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We show how to compute a Macaulay dual class in local cohomologyH    n    ([  V  ]   )$\hspace*{-0.56905pt}H^n_\mathfrak {m}(\mathbb {F}[V])\hspace*{-0.85358pt}$    for an given idealI  (    )  ⊂    [  V  ]    $I(\mathcal {S}) \subset \mathbb {F}[V]$    of generalized invariants associated to a set of reflections  ⊂  GL  (  n  ,    )    $\mathcal {S} \subset \text{GL}(n, \mathbb {F})$    that generate a finite subgroupG  (    )  ≤  GL  (  n  ,    )    $G(\mathcal {S}) \le \text{GL}(n, \mathbb {F})$   . We then apply this in the case that the coinvariant algebra    [  V  ]    G  (    )$\mathbb {F}[V]_{G(\mathcal {S})}$    is a fixed point free Poincaré duality algebra to obtain a Macaulay dual for the Hilbert ideal  (  G  (    )  )    $\mathfrak {h}(G(\mathcal {S}))$    ofG  (    )    $G(\mathcal {S})$   .

Ideals of generalized invariants, local cohomology and Macaulay's double duality theorem