Imaginary Verma modules for the twisted affine Nappi-Witten Lie algebra \(\widetilde{H}_4[\sigma]\) (Q2073169)

The Nappi-Witten Lie algebra \(H_4\) is a four-dimensional Lie algebra over \(\mathbb{C}\) generated by \(a, b, c, d\) with the following Lie bracket \([a, b] = c, [d, a] = a, [d, b] = -b, [c, H_4] = 0.\) The affine Nappi-Witten Lie algebra is defined as \(\hat{H}_4=H_4\otimes\mathbb{C}[t, t^{-1}] \oplus \mathbb{C}K\) equipped with some Lie bracket. The twisted affine Nappi-Witten Lie algebra \(\hat{H}_4[\sigma]\) is a subalgebra of \(\hat{H}_4\). The direct sum of \(\hat{H}_4[\sigma]\) with \(\mathbb{C}D\) is denoted by \(\tilde{H}_4[\sigma]\), which is also called a twisted affine Nappi-Witten Lie algebra since \([K,D]=0,[D,x\otimes t^n]=nx\otimes t^n\). In this paper, the author gave an irreducible criterion for the imaginary Verma modules for the twisted affine Nappi-Witten Lie algebra \(\tilde{H}_4[\sigma]\). When the imaginary Verma module is reducible, the author determined the explicit characterizations of the maximal submodule, imaginary vacuum vectors, and the composition structure.