Induced representations and classification for $GSp(2,F)$ and $Sp(2,F)$

Abstract

Induced representations and classification for GSp(2, F) and Sp(2, F) Mémoires de la S. M. F. 2 e série, tome 52 (1993), p. 75-133 <http://www.numdam.org/item?id=MSMF_1993_2_52__75_0> © Mémoires de la S. M. F., 1993, tous droits réservés. L'accès aux archives de la revue « Mémoires de la S. M. F. » (http://smf. emath.fr/Publications/Memoires/Presentation.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

Regular poles for the p-adic group $GSp_4$

“…, and since (σ1 GSp4(k) ) ψ is one-dimensional, we have (σ1 GSp4(k) Tables Table 1 displays the regular poles of the nonsupercuspidal representations [10], which have Jacquet module length of at most 2, in terms of the poles of Tate L -functions. The last column shows the expected exceptional poles from the local Langlands conjecture.…”

“…Also, for simplicity, we take µ = 1 . Irreducible and admissible representations of GSp 4 (k) , which has Jacquet module length of less than or equal to 2, are given in Table 1 due to the Sally-Tadic classification in [10]. In this table, nonsupercuspidal representations are named as IIIa, IIIb, IVa, IVb, IVc, IVd, Va, Vb, Vc, Vd, VIb, VIc, VII, VIIIa, VIIIb, IXa,IXb, X, XIa, and XIb.…”

Sign changes of Hecke eigenvalues of Siegel cusp forms of degree 2

“…Then, from We wish to remark here that we are able to prove the above proposition only by using Proposition 1.1. We cannot carry out the above proof by merely using the unitary classification from [10], [11]. Also, note that the hypothesis of the theorem (at least one of a or b is real) forces the local representation to be either the complementary series (type (C)) or the limits of complementary series -β = 1/2 that gives the type (SK) and β = 0 gives the tempered with at least one of a or b equal to ±1.…”

The residual spectrum of an inner form of 𝑆𝑝₈ supported in the minimal parabolic subgroup

“…It is irreducible by [29], [30], [27], and [28]. Hence, the H 1 (k v ) normalized intertwining operator…”

“…For the third claim see the comment on the proof of the previous proposition and also [24]. For the last claim one uses the irreducibility of certain induced representations for Sp 4 (k v ) given in [30], [27] and [28].…”