$A_{4}+A_{1}\subset E_{7}$. By [DE09, Sec 4], $\mathbb{O}$ is induced from the
orbit $\mathbb{O}_{L}=\\{0\\}$ of the Levi $L=L(A_{4}+A_{1};1,2,3,4,6)$. By
Proposition 4.1.2

$\gamma_{0}(\mathbb{O})=\gamma_{0}(\mathbb{O}_{L})=\rho(\mathfrak{l})=\frac{1}{2}(2,2,2,2,-7,2,-1)$

Conjugating by $W$ we get

$\gamma_{0}(\mathbb{O})=\frac{1}{2}(1,0,0,1,0,1,0).$

  * •

$A_{4}+A_{1}\subset E_{8}$. By [DE09, Sec 4], $\mathbb{O}$ is induced from the
rigid orbit $\mathbb{O}_{L}=A_{1}\times\\{0\\}$ of the Levi
$L=L(E_{6}+A_{1};1,2,3,4,5,6,8)$. The infinitesimal character
$\gamma_{0}(\mathbb{O}_{L})$ was computed in [LMM21]. It is
$\rho(\mathfrak{l})-\varpi$, where $\varpi$ is the fundamental weight for
$\mathfrak{l}$ corresponding to the central node in the $E_{6}$ factor. In
fundamental weights coordinates for $\mathfrak{g}$

$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}(2,2,2,0,2,2,-13,2).$

Applying Proposition 4.1.2 and conjugating by $W$ we get

$\gamma_{0}(\mathbb{O})=\frac{1}{2}(1,0,0,1,0,1,0,2).$

$\mathbb{O}$ | rigid? | $\gamma_{0}(\mathbb{O})$
---|---|---
$\\{0\\}$ | yes | $(1,1)$
$A_{1}$ | yes | $\frac{1}{3}(3,1)$
$\widetilde{A}_{1}$ | yes | $\frac{1}{2}(1,1)$
Table 4. Unipotent infinitesimal characters attached to birationally rigid orbits: type $G_{2}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | rigid? | $\gamma_{0}(\mathbb{O})$
---|---|---
$\\{0\\}$ | yes | $(1,1,1,1)$
$A_{1}$ | yes | $\frac{1}{2}(1,1,2,2)$
$\widetilde{A}_{1}$ | yes | $(1,0,1,1)$
$A_{1}+\widetilde{A}_{1}$ | yes | $(1,0,1,0)$
$A_{2}+\widetilde{A}_{1}$ | yes | $\frac{1}{4}(1,1,2,2)$
$\widetilde{A}_{2}+A_{1}$ | yes | $\frac{1}{3}(1,1,1,1)$
Table 5. Unipotent infinitesimal characters attached to birationally rigid orbits: type $F_{4}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | rigid? | $\gamma_{0}(\mathbb{O})$
---|---|---
$\\{0\\}$ | yes | $(1,1,1,1,1,1)$
$A_{1}$ | yes | $(1,1,1,0,1,1)$
$3A_{1}$ | yes | $\frac{1}{2}(1,1,1,1,1,1)$
$2A_{2}+A_{1}$ | yes | $\frac{1}{3}(1,1,1,1,1,1)$
Table 6. Unipotent infinitesimal characters attached to birationally rigid orbits: type $E_{6}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | rigid? | $\gamma_{0}(\mathbb{O})$
---|---|---
$\\{0\\}$ | yes | $(1,1,1,1,1,1,1)$
$A_{1}$ | yes | $(1,1,1,0,1,1,1)$
$2A_{1}$ | yes | $(1,1,1,0,1,0,1)$
$(3A_{1})^{\prime}$ | yes | $\frac{1}{2}(1,1,1,1,1,1,2)$
$4A_{1}$ | yes | $\frac{1}{2}(1,1,1,1,1,1,1)$
$A_{2}+A_{1}$ | no | $(1,0,0,1,0,1,0)$
$A_{2}+2A_{1}$ | yes | $(1,0,0,1,0,0,1)$
$2A_{2}+A_{1}$ | yes | $\frac{1}{3}(1,1,1,1,1,1,1)$
$(A_{3}+A_{1})^{\prime}$ | yes | $\frac{1}{2}(1,1,0,1,0,1,1)$
$A_{4}+A_{1}$ | no | $\frac{1}{2}(1,0,0,1,0,1,0)$
Table 7. Unipotent infinitesimal characters attached to birationally rigid orbits: type $E_{7}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | rigid? | $\gamma_{0}(\mathbb{O})$
---|---|---
$\\{0\\}$ | yes | $(1,1,1,1,1,1,1,1)$
$A_{1}$ | yes | $(1,1,1,0,1,1,1,1)$
$2A_{1}$ | yes | $(1,1,1,0,1,0,1,1)$
$3A_{1}$ | yes | $\frac{1}{2}(1,1,1,1,1,1,2,2)$
$4A_{1}$ | yes | $\frac{1}{2}(1,1,1,1,1,1,1,1)$
$A_{2}+A_{1}$ | yes | $(1,0,0,1,0,1,0,1)$
$A_{2}+2A_{1}$ | yes | $(1,0,0,1,0,0,1,1)$
$A_{2}+3A_{1}$ | yes | $\frac{1}{2}(1,1,1,0,1,1,1,1)$
$2A_{2}+A_{1}$ | yes | $\frac{1}{3}(1,1,1,1,1,1,1,3)$
$A_{3}+A_{1}$ | yes | $\frac{1}{2}(1,1,0,1,0,1,1,2)$
$2A_{2}+2A_{1}$ | yes | $\frac{1}{3}(1,1,1,1,1,1,1,1)$
$A_{3}+2A_{1}$ | yes | $\frac{1}{2}(1,1,1,0,1,0,1,1)$
$D_{4}(a_{1})+A_{1}$ | yes | $(0,0,0,1,0,0,1,0)$
$A_{3}+A_{2}+A_{1}$ | yes | $\frac{1}{2}(1,0,0,1,0,1,1,1)$
$A_{4}+A_{1}$ | no | $\frac{1}{2}(1,0,0,1,0,1,0,2)$
$2A_{3}$ | yes | $\frac{1}{4}(1,1,1,1,1,1,1,1)$
$A_{4}+A_{3}$ | yes | $\frac{1}{5}(1,1,1,1,1,1,1,1)$
$A_{5}+A_{1}$ | yes | $\frac{1}{6}(2,2,,1,1,1,1,1,1)$
$D_{5}(a_{1})+A_{2}$ | yes | $\frac{1}{4}(1,1,1,0,1,1,1,1)$
Table 8. Unipotent infinitesimal characters attached to birationally rigid
orbits: type $E_{8}$. Special unipotent characters are highlighted in blue.

####  4.3.2. Birationally rigid covers

In this section, we will produce a complete list of unipotent ideals attached
to birationally rigid covers $\widetilde{\mathbb{O}}$ of birationally induced
orbits. In addition to (1)-(4) of Section 4.3.1, we will repeatedly carry out
the following computations:

  1. (5)

Given a nilpotent orbit $\mathbb{O}$, determine the codimension 2 orbits
$\mathbb{O}_{k}\subset\overline{\mathbb{O}}$, their singularities, and the
normalizations thereof. These are evident from the incidence diagrams in
[FJLS15, Sec 13].

  2. (6)

Given a nilpotent orbit $\mathbb{O}$, determine the fundamental group
$\pi_{1}(\mathbb{O})$. See [CM93, Sec 6.1] for classical types and [CM93, Sec
8.4] for exceptional.

  3. (7)

Determine the reductive part $\mathfrak{r}$ of the centralizer of
$e\in\mathbb{O}$. See [CM93, Sec 6.1] for classical types and [CM93, Sec 13.1]
for exceptional.

  4. (8)

Given a nilpotent orbit $\mathbb{O}$ in a Levi subalgebra
$\mathfrak{l}\subset\mathfrak{g}$, compute the induced orbit
$\operatorname{Ind}^{\mathfrak{g}}_{\mathfrak{l}}\mathbb{O}$. If
$\mathfrak{g}$ is exceptional, we use [DE09, Sec 4]. If $\mathfrak{g}$ is
classical, we use the well-known formulas involving partitions, see [CM93, Sec
7.3].

  5. (9)

Given a singularity
$\Sigma_{k}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ of type
$A_{1}$, determine the integer $c_{k}$ (cf. (3.10.2)). By definition

$c_{k}=2|\operatorname{Pic}(\mathbb{O}_{M_{k}})||\operatorname{Pic}(\mathbb{O})|^{-1}=2|\pi_{1}(\mathbb{O}_{M_{k}})_{\mathrm{ab}}||\pi_{1}(\mathbb{O})_{\mathrm{ab}}|^{-1},$

where $H_{\mathrm{ab}}$ denotes the abelianization of $H$.

  6. (10)

For a given Levi subgroup $M\subset G$ determine a set of generators
$\tau_{i}$ for the free abelian group $\mathfrak{X}(M)$. If $M$ is standard,
then we can take $\tau_{i}$ to be the fundamental weights for $G$
corresponding to the simple roots not contained in $M$. More generally, if
$\beta_{1},...,\beta_{m}$ are simple roots for $M$, then $\mathfrak{X}(M)$ is
identified with the lattice

$\\{\lambda\in\Lambda\mid\langle\lambda,\beta_{i}^{\vee}\rangle=0,\ 1\leq
i\leq m\\}.$

The co-roots $\beta_{i}^{\vee}$ can be computed by hand or using the atlas
software.

We now begin the calculations:

##### Type $G_{2}$

.

$\alpha_{\mathrlap{1}}$$\alpha_{\mathrlap{2}}$

  * •

$G_{2}(a_{1})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{1};2)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{1};2),\\{0\\})$

There is one codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{1};2)$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(1,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(1,0)$. Now by Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=\frac{1}{2}(-3,2)+\frac{1}{2}(1,0)=(-1,1)$

Conjugating by $W$, we get

$\gamma_{0}(\widehat{\mathbb{O}})=(1,0)$

##### Type $F_{4}$

.

$\alpha_{\mathrlap{1}}$$\alpha_{\mathrlap{2}}$$\alpha_{\mathrlap{3}}$$\alpha_{\mathrlap{4}}$

  * •

$A_{2}$. Note that $\mathbb{O}$ is even and $L_{\mathbb{O}}=L(C_{3};2,3,4)$.
Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(C_{3};2,3,4),\\{0\\}).$

There is one codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(C_{3};2,3,4)$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(1,0,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(1,0,0,0)$. Now by Proposition
4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=(-3,1,1,1)+\frac{1}{2}(1,0,0,0)=\frac{1}{2}(-5,2,2,2).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,1,0,2).$

  * •

$B_{2}$. There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. We have $\mathfrak{r}=A_{1}$,
and therefore $\dim\mathfrak{P}^{X}=1$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(C_{3};2,3,4),(2,1^{4}))\\},$

and therefore by Lemma 3.10.2

$(L,\mathbb{O}_{L})=(L(C_{3};2,3,4),(2,1^{4})).$

It follows that $M_{1}=L$, and

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(C_{3};2,3,4)$ | $(2,1^{4})$ | $2$

Note that $\tau_{1}=(1,0,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{4}\tau_{1}=\frac{1}{4}(1,0,0,0)$. In standard coordinates
on $\mathrm{Sp}(6)$, $\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}(5,3,1)$, see
[LMM21, Prop 8.2.3]. So in our coordinates

$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{4}(-9,2,4,4)$

Now by Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\gamma_{0}(\mathbb{O}_{L})+\delta_{1}=\frac{1}{4}(-9,2,4,4)+\frac{1}{4}(1,0,0,0)=\frac{1}{2}(-4,1,2,2)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(0,1,0,2)$

  * •

$C_{3}(a_{1})$. There are two codimension orbits in the closure of
$\mathbb{O}$, and the corresponding singularities are of types $m$ and
$2A_{1}$. Hence, there is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Note that $\mathfrak{r}=A_{1}$,
and therefore $\dim\mathfrak{P}^{X}=1$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(B_{3};1,2,3),(2^{2},1^{2}))\\},$

and so by Lemma 3.10.2

$(L,\mathbb{O}_{L})=(L(B_{3};1,2,3),(2^{2},1^{2})).$

Thus, $M_{1}=L$, and

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(B_{3};1,2,3)$ | $(2^{2},1^{2})$ | $1$

Note that $\tau_{1}=(0,0,0,1)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,1)$. In standard coordinates
on $\mathrm{SO}(7)$, $\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}(3,2,1)$, see
[LMM21, Prop 8.2.3]. So in our coordinates

$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}(1,1,2,-6).$

Now by Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\gamma_{0}(\mathbb{O}_{L})+\delta_{1}=\frac{1}{2}(1,1,2,-6)+\frac{1}{2}(0,0,0,1)=\frac{1}{2}(1,1,2,-5).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,0,1,1).$

  * •

$F_{4}(a_{3})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{1}+A_{2};1,3,4)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{1}+A_{2};1,3,4),\\{0\\})$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{1}+A_{2};1,3,4)$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,1,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}0,1,0,0)$. Now by Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=\frac{1}{2}(2,-3,2,2)+\frac{1}{2}(0,1,0,0)=(1,-1,1,1)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(0,0,1,0)$

##### Type $E_{6}$

.

