Principal differential ideals and a generic inverse differential galois problem for GL_n

We characterize the principal differential ideals of a polynomial ring in n² indeterminates with coefficients in the ring of differential polynomials in n² indeterminates and derivation given by a "general" element of Lie(GL_n) and use this characterization to construct a generic Picard-Vessiot extension for GL_n. In the case when the differential base field has finite transcendence degree over its field of constants we provide necessary and sufficient conditions for solving the inverse differential Galois problem for these groups via specialization from our generic extension.