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Download PDF by David Ginzburg, Stephen Rallis, David Soudry: The Descent Map from Automorphic Representations of Gl (n)

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By David Ginzburg, Stephen Rallis, David Soudry

ISBN-10: 9814304980

ISBN-13: 9789814304986

Complaints of the Intl convention held to honor the sixtieth birthday of A.M. Naveira. convention was once held July 8-14, 2002 in Valencia, Spain. For graduate scholars and researchers in differential geometry 1. creation -- 2. On definite residual representations -- three. Coefficients of Gelfand-Graev style, of Fourier-Jacobi kind, and descent -- four. a few double coset decompositions -- five. Jacquet modules of parabolic inductions: Gelfand-Graev characters -- 6. Jacquet modules of parabolic inductions: Fourier-Jacobi characters -- 7. The tower estate --8. Non-vanishing of the descent I -- nine. Non-vanishing of the descent II -- 10. international genericity of the descent and worldwide integrals -- eleven. Langlands (weak) functorial raise and descent

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Let ϕσ be a cusp form in the space of σ. Define ϕσ (g)E ψℓ,w0 (g, fτ¯,¯s )dg. 19) GF \GA This the L2 -pairing between the complex conjugate of σ and σψℓ,w0 (Rτ¯,¯s ). 18). ,τℓ , the extension E/F are all unramified, and ψ is normalized (outside S). 14). It will be convenient to re-denote F Jψφℓ ,γ (E(·, fτ¯,¯s ))(h) = F Jψφℓ ,γ (E(h, fτ¯,¯s )). 21) ˆ ℓ (A). 16), F Jψφℓ ,γ (E(h, fτ¯,¯s ) is an automorphic function on GA = H point, we should remember that, in case H is metaplectic, ψ enters in the data of fτ¯,¯s , and hence in the data of the Eisenstein series; denote it now by E(h, fτ¯,¯s,ψ ).

Note that, by definition, for 1 ≤ i ≤ r, + w(Vϕ (ξ ∗ (i))) = Vξ(ϕ) (i). 4. Let w and ξ be as above. 24). 39) i = jw + 1, . . 40) ξ(jw + 1) < ξ(jw + 2) < · · · < ξ(r). 41) + + Proof. Let i ≤ jw − 1. 42) − and for y ∈ Vξ(ϕ) (i), u(x)(y) = y + x′ (v), − − where x′ ∈ HomE (Vξ(ϕ) (i), Vξ(ϕ) (i + 1)) is the dual of x (with respect to the form b). Since i ≤ jw − 1, it is clear that the subgroup of elements u(x), as above, is generated by positive root subgroups inside Dk , and hence w−1 u(x)w is upper unipotent.

Imr ) is holomorphic at z¯, for all v ∈ S. This shows that A(w)(ϕτ¯,¯s ) is holomorphic at z¯. 1 applies to similar Eisenstein series on the double cover GLm (A). 5in master The Descent Map from Automorphic Representations of GL(n) to Classical Groups and the multiplication is given by (g1 , ε1 )(g2 , ε2 ) = (g1 g2 , ε1 ε2 (det g1 , det g2 )), where (x, y) is the Hilbert symbol in Fv . Let τ1 , . . 1. , ar ), 1) to γψ (det a1 · . . · det ar ) ⊗ri=1 | det ai |si τi (ai ), ai ∈ GLmi (A). ,m (A) (τ1 | det ·|s1 ⊗· · ·⊗τr | det ·|sr ).

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The Descent Map from Automorphic Representations of Gl (n) to Classical Groups by David Ginzburg, Stephen Rallis, David Soudry


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