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.
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3$\begingroup$ For the combinatorial theory: perhaps Fulton's "Young Tableaux" $\endgroup$– Sam HopkinsCommented Sep 21, 2015 at 0:05
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3$\begingroup$ You seem to be asking quite a bit for one reference. The Grassmannian section in Griffiths-Harris is a fast introduction to the $A_n$-type Grassmannian. The paper by Bernstein-Gelfand-Gelfand gives a description of the cohomology ring that applies to all types. $\endgroup$– Jason StarrCommented Sep 21, 2015 at 1:02
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3$\begingroup$ Two books: 1) Manivel, Symmetric functions, Schubert polynomials and degeneracy loci, and 2) Hiller, Geometry of Coxeter groups. $\endgroup$– Leandro VendraminCommented Sep 21, 2015 at 1:46
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$\begingroup$ I am not sure to what extent this qualifies as "(for beginners)", but the first sections of the book by Baston and Eastwood on the Penrose transform cover most of the material you mention. $\endgroup$– Andreas CapCommented Sep 21, 2015 at 7:55
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$\begingroup$ Thanks all for the helpful references. I already knew some of them but definitely the comments and replies helped. $\endgroup$– KiuCommented Sep 26, 2015 at 15:00
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2 Answers
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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").
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The basics you can find in Joe Harris: Algebraic Geometry.
