Representations of the Klein Group Determined by Quadruples of Polynomials Associated with the Double Confluent Heun Equation

The canonical representation of the Klein group $K_4=\mathbb Z_2\oplus\mathbb Z_2$ on the space $\mathbb C^*=\mathbb C\setminus\{0\}$ induces a representation of this group on the ring $\mathscr L= C[z,z^{-1}]$, $z\in\mathbb C^*$, of Laurent polynomials and, as a consequence, a representation of the group $K_4$ on the automorphism group of the group $G=GL(4,\mathscr L)$ by means of the elementwise action. The semidirect product $\widehat G= G\ltimes K_4$ is considered together with a realization of the group $\widehat G$ as a group of semilinear automorphisms of the free $4$-dimensional $\mathscr L$-module $\mathscr M^4$. A three-parameter family of representations $\mathfrak R$ of $K_4$ in the group $\widehat G$ and a three-parameter family of elements $\mathfrak X\in\mathscr M^4$ with polynomial coordinates of degrees $2(\ell-1)$, $2\ell$, $2(\ell-1)$, and $2\ell$, where $\ell$ is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector $\mathfrak X$ is a fixed point of the corresponding representation $\mathfrak R$. An algorithm for calculating the polynomials that are the components of $\mathfrak X$ was obtained in a previous paper of the authors, in which it was proved that these polynomials give explicit formulas for automorphisms of the solution space of the doubly confluent Heun equation.