On the Stable Transfer for ${\mathrm{Sym}}^n$ Lifting of ${\mathrm{GL}}_2$

Abstract

Following the paradigm of \cite{MR3117742}, we are going to explore the stable transfer factors for ${\mathrm{Sym}}^n$ lifting from ${\mathrm{GL}}_2$ to ${\mathrm{GL}}_{n+1}$ over any local fields $F$ of characteristic zero with residue characteristic not equal to $2$. When $F=\mathbb{C}$ we construct an explicit stable transfer factor for any $n$. When $n$ is odd, we provide a reduction formula, reducing the question to the construction of the stable transfer factors when the $L$-morphism is the diagonal embedding from ${\mathrm{GL}}_2(\mathbb{C})$ to finitely many copies of ${\mathrm{GL}}_2(\mathbb{C})$ under mild assumptions on the residue characteristic of $F$. With the assumptions on the residue characteristic, the reduction formula works uniformly over any local fields of characteristic zero, except that for $p$-adic situation we need to exclude the twisted Steinberg representations.

Comments are welcome