We demonstrate a relationships between the representation theory of Borel subgroups and parabolic subgroups of general linear groups. In particular, we show that the representations of Borel subgroups could be computed from representations of certain maximal parabolic subgroups.

In this article we prove conditions under which a certain parabolic group cohomology space over a finite field F is a faithful module for the Hecke algebra of Katz modular forms over an algebraic closure of F. These results can e.g. be used to compute Katz modular forms of weight one with methods of linear algebra over F. MSC Classification: 11F25 (primary), 11F33, 11F67, 11Y40 (secondary).

In this communication, we introduce and study the various controllability problems for Neumann co-operative parabolic linear system involving Laplace operator with distributed or boundary controls and with observations belong to different spaces.