$\alpha_{\mathrlap{1}}$$\alpha_{2}$$\alpha_{\mathrlap{3}}$$\alpha_{\mathrlap{4}}$$\alpha_{\mathrlap{5}}$$\alpha_{\mathrlap{6}}$

  * •

$A_{2}$. Note that $\mathbb{O}$ is even and $L_{\mathbb{O}}=L(A_{5})$. Hence
by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{5}),\\{0\\})$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{5})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,1,0,0,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,1,0,0,0,0)$. Now by Proposition
4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=\frac{1}{2}(2,-9,2,2,2,2)+\frac{1}{2}(0,1,0,0,0,0)=(1,-4,1,1,1,1)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(1,0,0,1,0,1)$

  * •

$2A_{2}$. Note that $\mathbb{O}$ is even and $L_{\mathbb{O}}=L(D_{4})$. Hence
by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(D_{4}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{2}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{2}$ | $L(D_{4})$ | $\\{0\\}$ | -

We have

$\tau_{1}(1)=(1,0,0,0,0,0)\qquad\tau_{2}(1)=(0,0,0,0,0,1)$

So by Proposition 3.10.5, $\delta=\frac{1}{3}(1,0,0,0,0,1)$. Now by
Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=(-3,1,1,1,1,-3)+\frac{1}{3}(1,0,0,0,0,1)=\frac{1}{3}(-8,1,1,1,1,-8)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{3}(1,3,1,1,1,1).$

  * •

$D_{4}(a_{1})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(2A_{2}+A_{1})$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(2A_{2}+A_{1}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(2A_{2}+A_{1})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,0,0,1,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,1,0,0)$. Now by Proposition
4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=\frac{1}{2}(2,2,2,-5,2,2)+\frac{1}{2}(0,0,0,1,0,0)=(1,1,1,-2,1,1).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(0,0,0,1,0,0).$

  * •

$A_{5}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(D_{4}),(3,2^{2},1))\\}$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ which
maps to the orbit $\mathbb{O}_{1}=A_{4}+A_{1}$, and the corresponding
singularity is of type $A_{2}$. Since
$\pi_{1}(\mathfrak{L}_{1})=\pi_{1}(\mathbb{O}_{1})=1$, the monodromy action is
trivial. We have $\mathfrak{r}=A_{1}$, and therefore $\dim\mathfrak{P}^{X}=2$.
So by Lemma 3.10.2

$(L,\mathbb{O}_{L})=(L(D_{4}),(3,2^{2},1)).$

Therefore $M_{1}=L$, and we have

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{2}$ | $L(D_{4})$ | $(3,2^{2},1)$ | -

Since $M_{k}$ is standard, we have

$\tau_{1}=(1,0,0,0,0,0)\qquad\tau_{2}=(0,0,0,0,0,1)$

We claim that the map $\eta$ is given by $\eta(\tau_{i})=2\omega_{i}$ for
$i=1,2$. By [LMM21, Proposition 7.1.2], there is a short exact sequence

$0\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(Y)\stackrel{{\scriptstyle}}{{\to}}\operatorname{Cl}(Y)\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\mathbb{O}_{L})\stackrel{{\scriptstyle}}{{\to}}0$

We have
$\operatorname{Pic}(\mathbb{O}_{L})\simeq\pi_{1}(\mathbb{O}_{L})_{\mathrm{ab}}\simeq\mathbb{Z}_{2}^{2}$,
and $\operatorname{Pic}(Y)\simeq\operatorname{Pic}(G/P)\simeq\mathbb{Z}^{2}$,
and $\operatorname{Pic}^{a}(Y)$ is spanned by $\tau_{1}$ and $\tau_{2}$ under
this identification. Similarly to [LMM21, Lem 7.7.5], there is a short exact
sequence corresponding to the two irreducible components of the exceptional
divisor

$0\stackrel{{\scriptstyle}}{{\to}}\mathbb{Z}^{2}\stackrel{{\scriptstyle}}{{\to}}\operatorname{Cl}(Y)\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\mathbb{O})\stackrel{{\scriptstyle}}{{\to}}0.$

From an atlas computation, we see that
$\operatorname{Pic}(\mathbb{O})\simeq\pi_{1}(\mathbb{O})_{\mathrm{ab}}\simeq\mathbb{Z}_{3}$.
Write $T$ for the sublattice of $\operatorname{Cl}(Y)$ spanned by the
components of the exceptional divisor. It follows that
$\operatorname{Cl}(\mathbb{Z}^{2})\simeq\mathbb{Z}^{2}$, and moreover the
sublattice $3\operatorname{Pic}(Y)$ in $\operatorname{Cl}(Y)$ is contained in
$T$. The quotient of the lattice $\operatorname{Cl}(Y)$ by
$3\operatorname{Pic}(\mathbb{O})$ is
$\mathbb{Z}_{3}^{2}\times\mathbb{Z}_{2}^{2}$, and hence there is a short exact
sequence

$0\stackrel{{\scriptstyle}}{{\to}}3\operatorname{Pic}(Y)\stackrel{{\scriptstyle}}{{\to}}T\stackrel{{\scriptstyle}}{{\to}}\mathbb{Z}_{3}\times\mathbb{Z}_{2}^{2}\stackrel{{\scriptstyle}}{{\to}}0.$

Similarly, we have a sublattice
$T_{\Sigma}\subset\operatorname{Cl}(\mathfrak{S})\simeq\operatorname{Pic}(\mathfrak{S})$
spanned by the irreducible components of the exceptional divisor of
$\mathfrak{S}\stackrel{{\scriptstyle}}{{\to}}\Sigma$. We have a short exact
sequence

$0\stackrel{{\scriptstyle}}{{\to}}T_{\Sigma}\stackrel{{\scriptstyle}}{{\to}}\operatorname{Cl}(\mathfrak{S})\stackrel{{\scriptstyle}}{{\to}}\mathbb{Z}_{3}\stackrel{{\scriptstyle}}{{\to}}0.$

Therefore, $3\operatorname{Pic}(\mathfrak{S})$ is a sublattice of
$T_{\Sigma}$, and the cokernel of the embedding
$3\operatorname{Pic}(\mathfrak{S})\stackrel{{\scriptstyle}}{{\to}}T_{\Sigma}$
is isomorphic to $\mathbb{Z}_{3}$.

Let
$f:\operatorname{Pic}(Y)\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\mathfrak{S})$
be the restriction map, and let
$g:T\stackrel{{\scriptstyle}}{{\to}}T_{\Sigma}$ be the map which takes the
irreducible component of the exceptional divisor of
$Y\stackrel{{\scriptstyle}}{{\to}}X$ to its intersection with $\mathfrak{S}$,
an irreducible component of the exceptional divisor of
$\mathfrak{S}\stackrel{{\scriptstyle}}{{\to}}\Sigma$. Since $Y$ and
$\mathfrak{S}$ are $\mathbb{Q}$-terminal, there is a commutative diagram.

${0}$${3\operatorname{Pic}(\mathbb{O})}$${T}$${\mathbb{Z}_{3}\times\mathbb{Z}_{2}^{2}}$${0}$${0}$${3\operatorname{Pic}(\mathfrak{S})}$${T_{\Sigma}}$${\mathbb{Z}_{3}}$${0}$$\scriptstyle{3f}$$\scriptstyle{g}$$\scriptstyle{h}$

Note that
$\operatorname{Coker}(f)\simeq\operatorname{Coker}(3f)\simeq\operatorname{Ker}(h)\simeq\mathbb{Z}_{2}^{2}$,
and therefore $\eta(\tau_{i})$ and $\eta(\tau_{2})$ span a lattice in
$\operatorname{Pic}(\mathfrak{S})=\mathbb{Z}\langle\omega_{1},\omega_{2}\rangle$
with quotient isomorphic to $\mathbb{Z}_{2}^{2}$.

Since there is a unique pair $(L,\mathbb{O}_{L})$ in
$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$, $X$ admits a unique
$\mathbb{Q}$-factorial terminalization, up to isomorphism, see [LMM21, Lemma
7.2.4]. For an algebraic variety $Z$ with a projective morphism
$Z\stackrel{{\scriptstyle}}{{\to}}S$, write
$\operatorname{Pic}^{a}(Z)\subset\operatorname{Pic}(Z)$ for the semigroup of
relatively ample line bundles. By [BLPW16, Proposition 2.17] $\eta$ induces an
isomorphism between $\mathbb{R}_{>0}\operatorname{Pic}^{a}(Y)$ and the
fundamental domain for the $W$-action on
$\mathfrak{P}^{X}=\mathfrak{P}^{\mathfrak{S}}$, i.e.
$\mathbb{R}_{>0}\operatorname{Pic}^{a}(\mathfrak{S})$. We note that
$\operatorname{Pic}^{a}(Y)$ and $\operatorname{Pic}^{a}(\mathfrak{S})$ are
spanned by $\tau_{1},\tau_{2}$ and $\omega_{1},\omega_{2}$ respectively. So in
these bases both $\eta$ and $\eta^{-1}$ are given by 2-by-2 matrices with
nonnegative coefficients. Hence $\eta$ is given by a diagonal matrix, up to
permutations of bases. Since the cokernel of the map
$f:\operatorname{Pic}(Y)\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\mathfrak{S})$
is equal to $\mathbb{Z}_{2}^{2}$, it follows that $\eta(\tau_{1})=2\omega_{1}$
and $\eta(\tau_{2})=2\omega_{2}$.

Now we have $\delta=\frac{1}{6}(\tau_{1}+\tau_{2})=\frac{1}{6}(1,0,0,0,0,1)$.
By [LMM21, Prop 8.3],
$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}\rho(\mathfrak{l})$. Thus by
Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}\rho(\mathfrak{l})+\delta_{1}=\frac{1}{2}(-3,1,1,1,1,-3)+\frac{1}{6}(1,0,0,0,0,1)=\frac{1}{6}(-8,3,3,3,3,-8)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{6}(1,3,1,1,1,1).$

  * •

$E_{6}(a_{3})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(3A_{1};2,3,5)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(3A_{1};2,3,5),\\{0\\}).$

There are two codimension 2 leaves
$\mathfrak{L}_{1},\mathfrak{L}_{2}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$,
corresponding to the orbits $\mathbb{O}_{1}=A_{5}$ and
$\mathbb{O}_{2}=D_{5}(a_{1})$. The corresponding singularities are of types
$A_{1}$ and $A_{2}$, respectively. We will compute the pairs
$(M_{k},\mathbb{O}_{M_{k}})$ using Lemma 3.10.3. First, consider the pair

$(M_{1},\mathbb{O}_{M_{1}})=(L(A_{5};2,3,5,\theta_{1},\theta_{2}),(3^{2}))$

where $\theta_{1}$ and $\theta_{2}$ are the roots

$\theta_{1}=-\alpha_{2}-\alpha_{3}-\alpha_{4}-\alpha_{5}-\alpha_{6}\qquad\theta_{2}=\alpha_{1}+\alpha_{2}+\alpha_{3}+2\alpha_{4}+\alpha_{5}+\alpha_{6}.$

An atlas computation shows that this indeed defines of Levi (of the stated
type). By construction, $L\subset M_{1}$, and
$\mathbb{O}_{M_{1}}=\operatorname{Ind}^{M_{1}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{2})=1=\dim\mathfrak{P}_{2}$. Note that
$\overline{\mathbb{O}}_{M_{1}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{1},2}=\mathbb{O}_{(3,2,1)}$, and the corresponding singularity
is of type $A_{2}$. Finally, note that

$\displaystyle\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},2}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{1}}(\operatorname{Ind}^{M_{1}}_{L(A_{2}+A_{1};2,5,\theta_{1})}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(A_{2}+A_{1};2,5,\theta_{1})}\\{0\\}$

Up to conjugation by $G$, there is a unique Levi in $G$ of type $A_{2}+A_{1}$.
So the final induction coincides with $D_{5}(a_{1})=\mathbb{O}_{2}$, see
[DE09, Sec 4]. It now follows from Lemma 3.10.3 that the pair
$(M_{1},\mathbb{O}_{M_{1}})$ is adapted to $\mathfrak{L}_{1}$.

