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A method of proving non-unitarity of representations of $$p$$-adic groups. I. (English) Zbl 1215.22009
For a connected reductive group $$G$$ over a $$p$$-adic field $$F$$, the Steinberg representation $$St$$ and its Aubert dual, the trivial representation, are both unitary. Casselman proved that all other irreducible representations having the same infinitesimal character as $$St$$ are non-unitary.
In this paper, the authors give a partial generalization of Casselman’s result. They consider generalized Steinberg representations (defined below) of $$G_\ell=Sp(2\ell,F)$$ or $$SO(2\ell+1,F)$$.
Let $$\nu = |\det|_F$$. Let $$\rho$$ be an irreducible unitary cuspidal representation of $$GL(k,F)$$ and $$\sigma$$ an irreducible cuspidal representation of $$G_\ell$$. Suppose that $$\text{Ind}^{G_{k+\ell}}(\nu^\alpha \rho \otimes \sigma)$$ reduces for some $$\alpha \in (1/2)\mathbb{Z}_{>0}$$. Then the representation
$\Pi= \text{Ind}^{G_{(n+1)k+\ell}}(\nu^{\alpha + n} \rho \otimes \nu^{\alpha + n-1} \rho \otimes \cdots \otimes \nu^{\alpha } \rho \otimes \sigma)$
has the unique irreducible subrepresentation $$\pi$$ and the unique irreducible quotient $$\pi'$$. The representation $$\pi$$ is square-integrable (and hence unitary); it is called a generalized Steinberg representation. The representation $$\pi'$$ is the Aubert involution of $$\pi$$. This paper and its sequel prove that all irreducible subquotients of $$\Pi$$ different from $$\pi$$ and $$\pi'$$ are not unitarizable. The method of proof is based on the fact that the representation induced from a unitary representation is unitary, and therefore semi-simple.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 22E35 Analysis on $$p$$-adic Lie groups 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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