Parabolic extensions of a Hecke ring of the general linear group

One describes a wide class of polynomials with coefficients from a Hecke ring of the general linear group, decomposing into factors over the Hecke ring of an appropriate parabolic subgroup. One gives examples of such decompositions, used in the theory of modular forms, and also the decomposition of the denominator of the Hecke series of the general linear group, yielding another proof of the rationality of the Hecke series of the group GL_n.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 154, pp. 36–45, 1986.