An Introduction to Perverse Sheaves and Character Sheaves

After a brief review of derived categories, triangulated categories and t-structures, we shall consider the bounded derived category D(X,R) of the category of étale sheaves of R-modules on X (a k-scheme of finite type), where R is an l-torsion ring with l a prime number distinct from the characteristic of the field k. We shall construct a t-structure on the (triangulated) full subcategory D c(X,R) of D (X,R) consisting of constructible complexes of sheaves. The category of perverse (with respect to the middle perversity function) sheaves M(X,R) will be defined as the heart of that t-structure. Then we shall pass to the construction of D c(X, Q̄l), the bounded "derived" category of Q̄l-(constructible) sheaves on an algebraic variety X over an algebraically closed field in which the prime number l is invertible [BBD, 2.2.18] and describe the corresponding subcategory M(X) of perverse sheaves on X. In the second half, we shall define, and study some of the main properties of character sheaves over G, a reductive algebraic group defined over an algebraically closed field. Character sheaves on G, which were introduced by Lusztig in [Lu3] for a connected G (and extended by him more recently to disconnected G in [Lu8]), belong to the set of isomorphism classes of simple objects of M(G).