Next, define

$(M_{2},\mathbb{O}_{M_{2}})=(L(D_{4}),\mathbb{O}_{(3^{2},1^{2})}).$

By construction, $L\subset M_{2}$, and
$\mathbb{O}_{M_{2}}=\mathrm{Ind}^{M_{2}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{2})=2=\dim\mathfrak{P}_{2}$. Note that
$\overline{\mathbb{O}}_{M_{2}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{2},1}=\mathbb{O}_{(3,2^{2},1)}$, and the corresponding
singularity is of type $A_{1}$. Finally
$\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},1}=A_{5}=\mathbb{O}_{1}$, see
[DE09, Sec 4]. So by Lemma 3.10.3, the pair $(M_{2},\mathbb{O}_{M_{2}})$ is
adapted to $\mathfrak{L}_{2}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{5};2,3,5,\theta_{1},\theta_{2})$ | $(3^{2})$ | $1$
$2$ | $A_{2}$ | $L(D_{4})$ | $(3^{2},1^{2})$ | -

By definition $\tau_{1}(1)$ is (either) generator of the free abelian group

$\displaystyle\mathfrak{X}(M_{1})$
$\displaystyle=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0,\
\alpha\in\Delta(\mathfrak{m},\mathfrak{h})\\}$
$\displaystyle=\\{(x,0,0,y,0,z)\in\mathbb{Z}^{6}\mid x+y=y+z=0\\}$

So we may take $\tau_{1}(1)=(1,0,0,-1,0,0,1)$ and hence
$\delta_{1}=\frac{1}{2}(1,0,0,-1,0,1)$.

On the other hand, $M_{2}$ is standard. Hence

$\tau_{2}(1)=(1,0,0,0,0,0)\qquad\tau_{2}(2)=(0,0,0,0,0,1).$

We claim that $\eta_{2}(\tau_{i}(2))=\omega_{i}(2)$. We note that it suffices
to prove the analogous assertion for the orbit
$\mathbb{O}_{M}\subset\mathfrak{m}^{*}$, where $M=M_{1}$. Namely, we have
$M=L(A_{5};2,3,5,\theta_{1},\theta_{2})$, and
$\mathbb{O}_{M}=\mathbb{O}_{(3^{2})}$. We note that this orbit is Richardson,
and $(3A_{1},\\{0\\})$ is the only pair in $P_{\mathrm{rig}}(\mathbb{O}_{M})$.
Thus, by [LMM21, Lemma 7.2.4] $Y_{M}\simeq T^{*}(Q/P_{M})$ is the unique up to
an isomorphism $\mathbb{Q}$-terminalization of $X_{M}$. Arguing as in the case
of orbit $A_{5}\subset E_{6}$ we see that $\eta_{2}$ is diagonal with positive
integer coefficients in the bases $\tau_{1}(2),\tau_{2}(2)$ and
$\omega_{1}(2),\omega_{2}(2)$ respectively. Consider the sublattices $T$ and
$T_{\Sigma}$ of $\operatorname{Cl}(Y_{M})$ and
$\operatorname{Cl}(\mathfrak{S})$ spanned by the irreducible components of the
exceptional divisors of $Y_{M}\stackrel{{\scriptstyle}}{{\to}}X_{M}$ and
$\mathfrak{S}\stackrel{{\scriptstyle}}{{\to}}\Sigma$ respectively. Let
$f:\operatorname{Pic}(Y_{M})\stackrel{{\scriptstyle}}{{\to}}\operatorname{Pic}(\mathfrak{S})$
and $g:T\xrightarrow{\sim}T_{\Sigma}$ be the maps constructed as for the orbit
$A_{5}$. Note that both $Y_{M}$ and $\mathfrak{S}$ are smooth, and thus
$\operatorname{Pic}(Y_{M})\simeq\operatorname{Cl}(Y_{M})$ and
$\operatorname{Pic}(\mathfrak{S})\simeq\operatorname{Cl}(\mathfrak{S})$. Since
$\pi_{1}(\mathbb{O}_{M})\simeq\mathbb{Z}_{3}$, we have both
$T\subset\operatorname{Cl}(Y_{M})$ and
$T_{\Sigma}\subset\operatorname{Cl}(\mathfrak{S})$ are of index $3$, and
therefore $f$ is an isomorphism. Thus, both diagonal entries of $\eta$ must be
$1$, and thus $\eta_{2}(\tau_{i}(2))=\omega_{i}(2)$.

It follows that
$\delta_{2}=\frac{1}{3}(\tau_{2}(2)+\tau_{2}(2))=\frac{1}{3}(1,0,0,0,0,1)$.
Now by Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}+\delta_{2}$
$\displaystyle=\frac{1}{2}(-1,2,2,-3,2,-1)+\frac{1}{2}(1,0,0,-1,0,1)+\frac{1}{3}(1,0,0,0,0,1)$
$\displaystyle=\frac{1}{3}(1,3,3,-6,3,1)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{3}(0,1,1,0,1,0).$

##### Type $E_{7}$

.

$\alpha_{\mathrlap{1}}$$\alpha_{2}$$\alpha_{\mathrlap{3}}$$\alpha_{\mathrlap{4}}$$\alpha_{\mathrlap{5}}$$\alpha_{\mathrlap{6}}$$\alpha_{\mathrlap{7}}$

  * •

$(3A_{1})^{\prime\prime}$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(E_{6})$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(E_{6}),\\{0\\})$

There is a unique codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{6})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,0,0,0,0,0,1)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,0,0,1)$. Now by
Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta=(1,1,1,1,1,1,-8)+\frac{1}{2}(0,0,0,0,0,0,1)=\frac{1}{2}(2,2,2,2,2,2,-15)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(2,1,2,1,1,1,1)$

  * •

$A_{2}$. Note that $\mathbb{O}$ is even and $L_{\mathbb{O}}=L(D_{6})$. Hence
by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(D_{6}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(D_{6})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(1,0,0,0,0,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(1,0,0,0,0,0,0)$. Now by
Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta=\frac{1}{2}(-15,2,2,2,2,2,2)+\frac{1}{2}(1,0,0,0,0,0,0,0)=(-7,1,1,1,1,1,1).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(1,0,0,1,0,1,1).$

  * •

$A_{2}+3A_{1}$. Note that $\mathbb{O}$ is even and $L_{\mathbb{O}}=L(A_{6})$.
Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{6}),\\{0\\}).$

There is a unique codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{6})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,1,0,0,0,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,1,0,0,0,0,0)$. Now by
Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=(1,-6,1,1,1,1,1)+\frac{1}{2}(0,1,0,0,0,0,0)=\frac{1}{2}(2,-11,2,2,2,2,2)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,1,1,0,1,1,1).$

  * •

$(A_{3}+A_{1})^{\prime\prime}$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(D_{5};2,3,4,5,6)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(D_{5};2,3,4,5,6),\\{0\\}).$

There are two codimension 2 leaves
$\mathfrak{L}_{1},\mathfrak{L}_{2}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$,
corresponding to the orbits $\mathbb{O}_{1}=2A_{2}$ and
$\mathbb{O}_{2}=A_{3}$. Both singularities are of type $A_{1}$. We will
compute the pairs $(M_{k},\mathbb{O}_{M_{k}})$ using Lemma 3.10.3. First,
consider the pair

$(M_{1},\mathbb{O}_{M_{1}})=(L(D_{6}),(3,1^{9})).$

By construction, $L\subset M_{1}$ and
$\mathbb{O}_{M_{1}}=\mathrm{Ind}^{M_{1}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{1})=1=\dim\mathfrak{P}_{1}$.
$\overline{\mathbb{O}}_{M_{1}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{1},2}=(2^{2},1^{8})$, and the corresponding singularity is of
type $A_{1}$. Finally, note that
$\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},2}=A_{3}=\mathbb{O}_{2}$. It
now follows from Lemma 3.10.3 that $(M_{1},\mathbb{O}_{M_{1}})$ is adapted to
$\mathfrak{L}_{1}$.

Next, consider the pair

$(M_{2},\mathbb{O}_{M_{2}})=(L(D_{5}+A_{1};2,3,4,5,6,\theta),\\{0\\}\times(2))$

where $\theta$ is the highest root for $G$, i.e.

$\theta=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}$

An atlas computation shows that this indeed defines a Levi (of the stated
type). By construction, $L\subset M_{2}$ and of course
$\overline{\mathbb{O}}_{M_{2}}=\mathrm{Ind}^{M_{2}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{2})=1=\dim\mathfrak{P}_{2}$.
$\overline{\mathbb{O}}_{M_{2}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{2},1}=\\{0\\}$, and the corresponding singularity is of type
$A_{1}$. Finally, note that by [DE09, Sec 4]
$\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},1}=2A_{2}=\mathbb{O}_{1}$. So
by Lemma 3.10.3, $(M_{2},\mathbb{O}_{M_{2}})$ is adapted to
$\mathfrak{L}_{2}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(D_{6})$ | $(3,1^{9})$ | $2$
$2$ | $A_{1}$ | $L(D_{5}+A_{1};2,3,4,5,6,\theta)$ | $\\{0\\}\times(2)$ | $2$

Since $M_{1}$ is standard, $\tau_{1}(1)=(1,0,0,0,0,0,0)$ and hence
$\delta_{1}=\frac{1}{4}\tau_{1}(1)=\frac{1}{4}(1,0,0,0,0,0,0)$ by Proposition
3.10.6. On the other hand, $\tau_{1}(2)$ is (either) generator of the free
abelian group

$\displaystyle\mathfrak{X}(M_{2})$
$\displaystyle=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0,\
\alpha\in\Delta(\mathfrak{m}_{2},\mathfrak{h})\\}$
$\displaystyle=\\{(x,0,0,0,0,0,y)\in\mathbb{Z}^{7}\mid 2x+y=0\\}.$

So we may take $\tau_{1}(2)=(1,0,0,0,0,0,-2)$ and
$\delta_{2}=\frac{1}{4}\tau_{2}=\frac{1}{4}(1,0,0,0,0,0,-2)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}+\delta_{2}$
$\displaystyle=(-5,1,1,1,1,1,-4)+\frac{1}{4}(1,0,0,0,0,0,0)+\frac{1}{4}(1,0,0,0,0,0,-2)$
$\displaystyle=\frac{1}{2}(-9,2,2,2,2,2,-9)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(2,0,1,1,0,1,1)$

  * •

$D_{4}(a_{1})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{1}+A_{5})$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{1}+A_{5}),\\{0\\})$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{1}+A_{5})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,0,1,0,0,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,1,0,0,0,0)$. Now by
Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\rho(\mathfrak{l})+\delta_{1}=\frac{1}{2}(2,2,-9,2,2,2,2)+\frac{1}{2}(0,0,1,0,0,0,0)=(1,1,-4,1,1,1,1).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(0,0,0,1,0,0,1).$

  * •

$A_{3}+2A_{1}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{6}),3A_{1})\\}.$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. We have $\mathfrak{r}=2A_{1}$,
and thus $\dim\mathfrak{P}^{X}=1$. Now by Lemma 3.10.2, we have

$(L,\mathbb{O}_{L})=(L(E_{6}),3A_{1})$

Thus, $M_{1}=L$ and

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{6})$ | $3A_{1}$ | $1$

Note that $\tau_{1}=(0,0,0,0,0,0,1)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,0,0,1)$. Note that
$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}\rho(\mathfrak{l})$, see Table 6. Now
by Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}\rho(\mathfrak{l})+\delta_{1}=\frac{1}{2}(1,1,1,1,1,1,-8)+\frac{1}{2}(0,0,0,0,0,0,1)=\frac{1}{2}(1,1,1,1,1,1,-7)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,1,1,0,1,0,1).$

  * •

$D_{4}(a_{1})+A_{1}$. By [DE09, Sec 4]

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(A_{5};1,3,4,5,6),\\{0\\})\\}.$

There are two codimension orbits in $\overline{\mathbb{O}}$, and the
corresponding singularities are of types $A_{1}$ and $3A_{1}$. Therefore,
there are two codimension 2 leaves $\mathfrak{L}_{1}$,
$\mathfrak{L}_{2}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularities are both of type $A_{1}$. We have
$\mathfrak{r}=2A_{1}$, and thus $\dim\mathfrak{P}^{X}=2$. So by Lemma 3.10.2,
we have

$(L,\mathbb{O}_{L})=(L(A_{5};1,3,4,5,6),\\{0\\}).$

We now compute the pairs $(M_{k},\mathbb{O}_{M_{k}})$ using Lemma 3.10.3. Let

$(M_{1},\mathbb{O}_{M_{1}})=(L(E_{6}),A_{2})$

By construction, $L\subset M_{1}$, and as observed in the calculation for
$A_{2}\subset E_{6}$, $\mathbb{O}_{M_{1}}=\mathrm{Ind}^{M_{1}}_{L}\\{0\\}$.
Furthermore, $\dim\mathfrak{X}(\mathfrak{m}_{1})=1=\dim\mathfrak{P}_{1}$. Note
that $\overline{\mathbb{O}}_{M_{1}}$ contains a single orbit of codimension 2,
namely $\mathbb{O}_{M_{1},2}=3A_{1}$, and the corresponding singularity is of
type $A_{1}$. Finally, by [DE09, Sec 4]
$\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},2}=A_{3}+2A_{1}=\mathbb{O}_{2}$.
So by Lemma 3.10.3, $(M_{1},\mathbb{O}_{M_{1}})$ is adapted to
$\mathfrak{L}_{1}$.

