On the Notion of an Automorphic Representation
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چکیده
SupposeP is a parabolic subgroup ofGwith Levi factor M and σ = ⊗σv a cuspidal representation of M(A). Then Ind σ = ⊗vInd σv is a representation of G(A) which may not be irreducible, and may not even have a finite composition series. As usual an irreducible subquotient of this representation is said to be a constituent of it. For almost all v, Ind σv has exactly one constituent π◦ v containing the trivial representation of G(Ov). If Ind σv acts on Xv then π◦ v can be obtained by taking the smallest G(Fv)-invariant subspace Vv ofXv containing nonzero vectors fixed byG(Ov) together with the largest G(Fv)-invariant subspace Uv of Vv containing no such vectors and then letting G(Fv) act on Vv/Uv.
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