Orthogonality of Macdonald polynomials with unitary parameters

Abstract

For any admissible pair of irreducible reduced crystallographic root systems, we present discrete orthogonality relations for a finite-dimensional system of Macdonald polynomials with parameters on the unit circle subject to a truncation relation.

Appendix: Nondegeneracy of the Eigenvalues for exceptional root systems

In this appendix the nondegeneracy of the eigenvalues in Lemma 4.1 is verified for the exceptional root systems. Specifically, we will check that if for certain $\lambda ,\mu \in P_c$ the equality

$$\begin{aligned} \hat{m}_\omega (\rho _{\text {g}}+\lambda )= \hat{m}_\omega (\rho _{\text {g}}+\mu ) \end{aligned}$$

(6.1)

holds for all $\omega \in \hat{P}^+$ small (as an identity in $\text {g}$), then necessarily $\lambda =\mu$. This implies that the same holds true for the equality $E_\omega (\rho _{\text {g}}+\lambda )= E_\omega (\rho _{\text {g}}+\mu )$ in view of the triangularity of $E_\omega$ (2.9b) with respect to the monomial basis.

Since for $R$ exceptional the dual root system $R^\vee$ is isomorphic to $R$, the truncation relation in Remark 3.3 reads $t_\vartheta ^{h/2}t_\varphi ^{h/2}q_\varphi ^c=1$ (with $h$ being the Coxeter number of $R$). We write $\tilde{h}$ for the Coxeter number of the simply laced subsystem $W\varphi \subseteq R$ (so $\tilde{h}=h$ if $R$ is simply laced and $\tilde{h}=h/2$—in our situation—if $R$ is multiply laced). Let us furthermore denote the primitive root of unity $e^{2\pi i/\tilde{h}}$ by $\varepsilon$. Upon writing $\hat{m}_\omega (\rho _{\text {g}}+\lambda )=\sum _{\nu \in W\omega } q^{\langle \nu ,\lambda \rangle }\prod _{\alpha \in R^+}t_\alpha ^{\langle \nu ,\hat{\alpha }^\vee \rangle /2}$ and elimination of $t_\vartheta$ by means of the relations $t_\vartheta =\varepsilon q_\varphi ^{-c/\tilde{h}}$ if $R$ is simply laced or $t_\vartheta t_\varphi =\varepsilon q_\varphi ^{-c/\tilde{h}}$ if $R$ is multiply laced, both sides of the equality in Eq. (6.1) become Laurent polynomials in $t_\varphi$ with coefficients built of terms that are products of powers of $\varepsilon$ and $q$ (so the Laurent polynomials in question are of degree zero if $R$ is simply laced). For $R$ multiply laced both sides of Eq. (6.1) are equal as analytic functions in $\text {g}$ iff all coefficients of the corresponding Laurent polynomials in $t_\varphi$ match. (Indeed, the polar angles of $q=\exp \left( \frac{2\pi i}{u_\varphi (h_{\text {g}}+c)}\right)$ and $t_\varphi =q^{u_\varphi \text {g}_\varphi }=\exp \left( \frac{2\pi i \text {g}_\varphi }{ h_{\text {g}}+c}\right)$ are controlled by two independent parameters $\text {g}_\vartheta$ and $\text {g}_\varphi$, so by varying these parameters over the positive reals the tuple of the respective angles covers an open subset of $\left( 0,\frac{2\pi }{u_\varphi c}\right) \times \left( 0,\frac{2\pi }{\tilde{h}}\right)$.)

The expressions (for the coefficients of the Laurent polynomials in $t_\varphi$) on both sides of Eq. (6.1) are themselves polynomials in the primitive root of unity $\varepsilon$ of degree $\le \tilde{h}-1$ (possibly up to an overall factor $\varepsilon ^{1/2}$ when $\text {Ind}(R)> 1$), with coefficients that are sums of powers of $q$. To eliminate linear dependencies between these roots of unity, the powers $\varepsilon ^{\phi (\tilde{h})},\ldots ,\varepsilon ^{\tilde{h}-1}$—where $\phi$ refers to Euler’s totient function counting the number of coprimes not exceeding its argument—are expressed in terms of the basis $1,\varepsilon ,\ldots ,\varepsilon ^{\phi (\tilde{h})-1}$ via their residues modulo the cyclotomic polynomial $\Phi _{\tilde{h}}(\varepsilon )$ of degree $\phi (\tilde{h})$. Upon differentiating the coefficients with respect to $q$ and subsequently evaluating at $q=1$, a pairwise comparison of terms from both sides provides linear relations of the form $\langle \lambda -\mu ,v\rangle =0$ with $v\in Q^\vee$ (where we exploit the fact that the roots of unity $1,\varepsilon ,\ldots ,\varepsilon ^{\phi (\tilde{h})-1}$ are linearly independent over the rationals). By varying over the different coefficients and small weights $\omega \in \hat{P}$, we deduce this way that the equality in Eq. (6.1) implies that $\lambda -\mu$ must be orthogonal to $n (=\text {rank}(R)$) linearly independent vectors $v\in Q^\vee$ unless $R$ is of type $E_7$, whence $\mu$ must be equal to $\lambda$ in these cases.

