This paper has three main goals. First, we set up a general framework to
address the problem of constructing module bases for the equivariant cohomology
of certain subspaces of GKM spaces. To this end we introduce the notion of a
GKM-compatible subspace of an ambient GKM space. We also discuss
poset-upper-triangularity, a key combinatorial notion in both GKM theory and
more generally in localization theory in equivariant cohomology. With a view
toward other applications, we present parts of our setup in a general algebraic
and combinatorial framework. Second, motivated by our central problem of
building module bases, we introduce a combinatorial game which we dub poset
pinball and illustrate with several examples. Finally, as first applications,
we apply the perspective of GKM-compatible subspaces and poset pinball to
construct explicit and computationally convenient module bases for the
S1-equivariant cohomology of all Peterson varieties of classical Lie type,
and subregular Springer varieties of Lie type A. In addition, in the Springer
case we use our module basis to lift the classical Springer representation on
the ordinary cohomology of subregular Springer varieties to S1-equivariant
cohomology in Lie type A.Comment: 32 pages, 4 figure
