Lusztig’s Geometric Approach to Hall Algebras

The introduction of l-adic cohomology by M. Artin and Grothenieck was motivated by the Weil conjecture concerning the number of rational points of algebraic varieties over finite fields. Since its introduction in the early 60s, l-adic cohomology theory has been used in the representation theory of finite Chevalley groups in a “misterious way”. The first application of l-adic cohomology and intersection cohomology theory to reductive groups was by Lusztig in 1975 [L3]. The constuction of intersection cohomology complexes for certain algebraic varieties associated to reductive groups leads the celebrating work was by Deligne and Lusztig [DL] constructing a large class of virtual characters for finite Chevalley groups over algebraically field of characteritic zero. The spirit of the application of the l-adic cohomology groups is the Lefschetz fixed point theorem and Grothendieck trace formula. Such an approach will construct the virtual characters for finite Chevalley groups over all finite fields at once. Later on Lusztig in a serieous of papers realized virtual characters in terms simple persverse sheaves, complexes of representations [L4]. The concept of linear representation theory has ever changed, and the algebraic geometry seams to be play an inevitable role. Although the questions conerned are just simple algebraic question, such as computing the irreducible characters. Another celebrating work is Springer’s construction of representations of Weyl groups in term of l-adic cohomology groups of certan subvarieties the nilpotent varieties, realize the Young’s construction of irreducible representations of symestric groups in terms of partitions. Just as Springer summarized [Sp] To a diehard algebraist, it may seam disturbing that in order to solve a concrete problem..., one has invoke arcane theories such as l-adic cohomology and intersection cohomology..., whatever simplications the future might bring, the insights brought by these theories are there to stay. In this note Lusztig’s geometric construction of the Hall algebras are introduced with an amphasis on the connecting the geometric construction and the combinatorial construction. The advantage of the geometric construction has been summarized by Lusztig in his book. Lusztig’s geometric approach was not a just a coincident. Before Ringel [R1] has constructed the Hall algebraic for Dynkin quivers, lusztig has already constructed the homogeneous part of the quiver with one vertex and one loop in [L4]. In this case, the classes of nilpotent representations of dimension n are exactly the set of nilpotent orbits of the Lie algebra gln and the filtration numbers are used in [R1] are exactly the number of the rantional points of a fiber in a resolution of the orbit. Thus Lusztig’s geometric construction first appears in [L5] seams to be a natural generalization of [L4] to the quiver situation.