In studying the L-functions of induced representation, it is not easily come up with me the papers or books dealing the L-function of irreducible subrepresentation of induced representation, while the L-function of Langlands quotient is already defined. So I want to ask you how to define the L-function of it.
To explain this more precisely, let me recall some basic notions of it.
Let $F$ be a p-adic field and $G$ be an algebraic group over $F$. (we think $G$ as $GL_n(F)$ ) Let $P=MN$ be a parabolic subgroup of $G$ and $M$ and $N$ are its Levi and unipotent subgroups. If we let $\sigma$ be a irreducible supercuspidal representation of $M$ and extend it to $P$ by acting trivially of $N$, we can think the induced representation $I:=Ind_P ^G \sigma$.
Then we know that there is a unique irreducible subrepresentation $Z(\sigma)$ of $I$.
My question is this. How are the two standard L-functions $L(s,\sigma)$ and $L(s,Z(\sigma))$ related?
Furthermore, if $G$ is not general linear group but unitary group, and $E$ is quadratic extension of $F$, then how is the relationship between $L(s,BC(\sigma))$ and $L(s,BC (Z(\sigma)))$? (here $BC$ is the base change of the representation to quadratic extension field $E$)
References for this are greatly welcome!