The paper by G. Lusztig focuses on constructing a canonical basis for the quantized enveloping algebra $\mathbf{U}^+$, which is an algebra over the field of rational functions $\mathbf{Q}(v)$. This algebra is associated with a root system and specializes to the classical enveloping algebra $\mathbf{U}_1^+$ when $v = 1$. The main result is the construction of a canonical basis $\mathbf{B}$ of $\mathbf{U}^+$, which is defined through a series of steps involving PBW bases, a quotient module $\mathcal{L}$, and a $\mathbf{Z}[v^{-1}]$-basis $B$. The final step involves using the involution $^-: \mathbf{U}^+ \to \mathbf{U}^+$ and properties of quiver representations to establish the uniqueness of $\mathbf{B}$. Lusztig also discusses the geometric interpretation of the multiplication in $\mathbf{U}^+$ using a "convolution" operation on complexes in the derived category of constructible sheaves. This allows for the computation of local intersection cohomology and the Poincaré polynomial of certain closures. Additionally, he explores the analogous problem for the cyclic quiver, which corresponds to an affine Dynkin graph of type $A$, and shows that the closures of the corresponding strata are locally isomorphic to affine Schubert varieties. The canonical basis $\mathbf{B}$ has several remarkable properties, including that the product of two elements in $\mathbf{B}$ is a linear combination of elements in $\mathbf{B}$ with coefficients in $\mathbf{Z}[v, v^{-1}]$. It is also well-adapted to finite-dimensional representations of $\mathbf{U}$, providing a canonical basis for simple modules of the corresponding semisimple Lie algebra. The paper concludes with combinatorial formulas for dimensions and weight spaces, which are derived from the theory developed in the paper.The paper by G. Lusztig focuses on constructing a canonical basis for the quantized enveloping algebra $\mathbf{U}^+$, which is an algebra over the field of rational functions $\mathbf{Q}(v)$. This algebra is associated with a root system and specializes to the classical enveloping algebra $\mathbf{U}_1^+$ when $v = 1$. The main result is the construction of a canonical basis $\mathbf{B}$ of $\mathbf{U}^+$, which is defined through a series of steps involving PBW bases, a quotient module $\mathcal{L}$, and a $\mathbf{Z}[v^{-1}]$-basis $B$. The final step involves using the involution $^-: \mathbf{U}^+ \to \mathbf{U}^+$ and properties of quiver representations to establish the uniqueness of $\mathbf{B}$. Lusztig also discusses the geometric interpretation of the multiplication in $\mathbf{U}^+$ using a "convolution" operation on complexes in the derived category of constructible sheaves. This allows for the computation of local intersection cohomology and the Poincaré polynomial of certain closures. Additionally, he explores the analogous problem for the cyclic quiver, which corresponds to an affine Dynkin graph of type $A$, and shows that the closures of the corresponding strata are locally isomorphic to affine Schubert varieties. The canonical basis $\mathbf{B}$ has several remarkable properties, including that the product of two elements in $\mathbf{B}$ is a linear combination of elements in $\mathbf{B}$ with coefficients in $\mathbf{Z}[v, v^{-1}]$. It is also well-adapted to finite-dimensional representations of $\mathbf{U}$, providing a canonical basis for simple modules of the corresponding semisimple Lie algebra. The paper concludes with combinatorial formulas for dimensions and weight spaces, which are derived from the theory developed in the paper.