I'm a graduate student studying algebraic geometry. Recently, When I studying Hodge theory, I saw sl2-representation is used in Hodge theory. So I think that studying representation theory may be helpful for me. Can you recommand me some good text for studying representation theory, focused on materials helpful for algebraic geometry, and not so difficult to read.
I would recommend Introduction to Lie Algebras and Representation Theory by Humphreys. It covers all the basics of Lie algebras and their representations, though mostly in characteristic 0 and over an algebraically closed field. But then, going into any depth with the theory without these assumptions requires a lot of additional work anyway, and knowing the theory for this nice case is a good start.
Recently, Pavel Etingof published a book about his course on representation theory with his students. This is a description of the book:
"The goal of this book is to give a "holistic" introduction to representation theory, presenting it as a unified subject which studies representations of associative algebras and treating the representation theories of groups, Lie algebras, and quivers as special cases. Using this approach, the book covers a number of standard topics in the representation theories of these structures. Theoretical material in the book is supplemented by many problems and exercises which touch upon a lot of additional topics; the more difficult exercises are provided with hints.
The book is designed as a textbook for advanced undergraduate and beginning graduate students. It should be accessible to students with a strong background in linear algebra and a basic knowledge of abstract algebra."
This is a very enjoyable book to read. A version can be found here: link text. You get a bonus buying the book, where some nice historical remarks are added.