The paper by Robert Steinberg focuses on the study of irreducible representations of semisimple algebraic groups over fields of characteristic \( p \neq 0 \), particularly rational representations, and aims to determine all representations of corresponding finite simple groups. The main results include: 1. **Tensor Product Theorem**: Every irreducible rational projective representation of a semisimple algebraic group can be uniquely expressed as a tensor product of irreducible rational projective representations, each associated with a function from the set of simple roots to the nonnegative integers (high weight). 2. **Finite Simple Groups**: For finite simple algebraic groups over fields with \( q = p^n \) elements, every irreducible projective representation is the restriction of a rational representation of the corresponding infinite algebraic group. The number of such representations is \( q^l \), and each has a high weight \( \lambda \) where \( 0 \leq \lambda(a) \leq q - 1 \). 3. **Special Isogenies**: The paper discusses special isogenies for simple groups of types \( B_l \), \( C_l \), and \( F_4 \) in characteristic 2, and type \( G_2 \) in characteristic 3. These isogenies are used to construct and analyze representations of these groups. 4. **Nonnormal Forms**: The paper also explores nonnormal forms of simple groups, such as \( A_l^i \), \( D_l \), \( E_8^i \), and \( D_l^i \), and their covering groups. It shows that every irreducible \( \Gamma_q^l \)-module can be uniquely written as a tensor product of irreducible rational projective representations. 5. **Character Theory**: The paper provides a detailed analysis of the characters of irreducible modules, particularly for the module \( M_q \) with the highest possible high weight \( \omega^{q-1} \). It shows that the character of a semisimple element \( x \) on \( M_q \) is given by \( \chi(x) = \pm q^{d(x)} \). The paper is a comprehensive study of the representation theory of semisimple algebraic groups and their finite simple counterparts, using advanced techniques from algebraic geometry and group theory.The paper by Robert Steinberg focuses on the study of irreducible representations of semisimple algebraic groups over fields of characteristic \( p \neq 0 \), particularly rational representations, and aims to determine all representations of corresponding finite simple groups. The main results include: 1. **Tensor Product Theorem**: Every irreducible rational projective representation of a semisimple algebraic group can be uniquely expressed as a tensor product of irreducible rational projective representations, each associated with a function from the set of simple roots to the nonnegative integers (high weight). 2. **Finite Simple Groups**: For finite simple algebraic groups over fields with \( q = p^n \) elements, every irreducible projective representation is the restriction of a rational representation of the corresponding infinite algebraic group. The number of such representations is \( q^l \), and each has a high weight \( \lambda \) where \( 0 \leq \lambda(a) \leq q - 1 \). 3. **Special Isogenies**: The paper discusses special isogenies for simple groups of types \( B_l \), \( C_l \), and \( F_4 \) in characteristic 2, and type \( G_2 \) in characteristic 3. These isogenies are used to construct and analyze representations of these groups. 4. **Nonnormal Forms**: The paper also explores nonnormal forms of simple groups, such as \( A_l^i \), \( D_l \), \( E_8^i \), and \( D_l^i \), and their covering groups. It shows that every irreducible \( \Gamma_q^l \)-module can be uniquely written as a tensor product of irreducible rational projective representations. 5. **Character Theory**: The paper provides a detailed analysis of the characters of irreducible modules, particularly for the module \( M_q \) with the highest possible high weight \( \omega^{q-1} \). It shows that the character of a semisimple element \( x \) on \( M_q \) is given by \( \chi(x) = \pm q^{d(x)} \). The paper is a comprehensive study of the representation theory of semisimple algebraic groups and their finite simple counterparts, using advanced techniques from algebraic geometry and group theory.