Can you suggest books or lecture notes (for beginners) covering basic material about flag varieties and Grassmannians (of reductive groups), with emphasis on the usual flag variety, i.e. flag variety of $GL(n, \mathbb{C})$. Topics like: classical Plücker embedding and its generalizations for other reductive groups. Line bundles on flag varieties and Borel-Weil theorem. Cohomology of flag varieties. Definition and basic properties of Schubert cells and varieties. Also connections with symplectic geometry (i.e. coadjoint orbits) would also be nice.
[Math] Notes on flag varieties and Grassmannians for beginners
flag-varietiesgrassmannianstextbook-recommendation
Best Answer
The book "Flag varieties" by V. Lakshmibai and N. Gonciulea (Hermann 2001) covers several of these topics.
Michel Brion wrote lecture notes "Lectures on the geometry of flag varieties" which appeared in "Topics in Cohomological Studies of Algebraic Varieties", Impanga Lecture Notes, Ed. Piotr Pragacz, Birkhäuser Trends in Mathematics 2005, see also http://arxiv.org/abs/math/0410240
"Representations of algebraic groups" by J. C. Jantzen (2nd ed., AMS 2003) also might be worth a look (but might be tough to read for "beginners").