When $R$ is of type $E_7$, the relevant vectors $v\in Q^\vee$ turn out to span a hyperplane, viz. the equality in Eq. (6.1) now permits to conclude only that $\lambda -\mu$ must belong to the line perpendicular to this hyperplane. A comparison of the quadratic terms—obtained by first applying the differential operator $(q\frac{ \text {d}}{\text {d}q})^2$ to the coefficients of the expression on both sides of Eq. (6.1) and then evaluating at $q=1$—under the additional assumption that $\mu$ differs from $\lambda$ by a nonzero vector belonging to this perpendicular line, now entails a nonhomogeneous linear system for $\lambda$. When $c$ is not a multiple of $6$, its (two-dimensional) solution space does not intersect $P$, whence the equality in Eq. (6.1) still implies that $\lambda =\mu$ in this situation.

Below we identify for each exceptional root system (ordered by increasing rank), a minimal choice of small weights $\omega$ and the corresponding coefficients of $\hat{m}_\omega (\rho _{\text {g}}+\lambda )$ giving rise to a maximal system of linearly independent vectors $v\in Q^\vee$ that are orthogonal to $\lambda -\mu$ when Eq. (6.1) holds. Here the weights $\lambda$ (and $\mu$) will be expressed in the basis of fundamental weights $\lambda =\lambda _1\omega _1+\cdots +\lambda _n\omega _n$, and the relevant vectors $v\in Q^\vee$ will be represented by the components $(v_1, v_2, \ldots ,v_n)$ with respect to the dual basis of simple coroots (i.e. $v=v_1\alpha _1^\vee +\cdots +v_n\alpha _n^\vee$). In each case, the normalization of the root system, the choice of the positive subsystem, and the numbering of the elements of the simple and fundamental bases will follow the conventions of the tables in Bourbaki [2]. We end the appendix by providing some details regarding the additional analysis of the quadratic terms required to rule out the degeneracies when $R$ is of type $E_7$.

1.1 Type $G$

The quasi-minuscule weight $\omega$ of $\hat{R}$ is equal to $\varphi ^\vee$ if $\hat{R}=R^\vee$ and equal to $\vartheta$ is $\hat{R}=R$. For $R$ of type $G_2$, the corresponding monomials $\hat{m}_{\omega }(\rho _{\text {g}}+\lambda )$ are of the form $\hat{m}_{\omega }(\rho _{\text {g}}+\lambda )=\hat{m}_{\omega }^+(\rho _{\text {g}}+\lambda )+\overline{\hat{m}_{\omega }^+(\rho _{\text {g}}+\lambda )}$ with

$$\begin{aligned} \hat{m}_{\varphi ^\vee }^+(\rho _{\text {g}}+\lambda )&= t_\vartheta t_\varphi ^2q^{\lambda _1+2\lambda _2}+t_\vartheta t_\varphi q^{\lambda _1+\lambda _2}+t_\varphi q^{\lambda _2} \quad (\hat{R}=R^\vee ), \\ \hat{m}_{\vartheta }^+(\rho _{\text {g}}+\lambda )&= t_\vartheta ^2 t_\varphi q^{2\lambda _1+3\lambda _2}+t_\vartheta t_\varphi q^{\lambda _1+3\lambda _2}+t_\vartheta q^{\lambda _1} \quad (\hat{R}=R) . \end{aligned}$$

We have that $\tilde{h}=3$ and $\varepsilon =e^{2\pi i/3}$. Elimination of $t_\vartheta$ via the truncation relation $t_\vartheta t_\varphi =\varepsilon q_\varphi ^{-c/3}$ and calculation of the residues modulo the cyclotomic polynomial $\Phi _3(\varepsilon )=\varepsilon ^2+\varepsilon +1$ gives

$$\begin{aligned} \hat{m}_{\varphi ^\vee } (\rho _{\text {g}}+\lambda )&= \left( q^{\lambda _2} + \varepsilon q^{ \lambda _1 + 2\lambda _2 -\frac{c}{3}} \right) t_\varphi +\left( q^{-\lambda _2} -q^{-\lambda _1- 2 \lambda _2 + \frac{c}{3}} - \varepsilon q^{ -\lambda _1 - 2\lambda _2 + \frac{c}{3}} \right) t_\varphi ^{-1} \\&\quad + \left( - q^{ -\lambda _1 - \lambda _2 + \frac{c}{3}} + \varepsilon \left( q^{ \lambda _1 + \lambda _2 -\frac{c}{3}}-q^{ -\lambda _1 - \lambda _2 + \frac{c}{3}}\right) \right) \quad (\hat{R}=R^\vee ) \end{aligned}$$

