T \rightarrow \operatorname{U}(1)$ with the weight $d\chi/{2 \pi i} \in \mathfrak t^*$. We denote by $d\lambda$ the Lebesgue measure on $\mathfrak t^*$ that is normalized in such a way that any fundamental domain of the weight lattice $\Lambda^*$ has unit measure. Let us also choose a positive Weyl chamber $\mathfrak t^*_+ \subseteq \mathfrak t^*$; this determines a set of positive roots $\{ \alpha > 0 \} = \{ \alpha_1, \ldots, \alpha_R \} \subseteq \mathfrak t^*$. The set of negative roots is by definition $\{ -\alpha : \alpha > 0 \}$. We write $\mathfrak t^*_{>0}$ for the interior of the positive Weyl chamber, which contains the strictly dominant weights. We will often identify $\mathfrak k$ and its dual $\mathfrak k^*$, as well as $\mathfrak t$ and $\mathfrak t^*$, via some fixed $K$-invariant inner product $\braket{-,-}$ on $\mathfrak k$.

For the special unitary group $\operatorname{SU}(d)$, which we always take to be the group of unitary $d\times{}d$-matrices with unit determinant, we use the maximal torus consisting of diagonal matrices, on which the Weyl group $S_d$ acts by permuting diagonal entries. The Lie algebra ${\mathfrak{su}}(d)$ consists of anti-Hermitian matrices with trace zero, and our choice of invariant inner product is $\braket{X, Y} = -{\mathrm{tr}\left(X 