The Elliptic Weyl character formula: The ring
Last update: 02 March 2012
Let be a free module of rank with a positive definite symmetric bilinear form
and extend to a symmetric bilinear form on
and The motivation for this formula is ??????? Let
define (see [Kac, (12.7.2) and (12.7.3)])
which (modulo ) is exactly the sum over translates of by an dilate of the lattice
The expression is an element of
where are formal symbols indexed by infinite sums are allowed and
(see [Kac, Ex. 13.1] or [KP, §13.2]). This formula gives the product structure on the graded algebra
WHAT IS THE RIGHT COMBINATORIAL DESCRIPTION OF THIS BASE RING? PUT THE ACTION ON HERE?
Proof of the product formula for .
The action of on induced by the action on
is a set of representatives of the orbits on
it follows (as in [KP, Prop. 4.3(d-e)]) that
Because of the Weyl character formula
is a free module of rank 1 over
generated by In other words, as
BE SURE TO GET THE NORMALIZATION CONSTANT CORRECT ON THE LAST LINE!!!
Notes and References
These notes are taken from notes on the Elliptic Weyl character formula by Nora Ganter and Arun Ram.