Spherical unitary dual of general linear group over non-archimedean local field.

*(English)*Zbl 0554.20009Let F be a local non-archimedean field. The set of all equivalence classes of irreducible spherical representations of GL(n,F) is described in the first part of the paper. In particular, it is shown that each irreducible spherical representation is parabolically induced by an unramified character. Bernstein’s result on the irreducibility of the parabolically induced representations of GL(n,F) by irreducible unitary ones, and Ol’shanskij’s construction of complementary series give directly a description of all equivalence classes of irreducible unitary spherical representations of GL(n,F).

##### MSC:

20G05 | Representation theory for linear algebraic groups |

20G25 | Linear algebraic groups over local fields and their integers |

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

##### Keywords:

irreducible spherical representations; parabolically induced representations; complementary series##### References:

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