 
Summary: Seminar in Algebra and Number Theory Reflection Groups and Hecke Algebras
Fall 2005 P. Achar
Problem Set 1a
Due: September 6, 2005
1. Let W be a reflection group acting a real vector space V . Show that V admits an inner product
(i.e., a positivedefinite symmetric bilinear form) , with respect to which W consists of orthogonal
transformations.
2. Let k be an arbitrary field, and V a vector space over k. Let s : V V be a linear transformation
that fixes some hyperplane H pointwise.
(a) Suppose s2
= 1. Under what conditions does it follow that s is a reflection? (Hint: Consider the
characteristic of k.)
(b) More generally, s is called a pseudoreflection if, in addition to fixing H pointwise, s has a one
dimensional eigenspace with eigenvalue a root of unity. If some power of s is the identity map on
V , then under what conditions does it follow that s is a pseudoreflection?
3. Let V be a vector space. Recall that Humphreys' axioms for a root system V are:
(R1) R = {, } for all ;
(R2) s = .
Consider the following axiom
(R2 ) For each , there exists an element
