dual representation

《Cohomologie Locale Des Faisceaux Coherents (Sga 2): Seminaire De Geometrie Algebrique Du Bois Marie 1962 (Documents Mathematiques)》 · 第 27–28 页 · Section 3.1「The Character Table」的群表示约定及对偶构造（省略张量积与一般 Hom 空间的构造）

Suppose $V$ and $W$ are representations of $G$ over $R$. […] We write $V^*=\operatorname{Hom}_R(V,R)$ and the action is $(g\cdot f)(v)=f(g^{-1}v)$ for each $f:V\to R$ and $g\in G$. This representation is called the dual or contragredient representation of $V$.

《Introduction to Representation Theory》 · 第 37 页 · Section 2.14「Tensor products and duals of representations of Lie algebras」，Definition 2.14.2

Definition 2.14.2. The dual representation $V^*$ to a representation $V$ of a Lie algebra $\mathfrak g$ is the dual space $V^*$ to $V$ with $\rho_{V^*}(x)=-\rho_V(x)^*$.