Presenting the Affine Hecke algebra: Iwahori and Bernstein Presentations and the Path model
Last update: 05 April 2012
The inital data is
- a finite reflection group,
- is a free module (has basis )
- is a finite subgroup of generated by reflections.
is a matrix conjugate (in
HW: Show that if is a reflection in a finite subgroup
is a root of unity?
Our favourite example:
contains 3 reflection, and
the reflections in
The affine Weyl group (semidirect product presentation)
for and Then acts on
In our example:
are fundamental regions for acting on such that
are the walls of
are the walls of
- the corresponding reflections.
is presented by generators and such that is a subgroup,
The Dynkin, or Coxeter diagram of is the graph with
- and edges
(the graph of the "1-skeleton of
The affine Hecke algebra (Bernstein presentation)
is generated by
The affine Hecke algebra (Coxeterish presentation)
is presented by generators and with
Let A reduced word for
is a minimal length sequence
The periodic orientation is:
- If then is on the positive side of
- Parallel hyperplanes have parallel orientation.
For a reduced work
are bases of
The path model (the algebra of paths)
Four kinds of steps
An alcove walk is a sequence of steps such that
- the tail of the first step is in
- at every step, the head of each arrow is in the same alcove as the tail of the next.
Use the relations
to straighten any sequence of steps to a linear combination of alcove walks.
- Fix and
- Fix a minimal length walk
- and a minimal length walk
where the sum is over all alcove walks
is the ending alcove of
Notes and References
These notes are from lecture notes of Arun Ram (CBMS Lecture 1).