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Analytic Properties of Residual Eisenstein Series, I
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We partially generalize the results of [16] on the poles of degenerate, Siegel-parabolic Eisenstein series to residual-data Eisenstein series. In particular, for a, b integers greater than 1, we show that poles of the Eisenstein series induced from the Speh representation Δ(τ, b) on the Levi GLab of Sp2ab are located in the ‘segment’ of half integers Xb between a ‘right endpoint’ and its negative, inclusive of endpoints. The right endpoint is ±b/2, or (b-1)/2, depending on the analytic properties of the automorphic L-functions attached to τ. We study the automorphic forms Φ(b)i obtained as residues at the points s(b)i (defined precisely in the paper) by calculating their cuspidal exponents in certain cases. In the case of the ‘endpoint’ s(b)0 and ‘first interior point’ s(b)1 in the segment of singularity points, we are able to determine a set containing all possible cuspidal exponents of Φ(b)0 and Φ(b)1 precisely for all a and b. In these cases, we use the result of the calculation to deduce that the residual automorphic forms lie in L2(G(k)\G(A)). In a more precise sense, our result establishes a relationship between, on the one hand, the actually occurring cuspidal exponents of Φ(b)i, residues at interior points which lie to the right of the origin, and, on the other hand, the ‘analytic properties’ of the original residual-data Eisenstein series at the origin.
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