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.
Categories: math.RT math.NT