# Whittaker modules for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[\tau]$

## TL;DR

This paper investigates the structure and irreducibility of Whittaker modules for the twisted affine Nappi-Witten Lie algebra, providing classifications and characterizations of Whittaker vectors in various cases.

## Contribution

It establishes irreducibility criteria and classifies Whittaker vectors for different types of modules over the twisted affine Nappi-Witten Lie algebra.

## Key findings

- Irreducibility of $L_{ ext{ extit{psi}}, ext{ extit{xi}}}$ when $ ext{ extit{xi}}
eq 0$
- Complete determination of Whittaker vectors for $ ext{ extit{xi}}=0$
- Full characterization of Whittaker vectors in the singular case with $ ext{ extit{xi}}
eq 0$

## Abstract

The Whittaker module $M_{\psi}$ and its quotient Whittaker module $L_{\psi, \xi}$ for the twisted affine Nappi-Witten Lie algebra $\widehat{H}_{4}[\tau]$ are studied. For nonsingular type, it is proved that if $\xi\neq 0$, then $L_{\psi,\xi}$ is irreducible and any irreducible Whittaker $\widehat{H}_{4}[\tau]$-module of type $\psi$ with ${\bf k}$ acting as a non-zero scalar $\xi$ is isomorphic to $L_{\psi,\xi}$. Furthermore, for $\xi=0$, all Whittaker vectors of $L_{\psi, 0}$ are completely determined. For singular type, the Whittaker vectors of $L_{\psi, \xi}$ with $\xi \neq 0$ are fully characterized.