Invariant theory of \(\imath \)quantum groups of type AIII

We develop an invariant theory of quasi-split \(\imath \) $ı$ quantum groups \({\textbf {U}} _n^\imath \) ${\mathbf{U}}_n^{ı}$ of type AIII on a tensor space associated to \(\imath \) $ı$ Howe dualities. The first and second fundamental theorems for \({\textbf {U}} _n^\imath \) ${\mathbf{U}}_n^{ı}$ -invariants are derived.