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Osborne M.S., Warner G. the idea of Eisenstein structures (AP, 1981)(ISBN 0125292503)

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Extra resources for The theory of Eisenstein systems

Example text

R ) such that Gi. r, r G= i=1 as desired. // = 1,. . , r). 8, we shall need some additional preparation. Lemma Let ( P , S) be a r-percuspidal split parabolic subgroup of G with unipotent radical N and split component A. Suppose that {;:; ;[tl are sequences such that lim any,,an- = 1. n+ a0 Then yn is eventually in N. Proof Suppose that U N ,U,, and U N are neighborhoods of 1 in N,L, and N, respectively (Nbeing, as usual, the nilpotent subgroup of G opposed to N)-then U,-. U, . U Nis a neighborhood of 1 in G.

We shall therefore settle for a brief heuristical justification. , Eisenstein series associated with cusp forms. ] These functions are represented by convergent series in regions removed from the unitary spectrum. , in the scalar product formula, poles are encountered and residues obtained. In this way, new functions arise which behave like Eisenstein series but are not. The notion of Eisenstein system captures, through a list of axioms, the essential properties of such functions. One then uses Eisenstein systems as a vehicle for developing a residual theory of Eisenstein series which mirrors in every essential way its cuspidal counterpart.

Anyn. Since lim a; = -00, n-tm it is clear that we have lim aby'ab- ' = 1, n- m or still lim x, an(yny'yn- ')a,- ' x,- ' = 1.

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