The computation of $M_{2}$ is slightly more involved. Consider the pair

$(M_{2},\mathbb{O}_{M_{2}})=(L(A_{5}+A_{1};1,3,4,5,6,\theta),\\{0\\}\times(2))$

where $\theta$ is the root

$\theta=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+3\alpha_{4}+2\alpha_{5}+\alpha_{6}$

An atlas computation shows that this indeed defines a Levi subgroup (of the
stated type). By construction $L\subset M_{1}$ and clearly
$\mathbb{O}_{M_{2}}=\mathrm{Ind}^{M_{2}}_{L}\\{0\\}$.
$\overline{\mathbb{O}}_{M_{2}}$ contains a unique codimension 2 orbit, namely
$\mathbb{O}_{M_{2},1}=\\{0\\}$, and the corresponding singularity is of type
$A_{1}$. Finally,
$\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},1}=D_{4}(a_{1})=\mathbb{O}_{1}$.
So by Lemma 3.10.3 $(M_{2},\mathbb{O}_{M_{2}})$ is adapted to
$\mathfrak{L}_{2}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{6})$ | $A_{2}$ | $1$
$2$ | $A_{1}$ | $L(A_{5}+A_{1};1,3,4,5,6,\theta)$ | $\\{0\\}\times(2)$ | $1$

Since $M_{1}$ is standard, $\tau_{1}(1)=(0,0,0,0,0,0,1)$, and hence by
Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}(1)=\frac{1}{2}(0,0,0,0,0,0,1)$. On the other
hand, $\tau_{1}(2)$ is (either) generator of the free abelian group

$\displaystyle\mathfrak{X}(M_{2})$
$\displaystyle=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0,\
\forall\alpha\in\Delta(\mathfrak{m}_{2},\mathfrak{h})\\}$
$\displaystyle=\\{(0,x,0,0,0,0,y)\in\mathbb{Z}^{7}\mid x+y=0\\}$

So we may take $\tau_{2}=(0,1,0,0,0,0,-1)$ and hence by Proposition
3.10.6,$\delta_{2}=\frac{1}{2}\tau_{1}(2)=\frac{1}{2}(0,1,0,0,0,0,-1)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}+\delta_{2}$
$\displaystyle=\frac{1}{2}(2,-9,2,2,2,2,-5)+\frac{1}{2}(0,0,0,0,0,0,1)+\frac{1}{2}(0,1,0,0,0,0,-1)$
$\displaystyle=\frac{1}{2}(2,-8,2,2,2,2,-5).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(0,0,1,1,0,1,1).$

  * •

$A_{3}+A_{2}+A_{1}$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{4}+A_{2})$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{4}+A_{2}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{4}+A_{2})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,0,0,0,1,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,1,0,0)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}$
$\displaystyle=(1,1,1,1,-4,1,1)+\frac{1}{2}(0,0,0,0,1,0,0)$
$\displaystyle=\frac{1}{2}(2,2,2,2,-7,2,2)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,0,0,1,0,1,1).$

  * •

$A_{5}+A_{1}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{6}),2A_{2}+A_{1})\\},$

So by Lemma 3.10.2

$(L,\mathbb{O}_{L})=(L(E_{6}),2A_{2}+A_{1}),$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{6})$ | $2A_{2}+A_{1}$ | $3$

Note that $\tau_{1}=(0,0,0,0,0,0,1)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{6}\tau_{1}=\frac{1}{6}(0,0,0,0,0,0,1)$. Now by
Proposition 4.1.1

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{3}\rho(\mathfrak{l})+\delta_{1}=\frac{1}{3}(1,1,1,1,1,1,-8)+\frac{1}{6}(0,0,0,0,0,0,1)=\frac{1}{6}(2,2,2,2,2,2,-15)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{6}(2,1,2,1,1,1,1).$

  * •

$D_{5}(a_{1})+A_{1}$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{3}+A_{2};2,3,4,6,7)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{3}+A_{2};2,3,4,6,7),\\{0\\}).$

There are two codimension 2 leaves
$\mathfrak{L}_{1},\mathfrak{L}_{2}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$,
corresponding to the orbits $\mathbb{O}_{1}=A_{4}+A_{2}$ and
$\mathbb{O}_{2}=D_{5}(a_{1})$. The corresponding singularities are both of
type $A_{1}$. Next we will compute the pairs $(M_{k},\mathbb{O}_{M_{k}})$
using Lemma 3.10.3. First consider the pair

$(M_{1},\mathbb{O}_{M_{1}})=(L(D_{6}),(3^{3},1^{3})).$

By construction, $L\subset M_{1}$ and
$\mathbb{O}_{M_{1}}=\mathrm{Ind}^{M_{1}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{1})=1=\dim\mathfrak{P}_{1}$.
$\overline{\mathbb{O}}_{M_{1}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{1},2}=(3^{2},2^{2},1^{2})$, and the singularity is of type
$A_{1}$. Finally

$\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},2}=\operatorname{Ind}^{G}_{M_{1}}(\operatorname{Ind}^{M_{1}}_{L(A_{4};4,5,6,7)}\\{0\\})=\operatorname{Ind}^{G}_{L(A_{4};4,5,6,7)}\\{0\\}$

Up to conjugation by $G$, there is a unique Levi subgroup of Lie type $A_{4}$.
So the final induction above is the same as
$\operatorname{Ind}^{G}_{L(A_{4};1,2,3,4)}\\{0\\}$, which is
$D_{5}(a_{1})=\mathbb{O}_{2}$. It follows from Lemma 3.10.3 that
$(M_{1},\mathbb{O}_{M_{1}})$ is adapted to $\mathfrak{L}_{1}$.

Next, consider the pair

$(M_{2},\mathbb{O}_{M_{2}})=(L(A_{3}+A_{2}+A_{1};2,3,4,6,7,\theta),\\{0\\}\times\\{0\\}\times(2)),$

where $\theta$ is the highest root for $G$, i.e.

$\theta=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}.$

An atlas computation shows that this indeed defines a Levi subgroup (of the
correct Lie type). By construction, $L\subset M_{2}$ and clearly
$\mathbb{O}_{M_{2}}=\mathrm{Ind}^{M_{2}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{2})=1=\dim\mathfrak{P}_{2}$. There is a single
codimension 2 orbit in $\overline{\mathbb{O}}_{M_{2}}$, namely
$\mathbb{O}_{M_{2},1}=\\{0\\}$, and the singularity is of type $A_{1}$.
Finally
$\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},1}=A_{4}+A_{2}=\mathbb{O}_{1}$.
So by Lemma 3.10.3, $M_{2}$ is adapted to $\mathfrak{L}_{2}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(D_{6})$ | $(3^{3},1^{3})$ | $2$
$2$ | $A_{1}$ | $L(A_{3}+A_{2}+A_{1};2,3,4,6,7,\theta)$ | $\\{0\\}\times\\{0\\}\times(2)$ | $2$

Since $M_{1}$ is standard, $\tau_{1}(1)=(1,0,0,0,0,0,0)$ and so by Proposition
3.10.6, we have
$\delta_{1}=\frac{1}{4}\tau_{1}(1)=\frac{1}{4}(1,0,0,0,0,0,0)$. On the other
hand, $\tau_{1}(2)$ is (either) generator of the free abelian group

$\displaystyle\mathfrak{X}(M_{2})$
$\displaystyle=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0,\
\forall\alpha\in\Delta(\mathfrak{m}_{2},\mathfrak{h})\\}$
$\displaystyle=\\{(x,0,0,0,y,0,0)\in\mathbb{Z}^{7}\mid 2x+3y=0\\}$

Hence $\tau_{1}(2)=(3,0,0,0,-2,0,0)$ and
$\delta_{2}=\frac{1}{4}\tau_{1}(2)=\frac{1}{4}(3,0,0,0,-2,0,0)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}+\delta_{2}$
$\displaystyle=\frac{1}{2}(-3,2,2,2,-6,2,2)+\frac{1}{4}(1,0,0,0,0,0,0)+\frac{1}{4}(3,0,0,0,-2,0,0)$
$\displaystyle=\frac{1}{2}(-1,2,2,2,-7,2,2)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,0,0,1,0,0,1).$

  * •

$E_{7}(a_{5})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{1}+2A_{2};1,2,3,5,6)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{1}+2A_{2};1,2,3,5,6),\\{0\\}).$

There are two codimension 2 leaves $\mathfrak{L}_{1},\mathfrak{L}_{2}\subset
X$, corresponding to the orbits $\mathbb{O}_{1}=E_{6}(a_{3})$ and
$\mathbb{O}_{2}=D_{6}(a_{2})$. Both singularities are of type $A_{1}$. We will
compute the pairs $(M_{k},\mathbb{O}_{M_{k}})$ using Lemma 3.10.3. First
define

$(M_{1},\mathbb{O}_{M_{1}})=(L(E_{6}),D_{4}(a_{1}))$

By construction, $L\subset M_{1}$, and
$\mathbb{O}_{M_{1}}=\mathrm{Ind}^{M_{1}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{1})=1=\dim\mathfrak{P}_{1}$. Note that
$\overline{\mathbb{O}}_{M_{1}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{1},2}=A_{3}+A_{1}$, and the corresponding singularity is of
type $A_{1}$. Finally, by [DE09, Sec 4]

$\displaystyle\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},2}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{1}}(\operatorname{Ind}^{M_{1}}_{L(D_{5};1,2,3,4,5)}(3,2^{2},1^{3}))$
$\displaystyle=\operatorname{Ind}^{G}_{L(D_{5};1,2,3,4,5)}(3,2^{2},1^{3})$
$\displaystyle=D_{6}(a_{2})$ $\displaystyle=\mathbb{O}_{2}$

So by Lemma 3.10.3, $(M_{1},\mathbb{O}_{M_{1}})$ is adapted to
$\mathfrak{L}_{1}$.

Next define

$(M_{2},\mathbb{O}_{M_{2}})=(L(A_{1}+A_{5};1,2,3,5,6,\theta),\\{0\\}\times(2^{3})).$

where $\theta$ is the positive root

$\theta=\alpha_{2}+\alpha_{3}+2\alpha_{4}+\alpha_{5}+\alpha_{6}+\alpha_{7}$

An atlas computation shows that this indeed defines a Levi (of the stated
type). By construction, $L\subset M_{2}$, and
$\mathbb{O}_{M_{1}}=\mathrm{Ind}^{M_{1}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{2})=1=\dim\mathfrak{P}_{2}$. Note that
$\overline{\mathbb{O}}_{M_{2}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{2},1}=\\{0\\}\times(2^{2},1^{2})$, and the corresponding
singularity is of type $A_{1}$. Finally, by [DE09, Sec 4]

$\displaystyle\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},1}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{2}}(\operatorname{Ind}^{M_{2}}_{L(2A_{1}+A_{3};1,2,4,5,7)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(2A_{1}+3A_{1};1,2,4,5,7)}\\{0\\}$
$\displaystyle=E_{6}(a_{3})$ $\displaystyle=\mathbb{O}_{1}$

So by Lemma 3.10.3, $(M_{2},\mathbb{O}_{M_{2}})$ is adapted to
$\mathfrak{L}_{2}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{6})$ | $D_{4}(a_{1})$ | $1$
$2$ | $A_{1}$ | $L(A_{1}+A_{5};1,2,3,5,6,\theta)$ | $\\{0\\}\times(2^{3})$ | $1$

Since $M_{1}$ is standard, $\tau_{1}(1)=(0,0,0,0,0,0,1)$ and hence by
Proposition 3.10.6
$\delta_{1}=\frac{1}{2}\tau_{1}(1)=\frac{1}{2}(0,0,0,0,0,0,1)$. On the other
hand, $\tau_{1}(2)$ is (either) generator of the free abelian group

$\displaystyle\mathfrak{X}(M_{2})$
$\displaystyle=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0,\
\forall\alpha\in\Delta(\mathfrak{m},\mathfrak{h})\\}$
$\displaystyle=\\{(0,0,0,x,0,0,y)\in\mathbb{Z}^{7}\mid 2x+y=0\\}$

So we may take $\tau_{1}(2)=(0,0,0,1,0,0,-2)$ and
$\delta_{2}=\frac{1}{2}\tau_{1}(2)=\frac{1}{2}(0,0,0,1,0,0,-2)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}+\delta_{2}$
$\displaystyle=\frac{1}{2}(2,2,2,-5,2,2,-2)+\frac{1}{2}(0,0,0,0,0,0,1)+\frac{1}{2}(0,0,0,1,0,0,-2)$
$\displaystyle=\frac{1}{2}(2,2,2,-4,2,2,-3)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(0,0,1,0,1,0,0).$

  * •

$E_{7}(a_{4})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(2A_{1}+A_{2};2,3,5,6)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(2A_{1}+A_{2};2,3,5,6),\\{0\\})$

There are three codimension 2 leaves
$\mathfrak{L}_{1},\mathfrak{L}_{2},\mathfrak{L}_{3}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$,
corresponding to the orbits $\mathbb{O}_{1}=A_{6}$,
$\mathbb{O}_{2}=D_{5}+A_{1}$, and $\mathbb{O}_{3}=D_{6}(a_{1})$. All three
singularities are of type $A_{1}$. We will compute the pairs
$(M_{k},\mathbb{O}_{M_{k}})$ using Lemma 3.10.3.

