Contragredient (Dual) Representation

Let $\pi:G\to \mathrm{GL}(V)$ be a representation on a finite-dimensional complex vector space $V$.

The dual space is $V^*:=\mathrm{Hom}_\mathbb{C}(V,\mathbb{C})$.

The contragredient (dual) representation $\pi^\vee:G\to \mathrm{GL}(V^*)$ is defined by $(\pi^\vee(g)\ell)(v):=\ell(\pi(g^{-1})v)\quad (\ell\in V^*,\,v\in V).$

In the letter: this is the representation denoted $\pi^e$, appearing in the functional-equation-style question.