Pure Mathematics

Rankin-Selberg Convolutions for So2L+1 X Gln: Local Theory by David Soudry

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By David Soudry

This paintings reviews the neighborhood thought for sure Rankin-Selberg convolutions for a standard functionality of measure of typical representations of over a neighborhood box. The neighborhood integrals converge in a half-plane and proceed meromorphically to the full aircraft. One major result's the lifestyles of neighborhood gamma and elements. The gamma issue is received as a proportionality issue of a practical equation chuffed by way of the neighborhood integrals. furthermore, Soudry establishes the multiplicativity of the gamma issue (first variable). a distinct case of this end result yields the unramified computation and contains a brand new suggestion now not offered prior to. This presentation, which includes specific proofs of the consequences, turns out to be useful to experts in automorphic kinds, illustration concept, and capabilities, in addition to to these in different parts who desire to observe those effects or use them in different circumstances.

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Extra info for Rankin-Selberg Convolutions for So2L+1 X Gln: Local Theory

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T£ = Ind^(7_£, where (r_^(m) = |detm|"*^£ be a holomorphic section in TT^. We assume that £ takes values in W( the standard Whittaker model of dugji)rl{u)du . 1) JYt This integral converges absolutely for 3ft(C) large enough, it has a holomorphic extension to the whole plane, which defines for g = 1 a Whittaker functional for wc for all < G C. l]. 2. Assume that £>n. Let Wa and WT be elements of W((r, ip) and W(T, tp~~l) respectively. Let a be a Schwartz function on Fn (row vectors).

Jxl***) firtm (zfanitM) *a(*)dSdg O 6. A(W^Tf8) CAN BE MADE CONSTANT ( N O N A R C H I M E D E A N CASE) In this chapter F is a nonarchimedean local field. Assumptions are as in section 4. 1 Proposition. Assume I > n. There is a choice of W G W(7r, ip) and £T,S € VpTt8, such that A(w,tTfS) = i, Vsec. Proof. 1) f _ JAnxVn f JX(ntt) W (xjn,t(m(a)u)) «- 1 (m(a))| det a | * + ^ T ) , ( i l ( y ) ; az)dxdadu Here tJ = m(J)u(y), J € Zn. Choose £T>5 to have support in Qn • V, where V is a small open compact subgroup of Hn, and such that fTj, is fixed by the elements of V.

The proofs run in analogy with the proofs of the Euler product expansions in the global case. 1) the proof of the Euler product expansion in case £ > n. We do this in a little different manner than in [G], so that the "translation" to the local field case is transparent. This can be seen in the case £ > n as well. 3). Finally, we conclude from a theorem of Bernstein, that A(W,£T}s) is a rational function in #~ # , where q is the number of elements in the residue field of F. 1. We assume, (in this section only), that Jb is a number field, A - its ring of adeles, W,T - irreducible, automorphic, cuspidal representations of G/(A) and GL n (A) respectively, V~ nontrivial character of k\A.

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