Let $W$ be a finite subgroup of $GL(V)$ and hence it acts on $V$. Now consider the contra gradient action of $W$ on $V^*$. Now how to show that the eigen value of this action is the reciprocals of the eigen values of its original action on $V$ ?

V -finite dimensional vector space over $\mathbb{C}$.

Thanks in Advance.

  • $\begingroup$ Please give a definition of a contra gradient! $\endgroup$ – Peter Jul 3 '14 at 16:10
  • $\begingroup$ @Peter contra gradient action is the natural action of $W$ on $V ^*$ induced by the action of $W$ on $V$. $\endgroup$ – GA316 Jul 3 '14 at 16:49

Decompose into generalized eigenspaces of $g\in{\rm GL}(V)$ as $V=\bigoplus V_\lambda$. Then $V^*= \bigoplus V_\lambda^*$, where we view $V_\lambda^*$ as the space of linear functionals in $V^*$ with support $\subseteq V_\lambda$.

If $f\in V_\lambda^*$ then what is $(g\cdot f)(v)$? Answer this, and you'll have $V_\lambda^*=(V^*)_{\lambda^{-1}}$.


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