Introduction to moment maps on flag
varieties
Arun Ram
Department of Mathematics
University of Wisconsin
Madison, WI 53706 USA
ram@math.wisc.edu
Moment maps-distilled
Let be a vector space. Projective space is
where .
Let be an inner product on . The map
Let be a torus,
acting on and assume is -invariant
Let . Then the moment map on projective space is
Weights and convexity
Let . A weight vector of weight is
such that for all .
Then so that
for weight vectors.
Let be weight vectors of weights , respectively. If then
is a reasonable assumption. Then
since distinct weights are orthogonal. So
More generally, given weight vectors with weights then
the convex hull of the points .
Flag varieties
Let be a Borel subgroup of a Kac-Moody group . The coset space
is the flag variety.
Let , the simple -module of highest weight . Let be the maximal torus of . Then
The group acts on and the -fixed points of are , . Then and the image of in is
The moment map on associated to is the restriction of to ,
since has weight .
The Schubert variety
where is the Bruhat-Chevalley order on . So
The Demazure module is
a -submodule, but not a -submodule of .
Loop groups
Then is controlled by
,
where is the affine Weyl group. Points of in are indexed by
labeled paths .
There is a "folding map" on paths so that
Let be a folded path without labels. Let
Question: What is the moment map image of .
The closed slices are generalizations of the MV-polytopes studied by Anderson, Kogan
and Kamnitzer.