The　aim　of　this　paper　is　to　study　representations　of　weak　Hopf　algebras.　We　first　prove　that　the　global　dimension　of　a　weak　Hopf　algebra　is　the　same　as　the　projective　dimension　of　its　left　unital　subalgebra　and　deduce　some　equivalent　conditions　for　its　semisimplicity.　Then　we　give　the　construction　of　a　typical　example　u(m,L)　of　finite　dimensional　semisimple　weak　Hopf　algebra,　which　associates　to　a　cyclic　quiver　A(L-1)((1)).　All　irreducible　modules　of　u(m,L)　are　classified.　The　structure　of　the　truncated　tensor　product　V　circle　plus　for　any　two　irreducible　(m,L)-modules　V　and　W　is　obtained.