For $\mathfrak{L}_{1}$, consider the standard pair

$(M_{1},\mathbb{O}_{M_{1}})=(L(D_{6}),(5,3^{2},1))$

By construction, $L\subset M_{1}$ and
$\mathbb{O}_{M_{1}}=\mathrm{Ind}^{M_{1}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{1})=1=\dim\mathfrak{P}_{1}$.
$\overline{\mathbb{O}}_{M_{1}}$ contains two codimension 2 orbits, namely
$\mathbb{O}_{M_{1},2}=(4^{2},3,1)$ and $\mathbb{O}_{M_{1},3}=(5,3,2^{2})$, and
the corresponding singularities are of type $A_{1}$. Note that

$\displaystyle\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},2}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{1}}(\operatorname{Ind}^{M_{1}}_{L(2A_{2};2,4,6,7)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(2A_{2};2,4,6,7)}\\{0\\}$

Up to conjugation by $G$, there is a unique Levi subgroup of type $2A_{2}$. So
the final induction is the same as
$\operatorname{Ind}^{G}_{L(2A_{2};1,3,5,6)}\\{0\\}$, which is
$D_{5}+A_{1}=\mathbb{O}_{2}$ by [DE09, Sec 4]. Similarly

$\displaystyle\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},3}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{1}}(\operatorname{Ind}^{M_{1}}_{L(A_{3};5,6,7)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(A_{3};5,6,7)}\\{0\\}$
$\displaystyle=\operatorname{Ind}^{G}_{L(A_{3};1,3,4)}\\{0\\}$
$\displaystyle=D_{6}(a_{1})$ $\displaystyle=\mathbb{O}_{3}$

It follows from Lemma 3.10.3 that $(M_{1},\mathbb{O}_{M_{1}})$ is adapted to
$\mathfrak{L}_{1}$.

For $\mathfrak{L}_{2}$, consider the non-standard pair

$(M_{2},\mathbb{O}_{M_{2}})=(L(D_{5}+A_{1};2,3,5,6,\theta_{1},\theta_{2}),(3^{3},1)\times(2))$

where $\theta_{1},\theta_{2}$ are the roots

$\theta_{1}=-\alpha_{2}-\alpha_{3}-\alpha_{4}-\alpha_{5}-\alpha_{6}\qquad\theta_{2}=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}$

An atlas computation shows that this indeed defines a Levi subgroup (of the
correct type). By construction, $L\subset M_{2}$. Computing the Cartan matrix
for $M_{2}$, we see that the simple roots for $L$ embed as the simple roots
for $M_{2}$ corresponding to the non-central nodes of the $D_{5}$ Dynkin
diagram. In particular,
$\mathrm{Ind}^{M_{2}}_{L}\\{0\\}=\mathbb{O}_{(3^{3},1)\times(2)}=\mathbb{O}_{M_{2}}$.
Furthermore, $\dim\mathfrak{X}(\mathfrak{m}_{2})=1=\dim\mathfrak{P}_{2}$.
$\overline{\mathbb{O}}_{M_{2}}$ contains two codimension 2 orbits, namely
$\mathbb{O}_{M_{2},1}=(3^{3},1)\times\\{0\\}$ and
$\mathbb{O}_{M_{2},3}=(3^{2},2^{2})\times(2)$, and the corresponding
singularities are of type $A_{1}$. We have

$\displaystyle\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},1}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{2}}(\operatorname{Ind}^{M_{1}}_{L(3A_{1}+A_{2};1,2,3,5,7)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(3A_{1}+A_{2};1,2,3,5,7)}\\{0\\}$
$\displaystyle=A_{6}$ $\displaystyle=\mathbb{O}_{1}$

Similarly

$\displaystyle\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},3}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{2}}(\operatorname{Ind}^{M_{1}}_{L(A_{3};1,3,4)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(A_{3};1,3,4)}\\{0\\}$
$\displaystyle=D_{6}(a_{1})$ $\displaystyle=\mathbb{O}_{3}$

So by Lemma 3.10.3 $(M_{2},\mathbb{O}_{M_{2}})$ is adapted to
$\mathfrak{L}_{2}$.

For $\mathfrak{L}_{3}$, consider the pair

$(M_{3},\mathbb{O}_{M_{3}})=(L(A_{1}+A_{2}+A_{3};2,3,5,6,\theta_{2},\theta_{3}),\mathbb{O}_{(2)}\times\\{0\\}\times\mathbb{O}_{(2^{2})})$

where $\theta_{3}$ is the negative root

$\theta_{3}=-\alpha_{2}-\alpha_{3}-\alpha_{4}-\alpha_{5}-\alpha_{6}-\alpha_{7}.$

An atlas computation shows that this indeed defines a Levi (of the stated
type). By construction $L\subset M_{3}$. Computing the Cartan matrix for
$M_{3}$, we see that the $A_{2}$ factor of $L$ embeds into the $A_{2}$ factor
of $M_{3}$ and the $2A_{1}$ factor of $L$ embeds into $A_{3}$ factor of
$M_{3}$. In particular,
$\mathrm{Ind}^{M_{3}}_{L}\\{0\\}=\mathbb{O}_{(2)\times\\{0\\}\times\mathbb{O}_{(2^{2})}}=\mathbb{O}_{M_{3}}$.
Furthermore, $\dim\mathfrak{X}(\mathfrak{m}_{3})=1=\dim\mathfrak{P}_{3}$.
$\overline{\mathbb{O}}_{M_{3}}$ contains two codimension 2 orbits, namely
$\mathbb{O}_{M_{3},1}=\\{0\\}\times\\{0\\}\times\mathbb{O}_{(2^{2})}$ and
$\mathbb{O}_{M_{3},2}=\mathbb{O}_{(2)}\times\\{0\\}\times\mathbb{O}_{(2,1^{2})}$.Both
singularities are of type $A_{1}$. We have

$\displaystyle\operatorname{Ind}^{G}_{M_{3}}\mathbb{O}_{M_{3},1}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{3}}(\operatorname{Ind}^{M_{3}v}_{L(3A_{1}+A_{2};1,2,3,5,7)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(3A_{1}+A_{2};1,2,3,5,7)}\\{0\\}$
$\displaystyle=A_{6}$ $\displaystyle=\mathbb{O}_{1}$

Similarly

$\displaystyle\operatorname{Ind}^{G}_{M_{3}}\mathbb{O}_{M_{3},2}$
$\displaystyle=\operatorname{Ind}^{G}_{M_{3}}(\operatorname{Ind}^{M_{3}}_{L(2A_{2};1,3,5,6)}\\{0\\})$
$\displaystyle=\operatorname{Ind}^{G}_{L(2A_{2};1,3,5,6)}\\{0\\}$
$\displaystyle=D_{5}+A_{1}$ $\displaystyle=\mathbb{O}_{2}$

So by Lemma 3.10.3, $(M_{3},\mathbb{O}_{M_{3}})$ is adapted to
$\mathfrak{L}_{3}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(D_{6})$ | $\mathbb{O}_{(5,3^{2},1)}$ | $2$
$2$ | $A_{1}$ | $L(D_{5}+A_{1};2,3,5,6,\theta_{1},\theta_{2})$ | $\mathbb{O}_{(3^{3},1)}\times\mathbb{O}_{(2)}$ | $2$
$3$ | $A_{1}$ | $L(A_{1}+A_{2}+A_{3};2,3,5,6,\theta_{2},\theta_{3})$ | $\mathbb{O}_{(2)}\times\\{0\\}\times\mathbb{O}_{(2^{2})}$ | $2$

Since $M_{1}$ is standard, $\tau_{1}(1)=(1,0,0,0,0,0,0)$ and so by Proposition
3.10.6, $\delta_{1}=\frac{1}{4}\tau_{1}(1)=\frac{1}{4}(1,0,0,0,0,0,0)$. By
definition, $\tau_{1}(2)$ is (either) generator of the free abelian group

$\displaystyle\mathfrak{X}(M_{2})$
$\displaystyle=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0,\
\forall\alpha\in\Delta(\mathfrak{m}_{2},\mathfrak{h})\\}$
$\displaystyle=\\{(x,0,0,0,0,0,y)\in\mathbb{Z}^{7}\mid 2x+y=0\\}$

So we may take $\tau_{1}(2)=(1,0,0,0,0,0,-2)$ and
$\delta_{2}=\frac{1}{4}\tau_{1}(2)=\frac{1}{4}(1,0,0,0,0,0,-2)$. By a similar
computation, we get $\tau_{1}(3)=(3,0,0,-2,0,0,2)$ and
$\delta_{3}=\frac{1}{4}(3,0,0,-2,0,0,2)$. Now by Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}+\delta_{2}+\delta_{3}$
$\displaystyle=\frac{1}{2}(-1,2,2,-4,2,2,-2)+\frac{1}{4}(1,0,0,0,0,0,0)+\frac{1}{4}(1,0,0,0,0,0,-2)+\frac{1}{4}(3,0,0,-2,0,0,2)$
$\displaystyle=\frac{1}{4}(3,4,4,-10,4,4,-4)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{4}(1,1,1,0,0,1,1).$

##### Type $E_{8}$

$\alpha_{\mathrlap{1}}$$\alpha_{2}$$\alpha_{\mathrlap{3}}$$\alpha_{\mathrlap{4}}$$\alpha_{\mathrlap{5}}$$\alpha_{\mathrlap{6}}$$\alpha_{\mathrlap{7}}$$\alpha_{\mathrlap{8}}$

  * •

$A_{2}$. Note that $\mathbb{O}$ is even and $L_{\mathbb{O}}=L(E_{7})$. Hence
by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(E_{7}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. Thus

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{7})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,0,0,0,0,0,0,1)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,0,0,0,1)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}$
$\displaystyle=\frac{1}{2}(2,2,2,2,2,2,2,-27)+\frac{1}{2}(0,0,0,0,0,0,0,1)=(1,1,1,1,1,1,1,-13).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(1,0,0,1,0,1,1,1).$

  * •

$2A_{2}$. Note that $\mathbb{O}$ is even and $L_{\mathbb{O}}=L(D_{7})$. Hence
by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(D_{7}),\\{0\\})$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(D_{7})$ | $\\{0\\}$ | $1$

Now $\tau_{1}=(1,0,0,0,0,0,0,0)$. So by Proposition 3.10.6, we have
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(1,0,0,0,0,0,0,0)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}$
$\displaystyle=\frac{1}{2}(-21,2,2,2,2,2,2,2)+\frac{1}{2}(1,0,0,0,0,0,0)$
$\displaystyle=(-10,1,1,1,1,1,1,1).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(1,0,0,1,0,0,1,0).$

  * •

$D_{4}(a_{1})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{1}+E_{6})$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{1}+E_{6}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
correponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{1}+E_{6})$ | $\\{0\\}$ | $1$

Thus $\tau_{1}=(0,0,0,0,0,0,1,0)$, and by Proposition 3.10.6,
$\delta_{1}=\frac{1}{2}(0,0,0,0,0,0,1,0)$. Now by Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}$
$\displaystyle=\frac{1}{2}(2,2,2,2,2,2,-17,2)+\frac{1}{2}(0,0,0,0,0,0,1,0)$
$\displaystyle=(1,1,1,1,1,1,-8,1).$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(0,0,0,1,0,0,1,1).$

  * •

$D_{4}(a_{1})+A_{2}$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{7})$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{7}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{7})$ | $\\{0\\}$ | $1$

Note that $\tau_{1}=(0,1,0,0,0,0,0,0)$, so by Proposition 3.10.6,
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,1,0,0,0,0,0,0)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta$
$\displaystyle=\frac{1}{2}(2,-15,2,2,2,2,2,2)+\frac{1}{2}(0,1,0,0,0,0,0,0)$
$\displaystyle=(1,-7,1,1,1,1,1,1)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(0,0,0,1,0,0,0,1).$

  * •

$D_{4}+A_{2}$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{6};1,3,4,5,6,7)$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{6};1,3,4,5,6,7),\\{0\\}).$

There are two codimension 2 leaves
$\mathfrak{L}_{1},\mathfrak{L}_{2}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$,
corresponding to the orbits $\mathbb{O}_{1}=A_{4}+A_{2}+A_{1}$ and
$\mathbb{O}_{2}=D_{5}(a_{1})+A_{1}$. Both singularities are of type $A_{1}$.
We will compute $(M_{k},\mathbb{O}_{M_{k}})$ using Lemma 3.10.3.