and

$$\begin{aligned} \hat{m}_{\vartheta } (\rho _{\text {g}}+\lambda )&= \left( -q^{ - \lambda _1 +\frac{c}{3}u_\varphi } + \varepsilon \left( q^{ -2 \lambda _1 - 3 \lambda _2 + \frac{2c}{3}u_\varphi } - q^{ - \lambda _1 +\frac{c}{3}u_\varphi }\right) \right) t_\varphi \\&\quad + \left( - q^{ 2 \lambda _1 + 3 \lambda _2 -\frac{2c}{3}u_\varphi } + \varepsilon \left( q^{ \lambda _1 -\frac{c}{3}u_\varphi } - q^{ 2 \lambda _1 + 3 \lambda _2 -\frac{2c}{3}u_\varphi }\right) \right) t_\varphi ^{-1}\\&\quad + \left( -q^{-\lambda _1- 3 \lambda _2 +\frac{c}{3}u_\varphi } + \varepsilon \left( q^{ \lambda _1 + 3 \lambda _2 -\frac{c}{3}u_\varphi } - q^{ - \lambda _1 - 3 \lambda _2 + \frac{c}{3}u_\varphi }\right) \right) \quad (\hat{R}=R ) . \end{aligned}$$

Differentiation with respect to $q$ of the coefficients of the Laurent polynomials in $t_\varphi$ on both sides of Eq. (6.1) and subsequent evaluation at $q=1$ leads—upon comparing the coefficients of $t_\varphi$ and $\varepsilon t_\varphi$ from both sides—to the relations $\lambda _2=\mu _2,\,\lambda _1+2\lambda _2=\mu _1+2\mu _2$ if $\hat{R}=R^\vee$ and $\lambda _1=\mu _1,\,\lambda _1+3\lambda _2=\mu _1+3\mu _2$ if $\hat{R}=R$. In other words, the equality in Eq. (6.1) implies that $\lambda -\mu$ must be orthogonal to $\alpha _2^\vee$ and $\alpha _1^\vee +2\alpha _2^\vee$ if $\hat{R}=R^\vee$ and to $\alpha _1^\vee$ and $\alpha _1^\vee +3\alpha _2^\vee$ if $\hat{R}=R$. In both cases, the equality in Eq. (6.1) therefore holds only when $\lambda =\mu$.

1.2 Type $F$

Proceeding as for $G_2$, we compute for $\omega \in \hat{P}^+$ quasi-minuscule $\hat{m}_{\omega }(\rho _{\text {g}}+\lambda )=\hat{m}_{\omega }^+(\rho _{\text {g}}+\lambda )+\overline{\hat{m}_{\omega }^+(\rho _{\text {g}}+\lambda )}$, with $\omega =\varphi ^\vee$ and

$$\begin{aligned} \hat{m}_{\varphi ^\vee }^+(\rho _{\text {g}}+\lambda )&= t_\vartheta ^3 t_\varphi ^5 q^{2 \lambda _1 +3 \lambda _2+2 \lambda _3+\lambda _4} + t_\vartheta ^3 t_\varphi ^4 q^{\lambda _1 +3 \lambda _2+2 \lambda _3+\lambda _4} + t_\vartheta ^3 t_\varphi ^3 q^{\lambda _1 +2 \lambda _2+2 \lambda _3+\lambda _4}\\&\quad + t_\vartheta ^2 t_\varphi ^3 q^{\lambda _1 +2 \lambda _2+\lambda _3+\lambda _4} + t_\vartheta ^2 t_\varphi ^2 q^{\lambda _1 +\lambda _2+\lambda _3+\lambda _4}\\&\quad + t_\vartheta t_\varphi ^3 q^{\lambda _1 +2 \lambda _2+\lambda _3} + t_\vartheta ^2 t_\varphi q^{\lambda _2+\lambda _3+\lambda _4}\\&\quad + t_\vartheta t_\varphi ^2 q^{\lambda _1 +\lambda _2+\lambda _3} + t_\vartheta t_\varphi q^{\lambda _2+\lambda _3} + t_\varphi ^2 q^{\lambda _1 +\lambda _2} + t_\varphi \left( q^{\lambda _1}+ q^{\lambda _2}\right) \end{aligned}$$

if $\hat{R}=R^\vee$, and with $\omega =\vartheta$ and

$$\begin{aligned}&\hat{m}_{\vartheta }^+ (\rho _{\text {g}}+\lambda ) = t_\vartheta ^5 t_\varphi ^3 q^{2 \lambda _1 +4 \lambda _2+3 \lambda _3+2\lambda _4} + t_\vartheta ^4 t_\varphi ^3 q^{2\lambda _1 +4\lambda _2+3 \lambda _3+\lambda _4} + t_\vartheta ^3 t_\varphi ^3 q^{2\lambda _1 +4 \lambda _2+2 \lambda _3+\lambda _4}\\&\quad \quad + t_\vartheta ^3t_\varphi ^2 q^{2\lambda _1 +2 \lambda _2+2\lambda _3+\lambda _4} + t_\vartheta ^3 t_\varphi q^{2\lambda _2+2\lambda _3+\lambda _4} + t_\vartheta ^2 t_\varphi ^2 q^{ 2\lambda _1 +2 \lambda _2+\lambda _3+\lambda _4}\\&\quad \quad + t_\vartheta ^2 t_\varphi q^{2\lambda _2+\lambda _3+\lambda _4} + t_\vartheta t_\varphi ^2 q^{2\lambda _1 +2\lambda _2+\lambda _3} + t_\vartheta ^2 q^{\lambda _3+\lambda _4} + t_\vartheta t_\varphi q^{2\lambda _2 +\lambda _3} + t_\vartheta \left( q^{\lambda _3} + q^{\lambda _4}\right) \end{aligned}$$

