Weyl denominator and anti-invariant orbit sums

Write $X=X^{\ast }(T_p)$ and define $\mathbb{Z}[X]$ to be the free abelian group on symbols $e^{\nu }$, with $e^{\mu }{e}^{\nu }=e^{\mu +\nu }$. Thus its elements have finite support. W acts by $w{e}^{\nu }=e^{w\nu }$. By [L1]–[L2], fundamental weights, extended trivially on the central factor, are characters of $T_p$; so is $\rho$. The formal algebra can also be viewed as finite integral combinations of characters: distinct group characters are linearly independent as functions. Indeed, a nontrivial relation of minimum positive length, translated by an element t and minus the original relation times one of its character values at t, gives a shorter nontrivial relation if two of its distinct characters differ at t. Such t exists by distinctness; a relation of length one is impossible since characters never vanish.

A weight is singular if its semisimple component lies on a root hyperplane; the corresponding reflection fixes the full weight by [L1]. Otherwise that component lies in an open chamber. The chamber theorem [L3] then gives a unique strictly dominant point in its W-orbit, and a trivial stabilizer: a fixing element fixes the chamber containing the component and is the identity. Central coordinates are unchanged. Here strictly dominant means all simple-coroot pairings are positive. If the root system is empty this condition is vacuous, W is trivial and every character is regular and strictly dominant.

Put $F=e^{\rho }{\prod }_{\alpha >0}(1-e^{-\alpha })$. The simple reflection $s_i$ permutes all positive roots except ${\alpha }_i$ and has $s_i\rho =\rho -{\alpha }_i$, by [L2]–[L3]. Thus $s_{i}F=e^{\rho -{\alpha }_i}(1-e^{{\alpha }_i}){\prod }_{\alpha >0,\alpha \mathrm{\ne }{\alpha }_i}(1-e^{-\alpha })=-F$. Since simple reflections generate W, $wF=\det (w)F$ for every w. Each factor is formal in the integral group algebra and no division or evaluation at a singular torus element is involved.

For an anti-invariant element $f={\sum }_{\mu }{a}_{\mu }{e}^{\mu }$, coefficient comparison gives $a_{w\mu }=\det (w)a_{\mu }$. If a reflection fixes $\mu$, then $a_{\mu }=-a_{\mu }$ in $\mathbb{Z}$, so $a_{\mu }=0$. Step 1.2 therefore partitions its support into regular orbits, each with one strictly dominant representative $\nu$ and no repetitions in $A_{\nu }={\sum }_w\det (w)e^{w\nu }$. Its contribution is exactly $a_{\nu }{A}_{\nu }$. These orbit sums are anti-invariant by reindexing, have disjoint supports, and have coefficient 1 at their strictly dominant representative. Hence they form a $\mathbb{Z}$-basis of all anti-invariant elements.

Expanding F gives ${\sum }_{S\subseteq {\mathrm{\Phi }}^{+}}(-1{)}^{\mathrm{\mid }S\mathrm{\mid }}{e}^{\rho -{\sum }_{\alpha \in S}\alpha }$. Every exponent is at most $\rho$ in root order, meaning their difference is a nonnegative integral sum of simple roots by [L3]. The coefficient at $\rho$ is exactly 1: a nonempty subset of positive roots has a nonzero sum by their one-sign coordinates and linear independence. All exponents lie in E, so their central component is zero.

Apply the basis of step 2.1 to the anti-invariant F from step 1.3. If a strictly dominant $\nu$ has a nonzero coefficient, it is itself in the support and step 2.2 gives $\rho -\nu ={\sum }_i{n}_i{\alpha }_i$ with $n_i\ge 0$ and $\nu \in E$. Set $\delta =\nu -\rho$. Its simple-coroot pairings are nonnegative, because those of $\nu$ are positive integers and those of $\rho$ equal 1. Therefore $({\alpha }_i,\delta )\ge 0$ for each i in the positive definite Euclidean metric of [L1]. But $-\mathrm{\parallel }\delta {\mathrm{\parallel }}^2=(\rho -\nu ,\delta )={\sum }_i{n}_i({\alpha }_i,\delta )\ge 0,$ forcing $\delta =0$ and $\nu =\rho$. The coefficient at $\rho$ in step 2.2 is 1, so $F=A_{\rho }$. This derives the identity without assuming any dominance assertion about $\rho -w\rho$.

Step 3.1 proves the denominator identity and step 2.1 proves the basis assertion. For empty roots, $\rho =0$, the empty product and $A_0$ both equal 1, and the orbit-sum basis is the full character basis, including all central characters. The construction needs $\rho$ to be a character only of $T_p$, not of the original torus T. All assertions concern finite integral combinations, not all functions on the torus. Choice enters through the supplied covering and compact root theory.