Eichler-Shimura isomorphism and vector-valued Hecke eigenpolynomials

Abstract

The Eichler-Shimura isomophism gives a relation between holomorphic cusp forms and period polynomials. In [21], Paşol and Popa introduced vector-valued period polynomials and studied various properties of such polynomials. In this paper, we find a basis for the space of vector-valued period polynomials consisting of Hecke eigenpolynomials, and relate the basis to a Miller-like basis of the space of weakly holomorphic cusp forms by calculation of the matrix representation of the Hecke operator T˜l. Furthermore, we find the relation between our Hecke eigenpolynomials and the even and odd parts of the period polynomials induced from holomorphic Hecke eigenforms, and examine algebracity of the Hecke eigenpolynomials.