if $\hat{R}=R$. In the present case $\tilde{h}=6,\,\varepsilon =e^{2\pi i/6}$, and elimination of $t_\vartheta$ via $t_\vartheta t_\varphi =\varepsilon q_\varphi ^{-c/6}$ yields modulo the cyclotomic polynomial $\Phi _6(\varepsilon )=\varepsilon ^2-\varepsilon +1$:

$$\begin{aligned} \hat{m}_{\varphi ^\vee }(\rho _{\text {g}}+\lambda )&= \Bigl (q^{ \lambda _1 + \lambda _2 } - q^{2\lambda _1 + 3\lambda _2 + 2 \lambda _3 + \lambda _4 - \frac{c}{2}} + \varepsilon q^{\lambda _1 + 2\lambda _2 + \lambda _3 -\frac{c}{6}} \Bigr ) t_\varphi ^2\\&\quad +\Bigl (q^{ \lambda _1} +q^{\lambda _2} -q^{ \lambda _1+2 \lambda _2+\lambda _3+\lambda _4 -\frac{c}{3}}-q^{\lambda _1+3 \lambda _2+2 \lambda _3+\lambda _4 -\frac{c}{2}}\\&\quad + \varepsilon \left( q^{\lambda _1+2 \lambda _2+\lambda _3+\lambda _4 -\frac{c}{3}}+q^{\lambda _1+\lambda _2+\lambda _3 -\frac{c}{6}}-q^{-\lambda _2-\lambda _3-\lambda _4 +\frac{c}{3}}\right) \Bigr ) t_\varphi \\&\quad \Bigl ( -q^{-\lambda _1-2 \lambda _2\!-\!2 \lambda _3\!-\!\lambda _4 \!+\!\frac{c}{2} }\!+\!q^{-\lambda _2\!-\!\lambda _3 +\frac{c}{6}}\!-\!q^{\lambda _1+\lambda _2+\lambda _3+\lambda _4 -\frac{c}{3} }\!-\!q^{\lambda _1\!+\!2 \lambda _2\!+\!2 \lambda _3+\lambda _4 -\frac{c}{2}} \\&\quad +\varepsilon \left( q^{\lambda _1+\lambda _2+\lambda _3+\lambda _4 - \frac{c}{3}}+q^{\lambda _2+\lambda _3 -\frac{c}{6} } - q^{-\lambda _2-\lambda _3 + \frac{c}{6}}-q^{-\lambda _1-\lambda _2-\lambda _3-\lambda _4 + \frac{c}{3}}\right) \Bigr )\\&\quad +\Bigl ( q^{-\lambda _1-\lambda _2-\lambda _3 + \frac{c}{6} }-q^{-\lambda _1-3 \lambda _2-2 \lambda _3-\lambda _4 + \frac{c}{2} } + q^{-\lambda _2} + q^{-\lambda _1} - q^{ \lambda _2+\lambda _3+\lambda _4 - \frac{c}{3} }\\&\quad + \varepsilon \left( -q^{ -\lambda _1-2 \lambda _2-\lambda _3-\lambda _4 + \frac{c}{3} }+q^{\lambda _2+\lambda _3+\lambda _4 - \frac{c}{3}}-q^{-\lambda _1-\lambda _2-\lambda _3 + \frac{c}{6}}\right) \Bigr ) t_\varphi ^{-1}\\&\quad + \Bigl ( -q^{-2 \lambda _1\!-\!3 \lambda _2\!-\!2 \lambda _3\!-\!\lambda _4 \!+\! \frac{c}{2}}\!+\!q^{-\lambda _1\!-\!2 \lambda _2-\lambda _3 \!+\! \frac{c}{6}}+q^{-\lambda _1-\lambda _2} \!-\!\varepsilon q^{-\lambda _1-2 \lambda _2-\lambda _3 \!+\! \frac{c}{6}} \Bigr ) t_\varphi ^{-2} \end{aligned}$$

