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

Abstract

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 of finite dimension at least two, on which each of A, $A^*$ is diagonalizable. It is known that A, $A^*$ act on V as a tridiagonal pair of q-Racah type, giving access to four familiar elements K, B, $K^\downarrow$, $B^\downarrow$ in $\mathrm{End}(V)$ that are used to compare the eigenspace decompositions for A, $A^*$ on V. We display an invertible $H \in \mathrm{End}(V)$ such that $L(X)=H^{-1} X H$ on V for all $X \in {\mathcal {O}}_q$. We describe what happens when one of K, B, $K^\downarrow$, $B^\downarrow$ is conjugated by H. For example $H^{-1}KH=a^{-1}A-a^{-2}K^{-1}$ where a is a certain scalar that is used to describe the eigenvalues of A on V. We use the conjugation results to compare the eigenspace decompositions for A, $A^*$, $L^{\pm 1}(A^*)$ on V. In this comparison we use the notion of an equitable triple; this is a 3-tuple of elements in $\mathrm{End}(V)$ such that any two satisfy a q-Weyl relation. Our comparison involves eight equitable triples. One of them is $a A - a^2 K$, $M^{-1}$, K where $M= (a K-a^{-1} B)(a-a^{-1})^{-1}$. The map M appears in earlier work of Bockting-Conrad (Linear Algebra Appl 437:242–270, 2012) concerning the double lowering operator $\psi$ of a tridiagonal pair.