Twisting finite-dimensional modules for the q-Onsager algebra $${\mathcal {O}}_q$$ via the Lusztig automorphism

The q-Onsager algebra ${\mathcal {O}}_q is defined by two generators A, A^* and two relations, called the q-Dolan/Grady relations. Recently, Baseilhac and Kolb (Transform Groups, 2020, https://doi.org/10.1007/s00031-020-09555-7 ) found an automorphism L of {\mathcal {O}}_q , that fixes A and sends A^* to a linear combination of A^* , A^2A^* , AA^*A , A^*A^2 . Let V denote an irreducible {\mathcal {O}}_q -module...