First, define

$(M_{1},\mathbb{O}_{M_{1}})=(L(E_{7}),A_{2}+3A_{1})$

By construction $L\subset M_{1}$ and by the calculation for
$A_{2}+3A_{1}\subset E_{6}$, we have
$\mathbb{O}_{M_{1}}=\mathrm{Ind}^{M_{1}}_{L}\\{0\\}$. Furthermore,
$\dim\mathfrak{X}(\mathfrak{m}_{1})=1=\dim\mathfrak{P}_{1}$. Note that
$\overline{\mathbb{O}}_{M_{1}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{1},2}=A_{2}+2A_{1}$, and the corresponding singularity is of
type $A_{1}$. Finally,
$\operatorname{Ind}^{G}_{M_{1}}\mathbb{O}_{M_{1},2}=D_{5}(a_{1})+A_{1}=\mathbb{O}_{2}$.
So by Lemma 3.10.3, $(M_{1},\mathbb{O}_{M_{1}})$ is adapted to
$\mathfrak{L}_{1}$.

Next, define

$(M_{2},\mathbb{O}_{M_{2}})=(L(A_{1}+A_{6};1,3,4,5,6,7,\theta),(2)\times\\{0\\})$

where $\theta$ is the highest root for $G$, i.e.

$\theta=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}+2\alpha_{8}$

An atlas computation shows that this indeed defines a Levi (of the stated
type). By construction $L\subset M_{2}$ and
$\mathbb{O}_{M_{2}}=\mathrm{Ind}^{M_{2}}_{L}\\{0\\}$. Of course,
$\dim\mathfrak{X}(\mathfrak{m}_{2})=1=\dim\mathfrak{P}_{2}$. Note that
$\overline{\mathbb{O}}_{M_{2}}$ contains a single codimension 2 orbit, namely
$\mathbb{O}_{M_{2},1}=\\{0\\}\times\\{0\\}$, and the corresponding singularity
is of type $A_{1}$. Finally

$\operatorname{Ind}^{G}_{M_{2}}\mathbb{O}_{M_{2},1}=A_{4}+A_{2}+A_{1}=\mathbb{O}_{1}.$

So by Lemma 3.10.3,$(M_{2},\mathbb{O}_{M_{2}})$ is adapted to
$\mathfrak{L}_{2}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{7})$ | $A_{2}+3A_{1}$ | $2$
$2$ | $A_{1}$ | $L(A_{1}+A_{6};1,3,4,5,6,7,\theta)$ | $(2)\times\\{0\\}$ | $2$

Since $M_{1}$ is standard, $\tau_{1}(1)=(0,0,0,0,0,0,0,1)$, so by Proposition
3.10.6 $\delta_{1}=\frac{1}{4}\tau_{1}(1)=\frac{1}{4}(0,0,0,0,0,0,0,1)$. On
the other hand, $\tau_{1}(2)$ is (either) generator of the free abelian group

$\displaystyle\mathfrak{X}(M_{2})$
$\displaystyle=\\{\lambda\in\Lambda\mid\langle\lambda,\alpha^{\vee}\rangle=0,\
\forall\alpha\in\Delta(\mathfrak{m}_{2},\mathfrak{h})\\}$
$\displaystyle=\\{(0,x,0,0,0,0,0,y)\in\mathbb{Z}^{8}\mid 3x+2y=0\\}.$

So we may take $\tau_{1}(2)=(0,-2,0,0,0,0,0,3)$ and
$\delta_{2}=\frac{1}{4}\tau_{1}(2)=\frac{1}{4}(0,-2,0,0,0,0,0,3)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta_{1}+\delta_{2}$
$\displaystyle=(1,-6,1,1,1,1,1,-3)+\frac{1}{4}(0,0,0,0,0,0,0,1)+\frac{1}{4}(0,-2,0,0,0,0,0,3)$
$\displaystyle=\frac{1}{2}(2,-13,2,2,2,2,2,-4)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,0,0,1,0,0,1,1)$

  * •

$D_{6}(a_{2})$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(D_{7}),(3,2^{4},1))\\}.$

There are two codimension orbits in the closure of $\mathbb{O}$, and the
corresponding singularities are of types $m$ and $A_{1}$. Therefore, there is
a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. We have $\mathfrak{r}=2A_{1}$,
and thus $\dim\mathfrak{P}^{X}=1$. So by Lemma 3.10.2

$(L,\mathbb{O}_{L})=(L(D_{7}),(3,2^{4},1)).$

Thus, $M_{1}=L$ and

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(D_{7})$ | $(3,2^{4},1)$ | $2$

Note that $\tau_{1}=(1,0,0,0,0,0,0,0)$, so by Proposition 3.10.6
$\delta_{1}=\frac{1}{4}\tau_{1}=\frac{1}{4}(1,0,0,0,0,0,0,0)$. By [LMM21, Prop
8.2.3] $\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}\rho(\mathfrak{l})$. Thus by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\frac{1}{2}\rho(\mathfrak{l})+\delta$
$\displaystyle=\frac{1}{4}(-21,2,2,2,2,2,2,2)+\frac{1}{4}(1,0,0,0,0,0,0,0)$
$\displaystyle=\frac{1}{2}(-10,1,1,1,1,1,1,1)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(1,0,0,1,0,0,1,0).$

  * •

$E_{6}(a_{3})+A_{1}$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{7}),A_{1}+2A_{2})\\}.$

There are two codimension orbits in the closure of $\mathbb{O}$, and the
corresponding singularities are $m$ and $A_{1}$. Hence, there is a single
codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. We have $\mathfrak{r}=A_{1}$,
and thus $\dim\mathfrak{P}^{X}=1$. So by Lemma 3.10.2

$(L,\mathbb{O}_{L})=(L(E_{7}),A_{1}+2A_{2}).$

Thus, $M_{1}=L$ and we have

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{7})$ | $A_{1}+2A_{2}$ | $1$

Note that $\tau_{1}=(0,0,0,0,0,0,0,1)$. So by Proposition 3.10.6,
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,0,0,0,1)$. By 7
$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{3}\rho(\mathfrak{l})$. So by Proposition
4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\frac{1}{3}\rho(\mathfrak{l})+\delta$
$\displaystyle=\frac{1}{6}(2,2,2,2,2,2,2,-27)+\frac{1}{2}(0,0,0,0,0,0,0,1)$
$\displaystyle=\frac{1}{3}(1,1,1,1,1,1,1,-12)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{3}(0,1,1,0,1,0,1,1).$

  * •

$E_{7}(a_{5})$. We have

$\mathcal{P}_{\mathrm{rig}}(\mathbb{O})=\\{(L(E_{7}),(A_{1}+A_{3})^{\prime}),(L(E_{6}+A_{1}),3A_{1}\times\\{0\\})\\}.$

There are two codimension orbits in the closure of $\mathbb{O}$, and the
corresponding singularities are of types $m$ and $2A_{1}$. Hence, there is a
single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$. We have $\mathfrak{r}=A_{1}$,
and thus $\dim\mathfrak{P}^{X}=1$. At this point, it is not clear which pair
in $\mathcal{P}_{\mathrm{rig}}(\mathbb{O})$ induces $\mathbb{O}$ birationally.
However, it turns out not to matter. In both cases, we get the same answer for
$\gamma_{0}(\widehat{\mathbb{O}})$.

Assume first that

$(L,\mathbb{O}_{L})=(L(E_{7}),(A_{1}+A_{3})^{\prime}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{7})$ | $(A_{1}+A_{3})^{\prime}$ | $1$

Thus $\tau_{1}=(0,0,0,0,0,0,0,1)$ and by Proposition 3.10.6
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,0,0,0,1)$.
$\gamma_{0}(\mathbb{O}_{L})$ was computed in the previous subsection in terms
of fundamental weights for $L$. It is not difficult to rewrite this in terms
of fundamental weights for $G$

$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}(1,1,0,1,0,1,1,-9)$

Now by Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\gamma_{0}(\mathbb{O}_{L})+\delta$
$\displaystyle=\frac{1}{2}(1,1,0,1,0,1,1,-9)+\frac{1}{2}(0,0,0,0,0,0,0,1)$
$\displaystyle=\frac{1}{2}(1,1,0,1,0,1,1,-8)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(0,0,1,0,1,0,0,1).$

Next, suppose

$(L,\mathbb{O}_{L})=(L(E_{6}+A_{1}),3A_{1}\times\\{0\\}).$

Then

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(E_{6}+A_{1})$ | $3A_{1}\times\\{0\\}$ | $1$

Thus $\tau_{1}=(0,0,0,0,0,0,1,0)$ and by Proposition 3.10.6
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,0,0,1,0)$.
$\gamma_{0}(\mathbb{O}_{L})$ was computed in Section 4.3.1 in terms of
fundamental weights for $L$. Rewriting again in terms of fundamental weights
for $G$

$\gamma_{0}(\mathbb{O}_{L})=\frac{1}{2}(1,1,1,1,1,1,-9,2).$

Now by Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\gamma_{0}(\mathbb{O}_{L})+\delta$
$\displaystyle=\frac{1}{2}(1,1,1,1,1,1,-9,2)+\frac{1}{2}(0,0,0,0,0,0,1,0)$
$\displaystyle=\frac{1}{2}(1,1,1,1,1,1,-8,2)$

Conjugating by $W$ we get once again

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{2}(0,0,1,0,1,0,0,1).$

  * •

$E_{8}(a_{7})$. Note that $\mathbb{O}$ is even and
$L_{\mathbb{O}}=L(A_{4}+A_{3})$. Hence by Lemma 3.10.1

$(L,\mathbb{O}_{L})=(L(A_{4}+A_{3}),\\{0\\}).$

There is a single codimension 2 leaf
$\mathfrak{L}_{1}\subset\operatorname{Spec}(\mathbb{C}[\mathbb{O}])$ and the
corresponding singularity is of type $A_{1}$.

$k$ | $\Sigma_{k}$ | $M_{k}$ | $\mathbb{O}_{M_{k}}$ | $c_{k}$
---|---|---|---|---
$1$ | $A_{1}$ | $L(A_{4}+A_{3})$ | $\\{0\\}$ | $1$

Thus $\tau_{1}=(0,0,0,0,1,0,0,0)$, so by Proposition 3.10.6
$\delta_{1}=\frac{1}{2}\tau_{1}=\frac{1}{2}(0,0,0,0,1,0,0,0)$. Now by
Proposition 4.1.1

$\displaystyle\gamma_{0}(\widehat{\mathbb{O}})$
$\displaystyle=\rho(\mathfrak{l})+\delta$
$\displaystyle=\frac{1}{2}(2,2,2,2,-9,2,2,2)+\frac{1}{2}(0,0,0,0,1,0,0,0)$
$\displaystyle=(1,1,1,1,-4,1,1,1)$

Conjugating by $W$ we get

$\gamma_{0}(\widehat{\mathbb{O}})=(0,0,0,0,1,0,0,0).$

  * •

$E_{8}(b_{6})$. $\widehat{\mathbb{O}}$ is birationally rigid by Proposition
3.9.2. The infinitesimal character $\gamma_{0}(\widehat{\mathbb{O}})$ was
computed in [LMM21, Ex 8.5.2]

