In a series of papers the authors and \textit{Ed Cline}, known as CPS, developed a theory of abstract Kazhdan-Lusztig theories for a highest weight category \(\mathcal C\). It is well-known that \(\mathcal C\) realizes as the category of \(S\)-modules for a finite dimensional quasi-hereditary algebra \(S\). If \(S\) is positively graded, CPS also developed a theory on graded Kazhdan-Lusztig theories for the category \(\mathcal C\) of finitely generated graded \(S\)-modules. The property of having a graded Kazhdan-Lusztig theory is closely related to the Koszul property of \(S\). More precisely, \(\mathcal C\) has a graded Kazhdan-Lusztig theory if and only if \(S\) is a Koszul algebra and \(\mathcal C\) has an (ungraded) Kazhdan-Lusztig theory [see CPS, Proc. Lond. Math. Soc., III. Ser. 68, No. 2, 294-316 (1994; Zbl 0819.20045)]. This paper considers the relation of the Koszul property of \(S\) with some automorphisms of \(S\). One of the main results in the paper states essentially that, given an algebra \(S\) and an automorphism \(\sigma\), if the Ext-algebra associated to \(S\) can be given its natural graded structure by means of the eigenvalues of the induced action of \(\sigma\), then necessarily \(S\) is (graded and is) a Koszul algebra. This result, applied to quasi-hereditary algebras \(S\), shows how the existence of certain automorphisms on \(S\) can lead not only to a graded structure on \(S\), but also, when the highest weight category \({\mathcal C}=\text{mod-}S\) has an ungraded Kazhdan-Lusztig theory, to a graded Kazhdan-Lusztig theory on the associated graded module category. In the paper, the above result is, in particular, applied to the category of \(\ell\)-adic perverse sheaves on the flag variety \(G/B\), where \(G\) is a semisimple algebraic group, and \(B\) is one of its Borel subgroups. This category is shown to be a highest weight category, and, in positive characteristic, the Frobenius morphism on \(G/B\) naturally induces an automorphism on the associated quasi-hereditary algebra, satisfying the required conditions. Therefore, the associated graded module category \(\mathcal C\) has a graded Kazhdan-Lusztig theory. In this way, the authors obtain a natural proof that the algebra associated to the category of \(\ell\)-adic perverse sheaves on \(G/B\) is Koszul. (This result has been proved by \textit{A. Beilinson, V. Ginzburg, W. Soergel} in a different way. See their paper [in J. Am. Math. Soc. 9, No. 2, 473-527 (1996)].) Moreover, since the category of \(\ell\)-adic perverse sheaves on \(G/B\) is known to be equivalent to the principal block \(\mathcal O_{\text{triv}}\) of the category \(\mathcal O\) of the associated complex Lie algebra, it is obtained, as a corollary, that the algebra associated to \(\mathcal O_{\text{triv}}\) is Koszul. (This result was first announced by \textit{A. Beilinson, V. Ginzburg} with an outline of a proof [see their preprint Mixed categories, Ext-duality, and representations (results and conjectures)]. In an earlier version of the above-mentioned preprint, Beilinson-Ginzburg-Soergel gave a proof entirely different from the proof presented in this paper).