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

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E n _ i , e _ n } . ( { e i , . . , e n , e _ n , . . ) Given a smooth representation A of GLn(F) and a complex number s, we consider the representation of Qn defined by m(a) u(b) H* | det a | * + i ^ A(a) We denote by p\iS the corresponding induced representation of Hn. Put for a G GLn(F), A » = A(a*) . We consider the intertwining operator on VPrt, defined for 3ft(s) > > 0 by M(itf n ,£ T| ,)(A,m) = / tr^uw"1 h,m)du M(wn,£T}S) lies in VPr0 lmm9, if n is even and Vpr0 x_9 if n is odd. 2.

S Bi£GLn(F) (JuAJY^^i*))^ |det . (»),r®|det-| -*) . We have extended ^>a to i? 1). We have as modules over GL n (F). e. the group of matrices I tr-y J. Then C\E is isomorphic to the fol- lowing subgroup of P/+i. -)-(; :;)i; e ^- ^'-'"J V>« on CXJE1 is mapped to ^a(c(y,z,a:)) = ^(z)^(y/) where y = I ' I. GLn(F) is embedded in Pi+i b y m n _ 1. \ W / . |'-<+* V; 1 ) (Indc means compact induction). ], Jc(*) has finite length as a Pi+i module, with a unique subrepresentation I n d ^ ^ 1 ^ , which does not appear further as a subquotient.

N, such that N i=i Prw/. 1) for W in A(W,(Ttt) *(0 > 0 (and »(«) > 0). = / f / du9Ji)il>~l{u)du) I JNt\Gt \JYt J JjC(t'n) = L Thus we assume that f^rt9{^PttnUA9))i)a{x)dxdg d9Jt) 1 „ Jir,A^t,nU}n{9))^a{x)dxdg . To justify the last equality, we have to show that the last integral converges absolutely in some appropriate domain. It turns out to be a vertical strip L^. Its exact shape will be given later. This will soon be proved in a separate series of lemmas. )= I I / JYt Jzt\GLt(F) JYi * c (n,m)|detm|-<+* ,hr,A^^tniiAVmu))'tl;a(x)dxdydrndu.