$\gamma_{0}(\widehat{\mathbb{O}})=\frac{1}{3}(1,1,0,0,0,1,0,1).$

$\mathbb{O}$ | $\gamma_{0}(\widehat{\mathbb{O}})$
---|---
$G_{2}(a_{1})$ | $(1,0)$
Table 9. Unipotent infinitesimal characters attached to birationally rigid covers: type $G_{2}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | $\gamma_{0}(\widehat{\mathbb{O}})$
---|---
$A_{2}$ | $\frac{1}{2}(1,1,0,2)$
$B_{2}$ | $\frac{1}{2}(0,1,0,2)$
$C_{3}(a_{1})$ | $\frac{1}{2}(1,0,1,1)$
$F_{4}(a_{3})$ | $(0,0,1,0)$
Table 10. Unipotent infinitesimal characters attached to birationally rigid covers: type $F_{4}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | $\gamma_{0}(\widehat{\mathbb{O}})$
---|---
$A_{2}$ | $(1,0,0,1,0,1)$
$2A_{2}$ | $\frac{1}{3}(1,3,1,1,1,1)$
$D_{4}(a_{1})$ | $(0,0,0,1,0,0)$
$A_{5}$ | $\frac{1}{6}(1,3,1,1,1,1)$
$E_{6}(a_{3})$ | $\frac{1}{3}(0,1,1,0,1,0)$
Table 11. Unipotent infinitesimal characters attached to birationally rigid covers: type $E_{6}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | $\gamma_{0}(\widehat{\mathbb{O}})$
---|---
$(3A_{1})^{\prime\prime}$ | $\frac{1}{2}(2,1,2,1,1,1,1)$
$A_{2}$ | $(1,0,0,1,0,1,1)$
$A_{2}+3A_{1}$ | $\frac{1}{2}(1,1,1,0,1,1,1)$
$(A_{3}+A_{1})^{\prime\prime}$ | $\frac{1}{2}(2,0,1,1,0,1,1)$
$D_{4}(a_{1})$ | $(0,0,0,1,0,0,1)$
$A_{3}+2A_{1}$ | $\frac{1}{2}(1,1,1,0,1,0,1)$
$D_{4}(a_{1})+A_{1}$ | $\frac{1}{2}(0,0,1,1,0,1,1)$
$A_{3}+A_{2}+A_{1}$ | $\frac{1}{2}(1,0,0,1,0,1,1)$
$A_{5}+A_{1}$ | $\frac{1}{6}(2,1,2,1,1,1,1)$
$D_{5}(a_{1})+A_{1}$ | $\frac{1}{2}(1,0,0,1,0,0,1)$
$E_{7}(a_{5})$ | $\frac{1}{2}(0,0,1,0,1,0,0)$
$E_{7}(a_{4})$ | $\frac{1}{4}(1,1,1,0,0,1,1)$
Table 12. Unipotent infinitesimal characters attached to birationally rigid covers: type $E_{7}$. Special unipotent characters are highlighted in blue. $\mathbb{O}$ | $\gamma_{0}(\widehat{\mathbb{O}})$
---|---
$A_{2}$ | $(1,0,0,1,0,1,1,1)$
$2A_{2}$ | $(1,0,0,1,0,0,1,0)$
$D_{4}(a_{1})$ | $(0,0,0,1,0,0,1,1)$
$D_{4}(a_{1})+A_{2}$ | $(0,0,0,1,0,0,0,1)$
$D_{4}+A_{2}$ | $\frac{1}{2}(1,0,0,1,0,0,1,1)$
$D_{6}(a_{2})$ | $\frac{1}{2}(1,0,0,1,0,0,1,0)$
$E_{6}(a_{3})+A_{1}$ | $\frac{1}{3}(0,1,1,0,1,0,1,1)$
$E_{7}(a_{5})$ | $\frac{1}{2}(0,0,1,0,1,0,0,1)$
$E_{8}(a_{7})$ | $(0,0,0,0,1,0,0,0)$
$E_{8}(b_{6})$ | $\frac{1}{3}(1,1,0,0,0,1,0,1)$
Table 13. Unipotent infinitesimal characters attached to birationally rigid
covers: type $E_{8}$. Special unipotent characters are highlighted in blue.

##  5\. Maximality of unipotent ideals

######  Theorem 5.0.1.

Let $G$ be a complex reductive algebraic group and let
$\widetilde{\mathbb{O}}$ be a $G$-equivariant nilpotent cover. Then the
unipotent ideal $I_{0}(\widetilde{\mathbb{O}})\subset U(\mathfrak{g})$ is
maximal.

###### Proof.

Replacing $G$ with a covering group if necessary, we can assume that $G$ is
simply connected. Hence, $G$ is of the form

$G\simeq T\times G_{1}\times...\times G_{n},$

where $T$ is a torus and $G_{1},...,G_{n}$ are simply connected simple groups.
It follow that $\widetilde{\mathbb{O}}$ is of the form

$\widetilde{\mathbb{O}}\simeq\widetilde{\mathbb{O}}_{1}\times...\times\widetilde{\mathbb{O}}_{n},$

where $\widetilde{\mathbb{O}}_{i}$ are $G_{i}$-equivariant covers of nilpotent
co-adjoint $G_{i}$-orbits. Now

$\mathcal{A}_{0}^{\widetilde{X}}\simeq\bigotimes_{i=1}^{n}\mathcal{A}_{0}^{\widetilde{X}_{1}}\qquad\Phi_{0}^{\widetilde{X}}=\bigotimes_{i=1}^{n}\Phi_{0}^{\widetilde{X}_{i}},$

where we identify $U(\mathfrak{g})$ with $U(\mathfrak{t})\otimes\bigotimes
U(\mathfrak{g}_{i})$ and map trivially from $\mathfrak{t}$. Consequently

$I_{0}(\widetilde{\mathbb{O}})=\sum_{i=1}^{n}U(\mathfrak{t})\otimes
U(\mathfrak{g}_{1})\otimes...\otimes U(\mathfrak{g}_{i-1})\otimes
I_{0}(\widetilde{\mathbb{O}}_{i})\otimes
U(\mathfrak{g}_{i+1})\otimes...\otimes U(\mathfrak{g}_{n}).$

This ideal is maximal if and only if each $I_{0}(\widetilde{\mathbb{O}}_{i})$
is maximal. In this way, we can reduce to the case in which $G$ is simple and
simply connected, i.e. $G$ is isomorphic to $\mathrm{SL}(n)$,
$\mathrm{Spin}(n)$, $\mathrm{Sp}(2n)$ (for some $n$) or a simply connected
simple group of exceptional type. For $G=\mathrm{SL}(n)$ or $\mathrm{Sp}(2n)$,
the maximality assertion follows from [LMM21, Thm 8.5.1]. For
$G=\mathrm{Spin}(n)$, we refer to Proposition A.1.2. For the exceptional
cases, we refer to Proposition A.2.1. ∎

In [LMM21, Prop 6.3.3], we show that the maximality of
$I_{0}(\widetilde{\mathbb{O}})$ is equivalent to the simplicity of
$\mathcal{A}_{0}^{\widetilde{X}}$. This, combined with Theorem 5.0.1, proves
the following result.

######  Corollary 5.0.2.

Let $\widetilde{\mathbb{O}}$ be a (finite, connected) cover of a complex
nilpotent orbit in a complex reductive Lie algebra. Then the canonical
quantization of $\mathbb{C}[\widetilde{\mathbb{O}}]$ is a simple algebra.

##  6\. Real groups

In this section, we take $G$ to be a real reductive Lie group. For
concreteness, we will use Knapp’s definition of ‘real reductive Lie group’,
see [Kna96, Chp VII, Sec 2]. In particular, we assume that $G$ has a Cartan
decomposition, finitely many connected components, and that the identity
component of $G$ is a finite cover of an algebraic group. Choose a maximal
compact subgroup $K\subset G$ and let $\mathfrak{g}$ denote the complexified
Lie algebra of $G$. Under our assumptions on $G$, we have the usual bijection
between irreducible admissible representations of $G$ (up to infinitesimal
equivalence) and irreducible admissible $(\mathfrak{g},K)$-modules (up to
isomorphism). In this setting, we propose the following deifnition.

######  Definition 6.0.1.

Let $\mathbb{O}\subset\mathfrak{g}^{*}$ be a nilpotent orbit which satisfies
the condition

(6.0.1) $\operatorname{Spec}(\mathbb{C}[\mathbb{O}])\text{ contains no
codimension 2 leaves}.$

Then a _unipotent representation_ of $G$ attached to $\mathbb{O}$ is an
irreducible $(\mathfrak{g},K)$-module $X$ such that
$\operatorname{Ann}(X)=I_{0}(\mathbb{O})$. Write
$\mathrm{Unip}_{\mathbb{O}}(G)$ for the set of equivalence classes of such
representations.

######  Remark 6.0.2.

Note: every birationally rigid orbit (in particular, every rigid orbit)
satisfies condition (6.0.1). This is immediate from Proposition 3.7.1.

We note that Definition 6.0.1 is a generalization of the notion of a _special
unipotent_ representation, due to Adams, Barbasch, and Vogan ([ABV92]).
Suppose $G$ is algebraic, and let $\mathfrak{g}^{\vee}$ denote the Langlands
dual Lie algebra. There is an order-reversing map, called Barbasch-Vogan
duality, from nilpotent orbits in $(\mathfrak{g}^{\vee})^{*}$ to nilpotent
orbits in $\mathfrak{g}^{*}$, see [BV85, Appendix]. Denote this map by
$\mathsf{D}_{\mathfrak{g}}$. If
$\mathbb{O}^{\vee}\subset(\mathfrak{g}^{\vee})^{*}$ is a nilpotent orbit
corresponding to an $\mathfrak{sl}(2)$-triple $(e^{\vee},f^{\vee},h^{\vee})$,
then the element
$\frac{1}{2}h^{\vee}\in\mathfrak{h}^{\vee}\simeq\mathfrak{h}^{*}$ is well-
defined modulo the natural action of $W$, and therefore determines an
infinitesimal character for $U(\mathfrak{g})$. Write
$I_{\mathrm{max}}(\frac{1}{2}h^{\vee})\subset U(\mathfrak{g})$ for the
(unique) maximal primitive ideal with this infinitesimal character. By [BV85,
Prop A2], the associated variety of $I_{\mathrm{max}}(\frac{1}{2}h^{\vee})$ is
$\overline{\mathsf{D}_{\mathfrak{g}}(\mathbb{O}^{\vee})}$.

According to [ABV92, Sec 27], the _weak Arthur packet_ associated to
$\mathbb{O}^{\vee}$ is the finite set of irreducible
$(\mathfrak{g},K)$-modules

$\mathrm{Unip}_{\mathbb{O}^{\vee}}^{\mathrm{ABV}}(G):=\\{X\in\mathrm{Irr}(\mathfrak{g},K)\mid\operatorname{Ann}(X)=I_{\mathrm{max}}(\frac{1}{2}h^{\vee})\\}.$

The elements of $\mathrm{Unip}_{\mathbb{O}^{\vee}}^{\mathrm{ABV}}(G)$ are
called _special unipotent representations_.

The following proposition explains the relationship between
$\mathrm{Unip}_{\mathbb{O}^{\vee}}^{\mathrm{ABV}}(G)$ and
$\mathrm{Unip}_{\mathbb{O}}(G)$.

######  Proposition 6.0.3.

Suppose $\mathbb{O}\subset\mathfrak{g}^{*}$ is a special nilpotent orbit which
satisfies (6.0.1). Then the following are true:

  * (i)

There is a unique nilpotent orbit
$\mathbb{O}^{\vee}\subset(\mathfrak{g}^{\vee})^{*}$ such that
$\mathsf{D}_{\mathfrak{g}}(\mathbb{O}^{\vee})=\mathbb{O}$.

  * (ii)

$\mathbb{O}^{\vee}=\mathsf{D}_{\mathfrak{g}^{\vee}}(\mathbb{O})$.

  * (iii)

$I_{0}(\mathbb{O})=I_{\mathrm{max}}(\frac{1}{2}h^{\vee})$.

In particular

$\mathrm{Unip}_{\mathbb{O}}(G)=\mathrm{Unip}_{\mathbb{O}^{\vee}}^{\mathrm{ABV}}(G)$

###### Proof.

Suppose $\mathsf{D}_{\mathfrak{g}}\mathbb{O}^{\vee}=\mathbb{O}$. By [LMM21,
Prop 9.2.1], there is a cover $\widetilde{\mathbb{O}}$ of $\mathbb{O}$ such
that

$I_{\mathrm{max}}(\frac{1}{2}h^{\vee})=I_{0}(\widetilde{\mathbb{O}})$

Since $\mathbb{O}$ satisfies (6.0.1), $[\mathbb{O}]=[\widetilde{\mathbb{O}}]$,
see the discussion preceding Theorem 4.0.2. Hence by Theorem 4.0.2
$I_{0}(\mathbb{O})=I_{0}(\widetilde{\mathbb{O}})$. This proves (iii). By the
Dynkin classification of nilpotent orbits, the map
$\mathbb{O}^{\vee}\mapsto\frac{1}{2}h^{\vee}$ is injective. So
$\mathbb{O}^{\vee}$ is uniquely determined by $I_{0}(\mathbb{O})$. This proves
(i). For (ii), we note that
$\mathsf{D}_{\mathfrak{g}}\circ\mathsf{D}_{\mathfrak{g}^{\vee}}$ restricts to
the identity map on special nilpotent orbits, see (d) of [BV85, Prop A2]. This
completes the proof.

∎

###  6.1. Unitarity

######  Theorem 6.1.1.

Suppose $G$ is a real form of a simple exceptional group and
$\mathbb{O}\subset\mathfrak{g}^{*}$ is a rigid nilpotent orbit. Then all
unipotent representations attached to $\mathbb{O}$ (cf. Definition 6.0.1) are
unitary.

###### Proof.

By Proposition 6.0.3, if $\mathbb{O}$ is special, then all unipotent
representations attached to $\mathbb{O}$ are special unipotent. The unitarity
of such representations was recently established in [AMLV]. So we restrict our
attention to non-special rigid orbits. For any orbit $\mathbb{O}$,
$\mathrm{Unip}_{\mathbb{O}}(G)$ is the set of irreducible
$(\mathfrak{g},K)$-modules $X$ such that

  1. (1)

The infinitesimal character of $X$ is $\gamma_{0}(\mathbb{O})$.

  2. (2)

The Gelfand-Kirillov dimension of $X$ is $\frac{1}{2}\dim(\mathbb{O})$.

