 
Summary: INTEGRAL REPRESENTATIONS
OF THE INFINITE DIHEDRAL GROUP
JAUME AGUAD´E, CARLES BROTO, AND LAIA SAUMELL
1. Introduction.
We want to study the representations of the infinite dihedral group D in GL2(R),
where R is either the valuation ring Z(p) of rational numbers with denominator prime
to p or the ring of padic integers Zp for some prime p.
The motivation for this research comes from the homotopy theory of classifying
spaces of KacMoody groups. Associated to each generalized Cartan matrix (see, for
instance, the introduction of [2]), one can define a (not necessarily finite dimensional)
Lie algebra which can be integrated in some way which we will not discuss here (see
[5]) to produce a topological group K called a KacMoody group. These topological
groups, and their classifying spaces, have been studied from a homotopical point of
view in several recent papers ([6],[3], [2], [1]). Like in the Lie group case, K has
a maximal torus T and a Weyl group W which acts on the Lie algebra of T as a
cristalographic group. However, in contrast to what happens in the Lie group case,
this Weyl group can be infinite. If we start with a nonsingular 2×2 Cartan matrix,
we have a (nonafine) KacMoody group of rank two and then the Weyl group is
infinite dihedral and we obtain a representation of D in GL2(Z) associated to K.
In [1] we have investigated the cohomology of the classifying spaces of these rank two
