1549380232-Automorphic_Forms_and_Applications__Sarnak_

LECTURE 1. FOURIER EXPANSIONS AND MULTIPLICITY ONE 107 and the resulting distributions are then stable under the involution g r- ...

108 J.W. COGDELL, £-FUNCTIONS FOR GLn ~ E V7f. Then we have two nonzero Whittaker functionals on V7f, namely Ai(~) = W'P, ( e). ...

LECTURE l. FOURIER EXPANSIONS AND MULTIPLICITY ONE 109 Therefore, for a given place v the local Kirillov models of any two irred ...

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Lecture 2. Eulerian integrals for GLn Let f ( T) again be a holomorphic cusp form of weight k on fJ for the full modular group w ...

112 J.W. COGDELL, £-FUNCTIONS FOR GLn 2.1. Eulerian integrals for GL2 Let us first consider the L-functions for cuspidal represe ...

LECTURE 2. EULERIAN INTEGRALS FOR GLn 113 In this way, to n and x we have associated a family of global Eulerian integrals with ...

114 J.W. COGDELL, £-FUNCTIONS FOR GLn Since"( E Pm+ 1 (k) and Pm+l normalizes Yn,m and stabilizes 'ljJ we may make the change of ...

LECTURE 2. EULERIAN INTEGRALS FOR GLn 115 so that cp' is a cuspidal function on Pm+ 1 (A). D z.From Lecture 1, we know that cusp ...

116 J.W. COGDELL, £-FUNCTIONS FOR GLn 2.2.2. The global integrals We now have the prerequisites for writing down a family of Eul ...

LECTURE 2. EULERIAN INTEGRALS FOR GLn 117 From this point, the fact that the integrals are Eulerian is a consequence of the uniq ...

118 J.W. COGDELL, £-FUNCTIONS FOR GLn where the unfolded global integral is with the h integral over Nm(A)\ GLm(A) and the x int ...

LECTURE 2. EULERIAN INTEGRALS FOR GLn 119 2.3.1. The mirabolic Eisenstein series To construct our Eisenstein series we return to ...

120 J .W. COGDELL, £-FUNCTIONS FOR GLn This allows us to obtain the analytic properties of the Eisenstein series from the Poisso ...

LECTURE 2. EULERIAN INTEGRALS FOR GLn 121 will be bounded in vertical strips away from the poles and satisfies the functional eq ...

122 J.W. COGDELL, L-FUNCTIONS FOR GLn the domain of integration also naturally factors we can decompose this last integral into ...

LECTURE 3 Local L-functions If (7r, V7r) is a cuspidal representation of GLn(A) and (7r', v11",) is a cuspidal rep- resentation ...

124 J.W. COGDELL, £-FUNCTIONS FOR GLn in the case m < n, where the h integration is over Nm(k)\ GLm(k) in both integrals and ...

LECTURE 3. LOCAL £-FUNCTIONS 125 Proposition 3.2. The local integrals '1i 1 (s; W, W') or '1i(s; W , W', <I?) satisfy the fol ...

126 J.W. COGDELL, £-FUNCTIONS FOR GLn Fina lly, in the case m < n one can show by a rather elementary although somewhat invol ...