if $\hat{R}=R^\vee$, and

$$\begin{aligned} \hat{m}_{\vartheta } (\rho _{\text {g}}+\lambda )&= \Bigl ( - q^{ -2 \lambda _2-2 \lambda _3-\lambda _4 + \frac{c}{2} u_\varphi } + \varepsilon \left( q^{ -2 \lambda _1-4 \lambda _2-3 \lambda _3-2 \lambda _4 + \frac{5c}{6}u_\varphi } -q^{-\lambda _3-\lambda _4 +\frac{c}{3}u_\varphi }\right) \Bigr ) t_\varphi ^2 \\&\quad +\Bigl ( q^{-\lambda _4 \!+\!\frac{c}{6} u_\varphi }\!+\!q^{-\lambda _3 \!+\! \frac{c}{6}u_\varphi }-q^{-2 \lambda _1\!-\!2 \lambda _2-2 \lambda _3-\lambda _4 +\frac{c}{2}u_\varphi }\!-\!q^{-2 \lambda _1-4 \lambda _2-3 \lambda _3-\lambda _4 + \frac{2c}{3}u_\varphi }\Bigr .\\&\quad \Bigl .+\varepsilon \left( q^{2 \lambda _1+2 \lambda _2+\lambda _3 -\frac{c}{6}u_\varphi } - q^{-\lambda _4 + \frac{c}{6}u_\varphi } - q^{-\lambda _3 + \frac{c}{6}u_\varphi } - q^{-2 \lambda _2-\lambda _3-\lambda _4 + \frac{c}{3}u_\varphi }\Bigr .\right. \\&\quad \left. \Bigl .+ q^{-2 \lambda _1-4 \lambda _2-3 \lambda _3-\lambda _4 +\frac{2c}{3}u_\varphi }\right) \Bigr ) t_\varphi \\&\quad + \Bigl ( - q^{2 \lambda _1+4 \lambda _2+2 \lambda _3+\lambda _4 -\frac{c}{2}u_\varphi } - q^{2 \lambda _1+2 \lambda _2+\lambda _3+\lambda _4 - \frac{c}{3}u_\varphi } + q^{-2 \lambda _2-\lambda _3 + \frac{c}{6}u_\varphi }\Bigr .\\&\quad \Bigl .- q^{-2 \lambda _1-4 \lambda _2-2 \lambda _3-\lambda _4 + \frac{c}{2}u_\varphi }\\&\quad \varepsilon \left( q^{2 \lambda _2+\lambda _3 -\frac{c}{6}u_\varphi } + q^{ 2 \lambda _1+2 \lambda _2+\lambda _3+\lambda _4 - \frac{c}{3}u_\varphi } - q^{-2 \lambda _1-2 \lambda _2-\lambda _3-\lambda _4 + \frac{c}{3}u_\varphi }\right. \\&\quad \left. - q^{-2 \lambda _2-\lambda _3 + \frac{c}{6}u_\varphi }\right) \Bigr )\\&\quad + \Bigl ( q^{-2 \lambda _1-2 \lambda _2-\lambda _3 + \frac{c}{6}u_\varphi }-q^{2 \lambda _1+2 \lambda _2+2 \lambda _3+\lambda _4 -\frac{c}{2}u_\varphi } - q^{2 \lambda _2+\lambda _3+\lambda _4 - \frac{c}{3}u_\varphi } \\&\quad +\varepsilon \left( q^{\lambda _3-\frac{c}{6}u_\varphi } + q^{\lambda _4 - \frac{c}{6}u_\varphi } + q^{2 \lambda _2+\lambda _3+\lambda _4 -\frac{c}{3}u_\varphi } - q^{-2 \lambda _1-2 \lambda _2-\lambda _3 + \frac{c}{6}u_\varphi }\right. \\&\quad \left. - q^{2 \lambda _1+4 \lambda _2+3 \lambda _3+\lambda _4 -\frac{2c}{3}u_\varphi } \right) \Bigr ) t_\varphi ^{-1}\\&\quad + \Bigl ( q^{2 \lambda _1+4 \lambda _2+3 \lambda _3+2 \lambda _4 - \frac{5c}{6}u_\varphi } - q^{2 \lambda _2+2 \lambda _3+\lambda _4 - \frac{c}{2}u_\varphi } - q^{ \lambda _3 + \lambda _4 - \frac{c}{3}u_\varphi }\\&\quad +\varepsilon \left( q^{ \lambda _3 + \lambda _4 - \frac{c}{3}u_\varphi } - q^{ 2 \lambda _1+4 \lambda _2+3 \lambda _3+2 \lambda _4 -\frac{5c}{6}u_\varphi } \right) \Bigr ) t_\varphi ^{-2} \end{aligned}$$

if $\hat{R}=R$. Comparison of the coefficients of $t_\varphi ,\,t_\varphi ^2,\,\varepsilon t_\varphi$ and $\varepsilon t_\varphi ^2$ on both sides of Eq. (6.1) now leads (upon differentiation at $q=1$) to the following linearly independent vectors $v\in Q^\vee$ that are orthogonal to $\lambda -\mu$ if the equality holds: $(1, 4, 3, 2),\,(1, 2, 2, 1),\,(2, 4, 3, 2)$ and $(1, 2, 1, 0)$ if $\hat{R}=R^\vee$, and $(4,6,4, 1),\,(0,2,2,1),\,(0,0,0,1)$ and $(2,4,2,1)$ if $\hat{R}=R$ (where—recall—the components are with respect to the basis of simple coroots of $R$).

