Kac Moody Weyl vector

For the finite minimal realization in Realization of a generalized cartan matrix, a Weyl vector is a functional $\rho \in {\mathfrak{h}}^{\ast }$ such that $\rho (h_i)=1$ for every simple coroot. It exists: the independent finite family $h_i$ extends to a basis of the finite-dimensional Cartan, and assigning value one on the $h_i$ and arbitrary values on the complementary basis defines such a functional.

If ${\rho }^{\mathrm{\prime }}$ is another choice, $\eta ={\rho }^{\mathrm{\prime }}-\rho$ vanishes on all $h_i$. The reflection formula of Simple reflections and the kac moody weyl group gives $s_i\eta =\eta$ for every $i$, hence $w\eta =\eta$ for each finite Weyl word. Therefore $w(\mathrm{\Lambda }+{\rho }^{\mathrm{\prime }})-{\rho }^{\mathrm{\prime }}=w(\mathrm{\Lambda }+\rho )-\rho \mathrm{.}$ A shifted alternant with exponent $w(\mathrm{\Lambda }+\rho )$ and a denominator with prefactor $e^{\rho }$ both acquire the same monomial factor $e^{\eta }$ when $\rho$ changes, so their quotient is unchanged whenever these formal expressions are defined. If the simple coroots span the Cartan, the choice is unique; otherwise the free complementary values account for exactly its nonuniqueness. Only a finite basis extension is used, not AC.