This paper by A. M. Perelomov generalizes the concept of coherent states, originally associated with the nilpotent group of Weyl, to arbitrary Lie groups. The paper investigates the construction and properties of coherent states for simple Lie groups. Coherent states are states that are closely related to the group's representation in a Hilbert space. The paper proposes a method that can be applied to any Lie group and is consistent with the group's action on the set of coherent states. The paper begins by introducing the concept of coherent states and their connection to the nilpotent group of Weyl. It then discusses the limitations of previous methods in generalizing coherent states to other Lie groups, particularly compact groups. The paper presents a new method for extending the concept of coherent states to any Lie group. The paper then describes the general properties of coherent states. It defines the coherent states as a set of states generated by the action of a group's representation on a fixed vector in the Hilbert space. The paper shows that these states are determined by points in the factor space G/H, where H is the stationary subgroup of the fixed vector. The paper also discusses the properties of the phase factor associated with coherent states. It shows that this phase factor is a one-dimensional unitary representation of the group H. If this representation is not trivial, the factor group of H on its commutant is non-trivial, and the character of this group determines the representation of H. The paper concludes by highlighting the importance of coherent states in quantum theory and their potential applications in various fields.This paper by A. M. Perelomov generalizes the concept of coherent states, originally associated with the nilpotent group of Weyl, to arbitrary Lie groups. The paper investigates the construction and properties of coherent states for simple Lie groups. Coherent states are states that are closely related to the group's representation in a Hilbert space. The paper proposes a method that can be applied to any Lie group and is consistent with the group's action on the set of coherent states. The paper begins by introducing the concept of coherent states and their connection to the nilpotent group of Weyl. It then discusses the limitations of previous methods in generalizing coherent states to other Lie groups, particularly compact groups. The paper presents a new method for extending the concept of coherent states to any Lie group. The paper then describes the general properties of coherent states. It defines the coherent states as a set of states generated by the action of a group's representation on a fixed vector in the Hilbert space. The paper shows that these states are determined by points in the factor space G/H, where H is the stationary subgroup of the fixed vector. The paper also discusses the properties of the phase factor associated with coherent states. It shows that this phase factor is a one-dimensional unitary representation of the group H. If this representation is not trivial, the factor group of H on its commutant is non-trivial, and the character of this group determines the representation of H. The paper concludes by highlighting the importance of coherent states in quantum theory and their potential applications in various fields.