Weyl Jacobian

Let $G$ be a compact connected Lie group with maximal torus $T$, let $\mathrm{\Phi }=\mathrm{\Phi }(G,T)$ be its root system (Roots of a compact connected Lie group), and fix a positive system ${\mathrm{\Phi }}^{+}\subseteq \mathrm{\Phi }$ (Positive systems and simple roots). The Weyl Jacobian is the continuous function $J:T\to \mathbb{R},J(t)={\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}\mid 1-\alpha (t{)}^{-1}{\mid }^2\mathrm{.}$ Each factor is well defined because every root is an actual continuous character $T\to S^1$; its modulus is $1$, so $\mathrm{\mid }1-\alpha (t{)}^{-1}\mathrm{\mid }=\mathrm{\mid }1-\alpha (t)\mathrm{\mid }$ and the product is a nonnegative real number, vanishing exactly at those $t$ at which some positive root takes the value $1$. The function is globally defined without assuming that any half root $\alpha \mathrm{/}2$ or the vector $\rho$ is a character of $T$: only the honest characters $\alpha$ occur.

The exponent $-1$ in the first factor matches the standard normalisation of the Weyl integration formula (Weyl integration formula) in which the quotient measure on $G\mathrm{/}T$ is fixed by the coset Fubini identity with the right translation convention $F\mapsto F\circ R_h$; replacing $\alpha (t{)}^{-1}$ by $\alpha (t)$ does not change $J$, since $\mathrm{\mid }1-\alpha (t{)}^{-1}\mathrm{\mid }=\mathrm{\mid }1-\alpha (t)\mathrm{\mid }$ for $\mathrm{\mid }\alpha (t)\mathrm{\mid }=1$.

The product is finite because a root system is finite, and it does not depend on the numbering of the roots. If $\mathrm{\Phi }\mathrm{\ne }\mathrm{\varnothing }$, then $J(e)=0$. If $\mathrm{\Phi }=\mathrm{\varnothing }$ (in particular, when $G=T$ is a torus), the empty product is $J\equiv 1$. In all cases $J(t)=0$ exactly when some root takes the value $1$ at $t$; the zero set is the finite union of the kernels of the root characters, with the empty union understood as the empty set.

• The notation $J$ is standard for the density of the Weyl integration formula; it is also called the Weyl denominator density.
• The positive system is part of the data of the displayed product, but the next proposition proves that the product is independent of the choice of positive system and invariant under $W(G,T)$.
• The individual factors $\mathrm{\mid }1-\alpha (t{)}^{-1}{\mathrm{\mid }}^2$ are independent of any complexification convention: they are computed from the characters of $T$.