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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)

MSC:
22E50 Representations of Lie and linear algebraic groups over local fields
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