  In this paper we give a simple description of DT-invariants of double quivers
without potential as the multiplicity of the Steinberg character in some
representation associated with the quiver. When the dimension vector is
indivisible we use this description to express these DT-invariants as the
Poincar\'e polynomial of some singular quiver varieties. Finally we explain the
connections with previous work of Hausel-Letellier-Villegas where DT-invariants
are expressed as the graded multiplicities of the trivial representation of
some Weyl group in the cohomology of some non-singular quiver varieties
attached to an extended quiver.
