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We establish a method for calculating the Poincar\'e series of moduli spacesconstructed as quotients of smooth varieties by suitable non-reductive groupactions; examples of such moduli spaces include moduli spaces of unstablevector or Higgs bundles on a smooth projective curve, with a Harder-Narasimhantype of length two. To do so, we first prove a result concerning the smoothnessof fixed point sets for suitable non-reductive group actions on smoothvarieties. This enables us to prove that quotients of smooth varieties by suchnon-reductive group actions, which can be constructed using Non-Reductive GITvia a sequence of blow-ups, have at worst finite quotient singularities. Weconclude the paper by providing explicit formulae for the Poincar\'e series ofthese non-reductive GIT quotients.

Smoothness of non-reductive fixed point sets and cohomology of non-reductive GIT quotients