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Let F_q be the finite field of q elements. We define an action of PGL(2,q) on F_q[X] and study the distribution of the irreducible polynomials that remain invariant under this action for lower-triangular matrices. As a result, we describe the possible values of the coefficients of such polynomials and prove that, with a small finite number of possible exceptions, there exist polynomials of given degree with prescribed high-degree coefficients. The speaker is supported by TÜB ̇ITAK Project 114F432.