1.3 Type $E$

For $R$ of type $E_6$, one has that $\tilde{h}=h=12$, so $t_\vartheta =\varepsilon q^{-c/12}$ with $\varepsilon =e^{2\pi i/12}$, and the relevant cyclotomic polynomial is $\Phi _{12}(\varepsilon )=\varepsilon ^4-\varepsilon ^2+1$. We consider $\hat{m}_\omega (\rho _{\text {g}}+\lambda )$ with $\omega$ being equal either to the minuscule weight $\omega _6$ or to the quasi-minuscule weight $\omega _2=\varphi$. In the minuscule case the LHS of Eq. (6.1) becomes explicitly:

$$\begin{aligned} \hat{m}_{\omega _6}(\rho _{\text {g}}+\lambda )&= \varepsilon ^{11} \left( q^{\frac{1}{3} \left( -\lambda _1 -2 \lambda _3-\lambda _5 +\lambda _6\right) +\frac{c}{12}} + q^{\frac{1}{3} (-\lambda _1 +\lambda _3-\lambda _5 -2 \lambda _6) + \frac{c}{12}}\right) \\&\quad + \varepsilon ^{10} \left( q^{\frac{1}{3} (-\lambda _1 -2 \lambda _3-3 \lambda _4-\lambda _5 +\lambda _6 )+\frac{c}{6}} + q^{\frac{1}{3} (-\lambda _1 -2 \lambda _3-\lambda _5 -2 \lambda _6 )+\frac{c}{6}} \right) \\&\quad +\varepsilon ^9\left( q^{\frac{1}{3} (-\lambda _1-3 \lambda _2 -2 \lambda _3 -3 \lambda _4-\lambda _5 +\lambda _6) +\frac{c}{4}} + q^{\frac{1}{3} (-\lambda _1 -2 \lambda _3-3 \lambda _4-\lambda _5 -2 \lambda _6) +\frac{c}{4} } \right) \\&\quad +\varepsilon ^8 \Big (q^{ \frac{1}{3} (2 \lambda _1+3 \lambda _2 +4 \lambda _3+6 \lambda _4+5 \lambda _5 +4 \lambda _6) -\frac{2c}{3}} \\&\quad +q^{\frac{1}{3} (-\lambda _1-3 \lambda _2 -2 \lambda _3 -3 \lambda _4-\lambda _5 -2 \lambda _6)+\frac{c}{3} } +q^{\frac{1}{3} (-\lambda _1-2 \lambda _3 -3 \lambda _4 - 4 \lambda _5 -2 \lambda _6)+\frac{c}{3} }\Big )\\&\quad +\varepsilon ^7 \left( q^{\frac{1}{3} (2 \lambda _1+3 \lambda _2 +4 \lambda _3+6 \lambda _4+5 \lambda _5 +\lambda _6) -\frac{7c}{12}} +q^{\frac{1}{3} (-\lambda _1-3 \lambda _2 -2 \lambda _3 -3 \lambda _4-4 \lambda _5 -2 \lambda _6 )+\frac{5c}{12}}\right) \\&\quad +\varepsilon ^6 \left( q^{\frac{1}{3} (2 \lambda _1+3 \lambda _2 +4 \lambda _3+6 \lambda _4 +2 \lambda _5 +\lambda _6) -\frac{c}{2}} + q^{\frac{1}{3} (-\lambda _1-3 \lambda _2 -2 \lambda _3 -6 \lambda _4-4 \lambda _5 -2 \lambda _6) +\frac{c}{2} }\right) \\&\quad +\varepsilon ^5 \left( q^{\frac{1}{3} (-\lambda _1-3 \lambda _2 -5 \lambda _3 -6 \lambda _4-4 \lambda _5 -2 \lambda _6) +\frac{7c}{12}} +q^{\frac{1}{3} (2\lambda _1 + 3 \lambda _2 +4 \lambda _3 +3 \lambda _4+2 \lambda _5 +\lambda _6) - \frac{5c}{12} }\right) \\&\quad +\varepsilon ^4 \left( q^{\frac{1}{3} (-4 \lambda _1 -3 \lambda _2 -5 \lambda _3-6 \lambda _4-4 \lambda _5 -2 \lambda _6) +\frac{2c}{3}}\right. \\&\quad \left. +q^{\frac{1}{3} (2\lambda _1 + 4 \lambda _3 +3 \lambda _4+2 \lambda _5 +\lambda _6) -\frac{c}{3}} +q^{\frac{1}{3} (2 \lambda _1+3 \lambda _2 +\lambda _3 +3 \lambda _4+2 \lambda _5 +\lambda _6 )-\frac{c}{3} }\right) \\&\quad +\varepsilon ^3 \left( q^{\frac{1}{3} (2 \lambda _1+\lambda _3 +3 \lambda _4+2 \lambda _5 +\lambda _6) -\frac{c}{4} } + q^{\frac{1}{3} (-\lambda _1+3 \lambda _2 +\lambda _3 +3 \lambda _4+2 \lambda _5 +\lambda _6 )-\frac{c}{4}}\right) \\&\quad +\varepsilon ^2 \left( q^{\frac{1}{3} (-\lambda _1+\lambda _3 +3 \lambda _4+2 \lambda _5 +\lambda _6) -\frac{c}{6}} + q^{\frac{1}{3} (2 \lambda _1+\lambda _3 +2 \lambda _5 +\lambda _6)-\frac{c}{6}}\right) \\&\quad +\varepsilon \left( q^{\frac{1}{3} (2 \lambda _1 +\lambda _3-\lambda _5 +\lambda _6) -\frac{c}{12}} +q^{\frac{1}{3} (-\lambda _1+\lambda _3 +2 \lambda _5 +\lambda _6) -\frac{c}{12}}\right) \\&\quad + q^{\frac{1}{3} (-\lambda _1 +\lambda _3-\lambda _5 +\lambda _6) } + q^{\frac{1}{3} (-\lambda _1 -2 \lambda _3+2 \lambda _5 +\lambda _6) } + q^{\frac{1}{3} (2 \lambda _1+\lambda _3-\lambda _5 -2 \lambda _6 ) }, \end{aligned}$$

