Question on the residual representation

Question

Let $G=SO_n$ and fix a borel subgroup $P_0$ of $G$. Let $P=MN$ be a standard maximal parabolic subgroup $G$ and $\sigma$ a cuspidal representation of $M$

Consider the normalized parabolic induced representation $\text{Ind}_P^G(\sigma|\cdot|^z)$ and for sufficiently large $z$, we can define Eisenstein series $E(z,\phi)$ for $\phi \in \text{Ind}_P^G(\sigma)$. Since $E(z,\phi)$ has a meromorphic continuation, let $z_0$ be a simple pole of $E(z,\phi)$. Put $\mathcal{E}(\phi,z_0)$ a residue of $E(z,\phi)$ at $z=z_0$.

I am wondering if there are two $\phi_1,\phi_2 \in \text{Ind}_P^G(\sigma)$ such that $\mathcal{E}(\phi_1,z_0)=\mathcal{E}(\phi_2,z_0)$, then $E(\phi_1,z)=E(\phi_2,z)$ as meromorphic functions on $\mathbb{C}$. Is it right?

Answer 1

In the positive cone/half-plane this is essentially never the case, because in that region the map "take residue at $z_o$" is an intertwining map with non-trivial kernel (and image is the smaller quotient repn generated by the residue) from the principal series generated by the Eisenstein series, to the repn generated by the residue. So when two Eisenstein series differ by something in the kernel, they'll have the same residue.

This is already visible with $PGL(2)$, where Eisenstein series have at most a pole at $s=1$ (in classical normalization) in the right half-plane, and the residues are constants. At any higher level, even constructed in a classical way, there are several different data (with fixed right $K$-type, etc.), and several linearly independent Eisenstein series, but just one residue possible.

In contrast, in the left half-plane and/or in other images of the positive cone, due to zeros of the denominator of the constant term(s), there are many poles of Eisenstein occurring at points where the principal series is irreducible. At those points, the "take residue" map is a $G$-isomorphism. In that case, your desired property holds.