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Ben McKay
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The second definition is a generalization of the first. It is not equivalent. To see the first definition arising from the second, take $P$ to be the frame bundle so $H=GL(V)$ with $V=\mathbb{R}^n$ Then take as representations the tensor product $V^{\otimes p} \otimes V^{* \otimes q}$.

Ben McKay
  • 26.3k
  • 7
  • 67
  • 102