In Goodman and Wallach's book Symmetry, Representations, and Invariants (2009 edition), on page 231 (proof of Lemma 5.1.5), the authors introduce an inner product $\langle\:,\:\rangle$ on the space $\mathbb C[x_1,\ldots,x_n]$, defined as follows: $$ \langle f,g\rangle := \partial(f)\overline{g}(0), $$ where $\partial(f)$ is the differential operator determined by the map $x_i \longmapsto \partial/\partial x_i$, and $\overline{g}$ is the polynomial whose coefficients are the complex conjugates of those in $g$. It follows that for multi-indices $\alpha,\beta \in \mathbb N^n$, we have $\langle x^\alpha,x^\beta\rangle = \alpha!$ if $\alpha = \beta$, and $0$ otherwise.
Is this inner product attested to elsewhere in the literature? Does it have a standard name? For example, is it at all related to the Hall inner product on symmetric polynomials? In recent research I appeal to the fact that $\langle \:,\:\rangle$ yields an invariant pairing under the action of $\operatorname{GL}_n$ on $\mathbb C[x_1,\ldots,x_n]$, but I would like to reference this properly.