Notes on Analytic Properties of Residual Eisenstein Series, I

TL;DR

This paper extends Kudla and Rallis's results on poles of Eisenstein series to residual-data cases, analyzing their residues, cuspidal exponents, and automorphic forms' properties, especially in relation to their L^2 integrability.

Contribution

It generalizes the understanding of Eisenstein series poles and residues, providing explicit descriptions of cuspidal exponents and their relation to automorphic L-functions.

Findings

Poles of Eisenstein series are located within specific segments of half integers.

Residues at certain points are shown to be in L^2 space.

Explicit cuspidal exponents are determined for key residual automorphic forms.

Abstract

We partially generalize the results of Kudla and Rallis 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 $\Delta(\tau,b)$ on the Levi $\mathrm{GL}_{ab}$ of $\mathrm{Sp}_{2ab}$ are located in the "segment" of half integers $X_{b}$ between a "right endpoint" and its negative, inclusive of endpoints. The right endpoint is $\pm b/2$, or $(b-1)/2$, depending on the analytic properties of the automorphic $L$-functions attached to $\tau$. We study the automorphic forms $\Phi_{i}^{(b)}$ obtained as residues at the points $s_i^{(b)}$ (defined precisely in the paper) by calculating their cuspidal exponents in certain cases. In the case of the "endpoint" $s_0^{(b)}$ and `first interior point' $s_1^{(b)}$ in the segment of singularity points, we are able to determine a set containing \textit{all possible} cuspidal exponents of $\Phi_0^{(b)}$ and $\Phi_1^{(b)}$ 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 $L^2(G(k)\backslash G(\mathbf{A}))$. In a more precise sense, our result establishes a relationship between, on the one hand, the actually occurring cuspidal exponents of $\Phi_i^{(b)}$, 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. This preprint is a longer version of the paper "Analytic Properties of Residual Eisenstein Series, I", with the details of some proofs added and some additional examples adduced in support of the main conjecture.