 
Summary: Seminar in Algebra and Number Theory Reflection Groups and Hecke Algebras
Fall 2005 P. Achar
Problem Set 3a
Due: October 18, 2005
In the study of groups and representations, it is often useful to have a method for taking a representation of
a subgroup and producing from it a representation of the larger group. Here we will develop such a method
for reflection groups, known as "truncated induction" or "MacDonaldLusztigSpaltenstein induction."
Let W be a reflection group acting on V , and let W be a subgroup of W that is also generated by reflections.
Assume that V is equipped with a W (and hence W ) invariant inner product , . Let V = {v V 
wv = v for all w W }. amd let V = (V )
. So W acts as an essential reflection group on V . (Note:
W is not necessarily a parabolic subgroupit may or may not be equal to the full stabilizer of V .) Let
S = Sym(V
), S = Sym((V )
). and S = Sym((V )
). Sk
, (S )k
, and (S )k
denote the subspaces of
homogeneous degreek polynomials in each of the preceding.
