Explicit formulas concerning eigenvectors of weakly non-unitary matrices

We investigate eigenvector statistics of the Truncated Unitary ensemble $\mathrm{TUE}(N,M)$ in the weakly non-unitary case $M=1$, that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when $M \geq N$, suggesting possible extensions to $\mathrm{TUE}(N,M)$ for all values of $M$.