Representations of shifted twisted quantum affine algebras

In this paper, we introduce and study shifted twisted quantum affine algebras which provide a twisted counterpart of the theory of shifted quantum affine algebras. The shifted twisted quantum affine algebra $\U_q^{\mu_+,\mu_-}(\hgs)$ is obtained from the Drinfeld current presentation of twisted quantum loop algebras by shifting the Cartan--Drinfeld currents $\phi_i^\pm(z)$ according to a coweight pair $(\mu_+,\mu_-)$. We prove that it admits a triangular decomposition and that, up to isomorphism, they depend only on the total shift $\mu=\mu_+ + \mu_-$. For each shift $\mu$, we define a category $\mathcal O_\mu$ of representations of $\U_q^\mu(\hgs) = \U_q^{0,\mu}(\hgs)$ and prove a rationality theorem for the Cartan currents: on every weight space, the two currents $\phi_i^+(z)$ and $\phi_i^-(z)$ are expansions of the same rational operator-valued function, whose degree is prescribed by $\alpha_i(\mu)$. As a consequence, we classify the simple objects of $\mathcal O_\mu$ by rational $\ell$-weights of the corresponding degrees. We then construct a deformed Drinfeld coproduct and use it to define a fusion product on the direct sum $\mathcal{O}^{sh}$ of the categories $\mathcal O_\mu$. This fusion product is compatible with $q$-characters. We also classify finite-dimensional simple modules in $\mathcal{O}^{sh}$ in terms of dominant rational $\ell$-weights, with a separate treatment of type $A_{2n}^{(2)}$. Finally, we construct restriction representations relating representations of twisted quantum affine Borel algebras to representations of shifted twisted quantum affine algebras, and establish a $q$-characters formula for simple finite-dimensional representations of shifted twisted quantum affine algebras in terms of the $q$-characters of the corresponding simple representations of the twisted quantum affine Borel algebra $\U_q(\bs)$.