Let \(X={\operatorname{c\!\!-\!\!Ind}}_J^G\eta\) and retain \(A_0=\Lambda(\varpi_K)\). Define the normalized shift
The original candidate is the quotient
This does not split before central normalization: a vector of \(X\) has finite support modulo compact \(J\), whereas a scalar uniformizer shifts determinant valuations. It has no nonzero eigenvector with the required central eigenvalue.
After imposing the already fixed central character, the relevant quotient is cyclic of order \(N\). Then \(T^N=1\) and the finite average
selects the same extension. The operator acts on a complex representation space, where division by \(N\) is valid.
Scope
This is a different recovery mechanism from a finite sector inside the original \(\eta\). The root is retained, not reconstructed from the compact packet alone.