The chapter discusses the concept of Koszul duality in the context of representation theory, focusing on the category $\mathcal{O}$ associated with a complex semisimple Lie algebra $\mathfrak{g}$. The main result, Theorem 1.1.1, states that there is an isomorphism of finite-dimensional $\mathbb{C}$-algebras: \[ \operatorname{End}_{\mathcal{O}} P \cong \operatorname{Ext}_{\mathcal{O}}^{\bullet}(L, L), \] where $P$ is the projective cover of the simple module $L$. This algebra is shown to be a Koszul ring, meaning it admits a graded projective resolution with generators in degrees one and higher. The chapter also explores the dual nature of this algebra, demonstrating that it is self-dual under certain conditions. The text further generalizes these results to other infinitesimal characters and parabolic subalgebras, leading to Theorem 1.1.3, which provides isomorphisms involving the endomorphism algebras and Ext-algebras of modules in the categories $\mathcal{O}_\lambda$ and $\mathcal{O}^\mathfrak{q}$. These results highlight the duality between these categories and their graded versions. Additionally, the chapter delves into the geometric interpretation of Koszul rings, showing how they arise naturally in the framework of mixed geometry, specifically through the study of perverse sheaves on stratified varieties. The main theorem in this section asserts that the category of perverse sheaves on a stratified variety is equivalent to the category of finite-dimensional modules over a finite-dimensional graded Koszul algebra. Finally, the chapter provides detailed proofs and definitions related to Koszul rings, including their quadratic structure, filtration properties, and the construction of the Koszul complex. It also discusses the quadratic dual of a Koszul ring and the relationship between the quadratic dual and the Ext-algebra, leading to the conclusion that the quadratic dual of a left finite Koszul ring is also Koszul.The chapter discusses the concept of Koszul duality in the context of representation theory, focusing on the category $\mathcal{O}$ associated with a complex semisimple Lie algebra $\mathfrak{g}$. The main result, Theorem 1.1.1, states that there is an isomorphism of finite-dimensional $\mathbb{C}$-algebras: \[ \operatorname{End}_{\mathcal{O}} P \cong \operatorname{Ext}_{\mathcal{O}}^{\bullet}(L, L), \] where $P$ is the projective cover of the simple module $L$. This algebra is shown to be a Koszul ring, meaning it admits a graded projective resolution with generators in degrees one and higher. The chapter also explores the dual nature of this algebra, demonstrating that it is self-dual under certain conditions. The text further generalizes these results to other infinitesimal characters and parabolic subalgebras, leading to Theorem 1.1.3, which provides isomorphisms involving the endomorphism algebras and Ext-algebras of modules in the categories $\mathcal{O}_\lambda$ and $\mathcal{O}^\mathfrak{q}$. These results highlight the duality between these categories and their graded versions. Additionally, the chapter delves into the geometric interpretation of Koszul rings, showing how they arise naturally in the framework of mixed geometry, specifically through the study of perverse sheaves on stratified varieties. The main theorem in this section asserts that the category of perverse sheaves on a stratified variety is equivalent to the category of finite-dimensional modules over a finite-dimensional graded Koszul algebra. Finally, the chapter provides detailed proofs and definitions related to Koszul rings, including their quadratic structure, filtration properties, and the construction of the Koszul complex. It also discusses the quadratic dual of a Koszul ring and the relationship between the quadratic dual and the Ext-algebra, leading to the conclusion that the quadratic dual of a left finite Koszul ring is also Koszul.