TITLE: Why the name O for category O?
QUESTION [12 upvotes]: What is the motivation behind naming the category O appearing in the theory of Lie algebras? Does O stand for something? 
Here is a question Why the BGG category O? that further confuses me. It seems like there is a notion of when a category is O, is it?

REPLY [9 votes]: From [Humphreys: Representations of semisimple Lie algebras in the BGG category O], notes for Chapter 1:

The letter chosen to label the category is the first letter of a
  Russian word meaning “basic”

which is основной.