Weight of a Siegel modular form (awaiting review)

The weight of a Siegel modular form $f$ of degree $g$ is the finite-dimensional complex representation $\rho$ of $\GL(g,\mathbb{C})$ that occurs in the modular transformation property of $f$ under the action of $\gamma = \left( \begin{array}{ll} a & b \\ c & d \end{array}\right)$ on the Siegel upper half space $\mathcal{H}_g$. That is, the weight is the representation $\rho$ in the transformation law $$ f\left( (a \tau + b)(c \tau + d)^{-1} \right) = \rho(c \tau + d) f(\tau) . $$

Irreducible representations of $\GL(g,\mathbb{C})$ are indexed by their highest weight.

In degree $2$, to a pair of non-negative integers $(k,j)$, one associates the irreducible representation $ \rm det^{k}(\rm St) \otimes {\rm Sym}^j(\rm St), $ where $\rm St$ denotes the standard $2$-dimensional representation of $\rm \GL(2,\C)$. The pair $(k,j)$ is called its weight, and when $j = 0$, the integer $k$ is called the weight. Note that the highest weight of this representation is $(j+k,k)$.