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Representation theory of \(\mathrm{GL}(n)\) over a \(p\)-adic division algebra and unitarity in the Jacquet-Langlands correspondence. (English) Zbl 1124.22005
Let \(A\) be a central division algebra of dimension \(d_A\) over the local non-archimedean field \(F\) of characteristic \(0\). The generalized Jacquet-Langlands correspondence connects irreducible smooth representations of the group \(\text{GL}(p,A)\) with certain representations of \(\text{GL}(pd_A,F)\). This correspondence can be extended in a natural way to mappings between the Grothendieck groups of such representations. In the present paper the author assumes the truth of the conjecture that unitary parabolic induction is irrreducible for the group \(\text{GL}(m,A)\). Under this assumption he can prove explicit formulas for the correspondence. In particular it is proved that in the correspondence between the Grothendieck groups of \(\text{GL}(pd_A,F)\) and \(\text{GL}(p,A)\) an irreducible unitary representation of \(\text{GL}(pd_A,F)\) is sent either to zero or to plus or minus an irreducible representation of \(\text{GL}(p,A)\). On the way to this result some results in the representation theory of the group \(\text{GL}(n)\) of a division algebra over a non-archimedean local field are developed.

22E50 Representations of Lie and linear algebraic groups over local fields
11F70 Representation-theoretic methods; automorphic representations over local and global fields
11S37 Langlands-Weil conjectures, nonabelian class field theory
22E35 Analysis on \(p\)-adic Lie groups
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