with $\varepsilon ^4=\varepsilon ^2-1,\,\varepsilon ^5=\varepsilon ^3-\varepsilon ,\,\varepsilon ^6=-1,\,\varepsilon ^7=-\varepsilon ,\,\varepsilon ^8=-\varepsilon ^2,\,\varepsilon ^9=-\varepsilon ^3,\,\varepsilon ^{10}=1-\varepsilon ^2$, and $\varepsilon ^{11}=\varepsilon -\varepsilon ^3$. Differentiation at $q=1$ of the coefficients of $\varepsilon ^0,\,\varepsilon ^1,\,\varepsilon ^2$ and $\varepsilon ^3$ on both sides of Eq. (6.1) produces the following four linearly independent vectors $v\in Q^\vee$: $(1, 0, 2, 1, 0, 0),\,(1, 0, 0, 0, 0, -1),\,(1, 0, 2, 2, 2, 1)$ and $(2, 2, 2, 3, 2, 1)$, respectively. A similar computation for $\omega =\omega _2=\varphi$ complements these with two more linearly independent vectors $v$: $(0, 1, 1, 1, 1, 0)$ and $(0, 1, 1, 3, 1, 0)$, stemming from the coefficients of $\varepsilon ^0$ and $\varepsilon ^3$.

For $R$ of type $E_7$, one has that $\tilde{h}=h=18$, so $t_\vartheta =\varepsilon q^{-c/18}$ with $\varepsilon =e^{2\pi i/18}$, and the corresponding cyclotomic polynomial is $\Phi _{18}(\varepsilon )=\varepsilon ^6-\varepsilon ^3+1$. We consider $\hat{m}_\omega (\rho _{\text {g}}+\lambda )$ with $\omega$ being equal either to the minuscule weight $\omega _7$ or to the quasi-minuscule weight $\omega _1=\varphi$. In the minuscule case we divide out an overall factor $\varepsilon ^{1/2}q^{-c/(2h)}$ from Eq. (6.1) before proceeding. The relevant linearly independent vectors $v\in Q^\vee$ are: $(2, 2, 3, 4, 3, 2, 2) (\varepsilon ^0$-term), $(1, 0, 0, 0, 0, -1, 0) (\varepsilon ^1$-term) and $(0, 1, 0, 2, 3, 2, 1) (\varepsilon ^5$-term) for $\omega =\omega _7$, and $(1, 1, 2, 2, 2, 1, 0) (\varepsilon ^0$-term), $(1, 0, 1, 2, 1, 1, 0) (\varepsilon ^1$-term) and $(1, 2, 2, 4, 2, 1, 0) (\varepsilon ^4$-term) for $\omega =\omega _1$.

For $R$ of type $E_8$, one has that $\tilde{h}=h=30$, so $t_\vartheta =\varepsilon q^{-c/30}$ with $\varepsilon =e^{2\pi i/30}$, and the corresponding cyclotomic polynomial is $\Phi _{30}(\varepsilon )=\varepsilon ^8+\varepsilon ^7-\varepsilon ^5-\varepsilon ^4-\varepsilon ^3+\varepsilon +1$. We consider $\hat{m}_\omega (\rho _{\text {g}}+\lambda )$ with $\omega$ being equal either to the quasi-minuscule weight $\omega _8=\varphi$ or to the only other small weight $\omega _1$. The relevant linearly independent vectors $v\in Q^\vee$ are for $\omega =\omega _8$: $(1, 1, 4, 5, 4, 2, 1, 1) (\varepsilon ^0$-term), $(2, 3, 6, 7, 5, 4, 2, 1) (\varepsilon ^1$-term), $(2, 3, 2, 4, 3, 2, 2, 0) (\varepsilon ^2$-term) and $(0, 0, 2, 2, 1, 0, 1, 0) (\varepsilon ^3$-term), and for $\omega =\omega _1$: $(7, 7, 30, 39, 29, 14, 6, 7) (\varepsilon ^0$-term), $(14, 21, 44, 51, 35, 28, 12, 5) (\varepsilon ^1$-term), $(16, 24, 16, 30, 23, 14, 15, 0) (\varepsilon ^2$-term) and $(-2, 0, 14, 15, 6, 2, 6, -1) (\varepsilon ^3$-term).

