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

摘要

The canonical representation of the Klein group K ~(4)= ℤ~(2)⊕ℤ~(2)on the space ℂ* = ℂ {0} induces a representation of this group on the ring L = C [z, z_(−1)], z ∈ 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, L ) by means of the elementwise action. The semidirect product Ĝ G = G K ~(4)is considered together with a realization of the group Ĝ as a group of semilinear automorphisms of the free 4-dimensional L -module M _(4). A three-parameter family of representations R of K ~(4)in the group Ĝ and a three-parameter family of elements X ∈ M _(4)with polynomial coordinates of degrees 2( ℓ − 1), 2 ℓ , 2( ℓ − 1), and 2 ℓ , where ℓ is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector X is a fixed point of the corresponding representation R. An algorithm for calculating the polynomials that are the components of 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.