Analytic Properties of Residual Eisenstein Series, I

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 GL_ab of Sp_2ab are located in the ‘segment’ of half integers X_b 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.