1.4 Type $E_7$ revisited

In the case that $R$ is of type $E_7$, it follows from the previous considerations that the equality in Eq. (6.1) can hold only if $\lambda -\mu$ is an integral multiple of the weight

$$\begin{aligned} \nu =2\omega _1+ 2\omega _2-\omega _3-\omega _4-\omega _5+2\omega _6 -\omega _7=\alpha _1+\alpha _2+\alpha _6 \end{aligned}$$

(6.2)

(which spans the orthogonal complement of the hyperplane spanned by the above vectors $v\in Q^\vee$ for this case). Substituting $\mu =\lambda +k \nu (k\in \mathbb Z$) and application of the operator $(q\frac{ \text {d}}{\text {d}q})^2$ to the coefficients on both sides of the equality entails a system of quadratic relations in $\lambda$ and $k$ (upon evaluation at $q=1$). In each of these relations the LHS cancels against the quadratic terms in $\lambda$ on the RHS (viz. the $k^0$-terms) and —more surprisingly—the quadratic terms in $k$ on the RHS also turn out to cancel against each other. From the remaining linear terms in $k$ we then deduce that the equality in Eq. (6.1) implies that either $k=0$ or that $\lambda$ must satisfy a nonhomogenous system of five linearly independent equations: $2\lambda _1+ 2\lambda _2+ 3\lambda _3+4\lambda _4+ 3\lambda _5+ 2\lambda _6+ 2\lambda _7=c (\varepsilon ^0$-term), $\lambda _1-\lambda _6=0 (\varepsilon ^1$-term) for $\omega =\omega _7$, and $\lambda _1+\lambda _2+2\lambda _3+ 2\lambda _4+ 2\lambda _5+\lambda _6= c/2 (\varepsilon ^0$-term), $\lambda _1+\lambda _3+ 2\lambda _4+ \lambda _5+ \lambda _6=c/3 (\varepsilon ^1$-term) and $2\lambda _2+\lambda _3+ 2\lambda _4+ \lambda _5 =c/3 (\varepsilon ^5$-term) for $\omega =\omega _1$. The intersection of its two-dimensional plane of solutions with the convex hull of $P_c$ is given by the triangle

$$\begin{aligned} \lambda ^{(0)} -s\nu -r\eta ,\quad |r|\le s\le \frac{c}{12}, \end{aligned}$$

(6.3)

where $\lambda ^{(0)} :=\frac{c}{6}(\omega _1+\omega _2+\omega _6),\nu$ is given by Eq. (5.7), and $\eta :=\omega _3-\omega _5$. Our condition that $c$ not be an integral multiple of $6$ when $R$ is of type $E_7$ guarantees that the intersection of the triangle with $P_c$ is empty, i.e. the equality in Eq. (5.6) can only hold if $k=0$ (so $\lambda =\mu$).

Remark 5.4

When $c$ is a multiple of $6$ the intersection of the triangle (5.8) with $P_c$ is given by weights of the form $\lambda ^{(0)} -k\nu -l\eta$ with $k,l\in \mathbb Z$ such that $|l|\le k\le [\frac{c}{12}]$. For instance, for $c=6$ the intersection consists only of $\lambda ^{(0)}$ (so degenerations are not possible in this case) whereas for proper multiples of $6$ a pair of weights $\lambda$ and $\mu$ in the triangle corresponding to the same value for $l$ and different values for $k$ may lead to equal expressions on both sides of Eq. (5.6) for all $\omega \in \hat{P}$ small. Explicit computations for a few multiples of $6$ suggest that for fixed $l$ and any $\omega \in \hat{P}$ small, the expression for $\hat{m}_\omega (\rho _{\text {g}}+\lambda ^{(0)} -k\nu -l\eta )$ is in fact independent of $k=|l|,\ldots ,[\frac{c}{12}]$. Such degenerations only occur for weights near the affine wall of $P_c$. Indeed, since $\langle \lambda ^{(0)} -s\nu -r\eta ,\varphi ^\vee \rangle \ge \langle \lambda ^{(0)}-\frac{c}{12}\nu ,\varphi ^\vee \rangle = \frac{11}{12}c$ for $|r|\le s\le \frac{c}{12}$, the degenerations in question are restricted to weights outside $P_{\tilde{c}}\subseteq P_c$ with $\tilde{c}=\lceil \frac{11}{12}c\rceil$.