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A sufficient condition for the irreducibility of a parabolically induced representation of \(GL(m,D)\). (Une condition suffisante pour l’irréductibilité d’une induite parabolique de \(GL(m,D)\).) (French. English summary) Zbl 1315.22019
Let \(F\) be a local field and \(D\) a finite dimensional division algebra of center \(F\). Let \(\mathrm{GL}(n,D)\) be the invertible \(n \times n\)-matrices with entries in \(D\).
The first half of the article recalls the classification of irreducible smooth representations of \(\mathrm{GL}(n,D)\). Accepting as building blocks certain (twists of) irreducible “cuspidal” representations, every irreducible representation is found by parabolic induction of tensor products of these for smaller copies \(\mathrm{GL}(n_1,D),\dots,\mathrm{GL}(n_r,D)\) of \(\mathrm{GL}(n,D)\) such that \(n_1 + \dots + n_r=n\). This construction gives a parametrization of all irreducible representations by “multisegments”, tuples of intervals of natural numbers attached to cuspidal representations. This yields two “dual” parametrizations, Langlands’s and Zelevinsky’s, depending on, given a multisegment, taking either the unique irreducible quotient or subrepresentation of the parabolic inductions.
The second half establishes the main theorem: If \(i\) and \(j\) are two multisegments that parametrize irreducible representations \(L(i)\) of \(\mathrm{GL}(n,D)\), given by Langlands’s parametrization, and \(Z(j)\) of \(\mathrm{GL}(m,D)\), given by Zelevinsky’s parametrization, then the parabolic induction of \(L(i) \otimes Z(j)\) from \(\mathrm{GL}(n,D) \times \mathrm{GL}(m,D)\) to \(\mathrm{GL}(n+m,D)\) is irreducible if (but not only if) \(i\) and \(j\) are not “juxtaposed”, a combinatorial criterion.
The final section applies the juxtaposition criterion to certain “ladder” representations especially suited to it, singled out by combinatorial conditions on their multisegment parametrizations. These include in particular the Speh representations whose products constitute all “rigid” unitary irreducible representations.
Reviewer: Enno Nagel (Paris)

22E50 Representations of Lie and linear algebraic groups over local fields
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[1] Aubert, Anne-Marie, Dualité dans le groupe de Grothendieck de la catégorie des représentations lisses de longueur finie d’un groupe réductif \(p\)-adique, Trans. Amer. Math. Soc., 347, 6, 2179-2189, (1995) · Zbl 0827.22005
[2] Badulescu, Alexandre Ioan; Renard, David, Functional analysis IX, 48, Zelevinsky involution and Moeglin-waldspurger algorithm for \({\rm GL}_n(D), 9-15, (2007),\) Univ. Aarhus, Aarhus · Zbl 1139.22012
[3] Badulescu, Alexandru Ioan, On \(p\)-adic speh representations · Zbl 1305.22016
[4] Badulescu, Alexandru Ioan, Un résultat d’irréductibilité en caractéristique non nulle, Tohoku Math. J. (2), 56, 4, 583-592, (2004) · Zbl 1064.22003
[5] Badulescu, Alexandru Ioan, Jacquet-Langlands et unitarisabilité, J. Inst. Math. Jussieu, 6, 3, 349-379, (2007) · Zbl 1159.22005
[6] Badulescu, Alexandru Ioan, Global Jacquet-Langlands correspondence, multiplicity one and classification of automorphic representations, Invent. Math., 172(2), 383-438, (2008) · Zbl 1158.22018
[7] Badulescu, Alexandru Ioan; Henniart, Guy; Lemaire, Bertrand; Sécherre, Vincent, Sur le dual unitaire de \({\rm GL}(r,D),\) Amer. J. Math., 132, 1365-1396, (2010) · Zbl 1205.22011
[8] Bernstein, I. N.; Zelevinsky, A. V., Induced representations of reductive \(p\)-adic groups. I, Ann. Sci. École Norm. Sup. (4), 10, 4, 441-472, (1977) · Zbl 0412.22015
[9] Bernstein, Joseph N., Lie group representations, II (College Park, Md., 1982/1983)\(, 1041, P\)-invariant distributions on \({\rm GL}(N)\) and the classification of unitary representations of \({\rm GL}(N) \)(non-Archimedean case), 50-102, (1984), Springer, Berlin · Zbl 0541.22009
[10] Boyer, Pascal, Monodromie du faisceau pervers des cycles évanescents de quelques variétés de Shimura simples, Invent. Math., 177, 239-280, (2009) · Zbl 1172.14016
[11] Chenevier, Gaëtan; Renard, David, Characters of speh representations and Lewis caroll identity, Represent. Theory, 12, 447-452, (2008) · Zbl 1163.22008
[12] Deligne, P.; Kazhdan, D.; Vignéras, M.-F., Representations of reductive groups over a local field, Représentations des algèbres centrales simples \(p\)-adiques, 33-117, (1984), Hermann, Paris · Zbl 0583.22009
[13] Lapid, Erez; Mínguez, Alberto, On a determinantal formula of tadić · Zbl 1288.22013
[14] Mínguez, Alberto, Correspondance de Howe explicite : paires duales de type II, Ann. Sci. Éc. Norm. Supér. (4), 41, 5, 717-741, (2008) · Zbl 1220.22014
[15] Mínguez, Alberto, Sur l’irréductibilité d’une induite parabolique, J. Reine Angew. Math., 629, 107-131, (2009) · Zbl 1172.22008
[16] Mínguez, Alberto; Sécherre, Vincent, Représentations banales de \({\rm GL}(m,D)\) · Zbl 1283.22014
[17] Mínguez, Alberto; Sécherre, Vincent, Représentations lisses modulo \(ℓ\) de \({\rm GL}(m,D)\) · Zbl 1293.22005
[18] Mœglin, C.; Waldspurger, J.-L., Sur l’involution de zelevinski, J. Reine Angew. Math., 372, 136-177, (1986) · Zbl 0594.22008
[19] Mœglin, C.; Waldspurger, J.-L., Le spectre résiduel de \({\rm GL}(n),\) Ann. Sci. École Norm. Sup. (4), 22, 4, 605-674, (1989) · Zbl 0696.10023
[20] Sécherre, Vincent, Proof of the tadić conjecture (U0) on the unitary dual of \({\rm GL}_m(D),\) J. Reine Angew. Math., 626, 187-203, (2009) · Zbl 1170.22009
[21] Tadić, Marko, Induced representations of \({\rm GL}(n,A)\) for \(p\)-adic division algebras \(A,\) J. Reine Angew. Math., 405, 48-77, (1990) · Zbl 0684.22008
[22] Tadić, Marko, On characters of irreducible unitary representations of general linear groups, Abh. Math. Sem. Univ. Hamburg, 65, 341-363, (1995) · Zbl 0856.22026
[23] Tadić, Marko, Representation theory of \({\rm GL}(n)\) over a \(p\)-adic division algebra and unitarity in the Jacquet-Langlands correspondence, Pacific J. Math., 223, 1, 167-200, (2006) · Zbl 1124.22005
[24] Zelevinsky, A. V., Induced representations of reductive \(p\)-adic groups. II. on irreducible representations of \({\rm GL}(n),\) Ann. Sci. École Norm. Sup. (4), 13, 2, 165-210, (1980) · Zbl 0441.22014
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