pp. 33-62 | Article