Recall that $\gamma_{0}(\mathbb{O})$ was computed, for all rigid $\mathbb{O}$,
in Section 4.3.1. The atlas command ‘all_parameters_gamma’ lists the Langlands
parameters of all irreducible $(\mathfrak{g},K)$-modules of a given
infinitesimal character, and the command ‘GK_dim’ computes the Gelfand-
Kirillov dimension of the representation corresponding to a given parameter.
Applying these commands in conjunction for all real forms of simple
exceptional groups, we determine that there are a total of 12 unipotent
representations attached to rigid nilpotent orbits. They are listed in Tables
14-18. Once the representations have been located, the command ‘is_unitary’
can be used to check unitarity. ∎

$\mathbb{O}$ | $\gamma_{0}(\mathbb{O})$ | $G$ | $\\#\mathrm{Unip}_{\mathbb{O}}(G)$
---|---|---|---
$A_{1}$ | $\frac{1}{3}(3,1)$ |  $G_{2}$ (split) | 1
$\widetilde{A}_{1}$ | $\frac{1}{2}(1,1)$ |  $G_{2}$ (split) | 0
Table 14. Non-special unipotents: type $G_{2}$ $\mathbb{O}$ | $\gamma_{0}(\mathbb{O})$ | $G$ | $\\#\mathrm{Unip}_{\mathbb{O}}(G)$
---|---|---|---
$A_{1}$ | $\frac{1}{2}(1,1,2,2)$ |  $F_{4}$ ($B_{4}$) | 0
$A_{1}$ | $\frac{1}{2}(1,1,2,2)$ |  $F_{4}$ (split) | 0
$A_{2}+\widetilde{A}_{1}$ | $\frac{1}{4}(1,1,2,2)$ |  $F_{4}$ ($B_{4}$) | 0
$A_{2}+\widetilde{A}_{1}$ | $\frac{1}{4}(1,1,2,2)$ |  $F_{4}$ (split) | 0
$\widetilde{A}_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1)$ |  $F_{4}$ ($B_{4}$) | 0
$\widetilde{A}_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1)$ |  $F_{4}$ (split) | 1
Table 15. Non-special unipotents: type $F_{4}$ $\mathbb{O}$ | $\gamma_{0}(\mathbb{O})$ | $G$ | $\\#\mathrm{Unip}_{\mathbb{O}}(G)$
---|---|---|---
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ (Hermitian) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ (quasi-split) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ ($F_{4}$) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ (split) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ (Hermitian) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ (quasi-split) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ ($F_{4}$) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ (split) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ (Hermitian) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ (quasi-split) | 3
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ ($F_{4}$) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{sc}}$ (split) | 1
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ (Hermitian) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ (quasi-split) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ ($F_{4}$) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1)$ |  $E_{6}^{\mathrm{ad}}$ (split) | 0
Table 16. Non-special unipotents: type $E_{6}$ $\mathbb{O}$ | $\gamma_{0}(\mathbb{O})$ | $G$ | $\\#\mathrm{Unip}_{\mathbb{O}}(G)$
---|---|---|---
$(3A_{1})^{\prime}$ | $\frac{1}{2}(1,1,1,1,1,1,2)$ |  $E_{7}^{\mathrm{sc}}$ (Hermitian) | 0
$(3A_{1})^{\prime}$ | $\frac{1}{2}(1,1,1,1,1,1,2)$ |  $E_{7}^{\mathrm{sc}}$ (quaternionic) | 0
$(3A_{1})^{\prime}$ | $\frac{1}{2}(1,1,1,1,1,1,2)$ |  $E_{7}^{\mathrm{sc}}$ (split) | 0
$(3A_{1})^{\prime}$ | $\frac{1}{2}(1,1,1,1,1,1,2)$ |  $E_{7}^{\mathrm{ad}}$ (Hermitian) | 0
$(3A_{1})^{\prime}$ | $\frac{1}{2}(1,1,1,1,1,1,2)$ |  $E_{7}^{\mathrm{ad}}$ (quaternionic) | 0
$(3A_{1})^{\prime}$ | $\frac{1}{2}(1,1,1,1,1,1,2)$ |  $E_{7}^{\mathrm{ad}}$ (split) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (Hermitian) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (quaternionic) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (split) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (Hermitian) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (quaternionic) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (split) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (Hermitian) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (quaternionic) | 1
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (split) | 1
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (Hermitian) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (quaternionic) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (split) | 0
$(A_{3}+A_{1})^{\prime}$ | $\frac{1}{2}(1,1,0,1,0,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (Hermitian) | 0
$(A_{3}+A_{1})^{\prime}$ | $\frac{1}{2}(1,1,0,1,0,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (quaternionic) | 0
$(A_{3}+A_{1})^{\prime}$ | $\frac{1}{2}(1,1,0,1,0,1,1)$ |  $E_{7}^{\mathrm{sc}}$ (split) | 0
$(A_{3}+A_{1})^{\prime}$ | $\frac{1}{2}(1,1,0,1,0,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (Hermitian) | 0
$(A_{3}+A_{1})^{\prime}$ | $\frac{1}{2}(1,1,0,1,0,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (quaternionic) | 0
$(A_{3}+A_{1})^{\prime}$ | $\frac{1}{2}(1,1,0,1,0,1,1)$ |  $E_{7}^{\mathrm{ad}}$ (split) | 0
Table 17. Non-special unipotents: type $E_{7}$ $\mathbb{O}$ | $\gamma_{0}(\mathbb{O})$ | $G$ | $\\#\mathrm{Unip}_{\mathbb{O}}(G)$
---|---|---|---
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,2,2)$ |  $E_{8}$ (quaternionic) | 0
$3A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,2,2)$ |  $E_{8}$ (split) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$4A_{1}$ | $\frac{1}{2}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (split) | 0
$A_{2}+3A_{1}$ | $\frac{1}{2}(1,1,1,0,1,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$A_{2}+3A_{1}$ | $\frac{1}{2}(1,1,1,0,1,1,1,1)$ |  $E_{8}$ (split) | 0
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1,3)$ |  $E_{8}$ (quaternionic) | 1
$2A_{2}+A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1,3)$ |  $E_{8}$ (split) | 1
$A_{3}+A_{1}$ | $\frac{1}{2}(1,1,0,1,0,1,1,2)$ |  $E_{8}$ (quaternionic) | 0
$A_{3}+A_{1}$ | $\frac{1}{2}(1,1,0,1,0,1,1,2)$ |  $E_{8}$ (split) | 0
$2A_{2}+2A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$2A_{2}+2A_{1}$ | $\frac{1}{3}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (split) | 1
$A_{3}+2A_{1}$ | $\frac{1}{2}(1,1,1,0,1,0,1,1)$ |  $E_{8}$ (quaternionic) | 0
$A_{3}+2A_{1}$ | $\frac{1}{2}(1,1,1,0,1,0,1,1)$ |  $E_{8}$ (split) | 0
$A_{3}+A_{2}+A_{1}$ | $\frac{1}{2}(1,0,0,1,0,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$A_{3}+A_{2}+A_{1}$ | $\frac{1}{2}(1,0,0,1,0,1,1,1)$ |  $E_{8}$ (split) | 0
$2A_{3}$ | $\frac{1}{4}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$2A_{3}$ | $\frac{1}{4}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (split) | 0
$A_{4}+A_{3}$ | $\frac{1}{5}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$A_{4}+A_{3}$ | $\frac{1}{5}(1,1,1,1,1,1,1,1)$ |  $E_{8}$ (split) | 1
$A_{5}+A_{1}$ | $\frac{1}{6}(2,2,1,1,1,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$A_{5}+A_{1}$ | $\frac{1}{6}(2,2,1,1,1,1,1,1)$ |  $E_{8}$ (split) | 0
$D_{5}(a_{1})$ | $\frac{1}{4}(1,1,1,0,1,1,1,1)$ |  $E_{8}$ (quaternionic) | 0
$D_{5}(a_{1})$ | $\frac{1}{4}(1,1,1,0,1,1,1,1)$ |  $E_{8}$ (split) | 0
Table 18. Non-special unipotents: type $E_{8}$

Below, we list the atlas parameters of the 12 unipotent representations
enumerated in the tables above.

  * •

$\mathbb{O}=A_{1}$, $\gamma_{0}(\mathbb{O})=\frac{1}{3}(3,1)$,
$G=G_{2}(\mathrm{split})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{(x=9,\lambda=(1,2),\nu=\frac{1}{3}(3,1))\\}.$

  * •

$\mathbb{O}=\widetilde{A}_{2}+A_{1}$,
$\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1)$, $G=F_{4}(\mathrm{split})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{(x=228,\lambda=(2,2,1,2),\nu=\frac{1}{3}(1,1,1,1))\\}$

  * •

$\mathbb{O}=2A_{2}+A_{1}$, $\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1,1,1)$,
$G=E_{6}(\mathrm{quasisplit})$.

$\displaystyle\mathrm{Unip}_{\mathbb{O}}(G)=\\{$
$\displaystyle((x=1790,\lambda=(1,2,1,1,1,1),\nu=\frac{1}{3}(1,1,1,1,1,1)),$
$\displaystyle(x=1778,\lambda=(1,2,0,2,1,0),\nu=\frac{1}{6}(2,2,-1,5,2,-1)),$
$\displaystyle(x=1777,\lambda=(0,2,1,2,0,1),\nu=\frac{1}{6}(-1,2,2,5,-1,2))\\}$

  * •

$\mathbb{O}=2A_{2}+A_{1}$, $\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1,1,1)$,
$G=E_{6}(\mathrm{split})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{(x=981,\lambda=(2,2,2,1,2,2),\nu=\frac{1}{3}(1,1,1,1,1,1))\\}$

  * •

$\mathbb{O}=2A_{2}+A_{1}$,
$\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1,1,1,1)$,
$G=E_{7}^{\mathrm{sc}}(\mathrm{quaternionic})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{(x=8920,\lambda=(2,1,3,1,-1,1,1),\nu=\frac{1}{3}(1,1,4,1,-2,1,1))\\}.$

  * •

$\mathbb{O}=2A_{2}+A_{1}$,
$\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1,1,1,1)$,
$G=E_{7}^{\mathrm{sc}}(\mathrm{split})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{((x=20925,\lambda=(2,2,2,1,2,2,1),\nu=\frac{1}{3}(1,1,1,1,1,1,1)))\\}.$

  * •

$\mathbb{O}=2A_{2}+A_{1}$,
$\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1,1,1,1,3)$,
$G=E_{8}(\mathrm{quaternionic})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{((x=66576,\lambda=(1,1,-4,4,3,2,-4,1),\nu=(1,1,-8,7,4,1,-5,0)/3))\\}.$

  * •

$\mathbb{O}=2A_{2}+A_{1}$,
$\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1,1,1,1,3)$,
$G=E_{8}(\mathrm{split})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{(x=320205,\lambda=(2,2,2,1,2,2,1,1),\nu=(1,1,1,1,1,1,1,3)/3))\\}.$

  * •

$\mathbb{O}=2A_{2}+2A_{1}$,
$\gamma_{0}(\mathbb{O})=\frac{1}{3}(1,1,1,1,1,1,1,1)$,
$G=E_{8}(\mathrm{split})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{(x=320205,\lambda=(2,2,2,1,2,2,1,2),\nu=(1,1,1,1,1,1,1,1)/3))\\}$

  * •

$\mathbb{O}=A_{4}+A_{4}$,
$\gamma_{0}(\mathbb{O})=\frac{1}{5}(1,1,1,1,1,1,1,1)$,
$G=E_{8}(\mathrm{split})$.

$\mathrm{Unip}_{\mathbb{O}}(G)=\\{((x=320205,\lambda=(2,2,2,2,1,2,2,2),\nu=(1,1,1,1,1,1,1,1)/5))\\}$

We conclude by remarking on two general patterns, which deserve further
consideration. First, none of the representations listed above are spherical.
This (perhaps) suggests the following general conjecture: if $G$ is a split
real group and $X$ is an irreducible unitary spherical representation, then
the associated variety of $X$ is the closure of a special nilpotent orbit. We
do not know of a counterexample (in classical or exceptional types). Second,
we note that if $\gamma_{0}(\mathbb{O})$ has an even integer in its
denominator, then $\mathrm{Unip}_{\mathbb{O}}(G)$ is empty. In these cases,
there should be interesting unitary representations of the appropriate two-
fold covers.

##  Appendix A Maximality computations

In Sections A.1 and A.2 below, we will prove the maximality of all unipotent
ideals for spin and exceptional groups. We will now summarize our approach.

Suppose $G$ is a complex reductive algebraic group with Lie algebra
$\mathfrak{g}$. Form the Langlands dual group $G^{\vee}$. If we fix a Cartan
subalgebra $\mathfrak{h}\subset\mathfrak{g}$, then $\mathfrak{g}^{\vee}$
contains a Cartan subalgebra $\mathfrak{h}^{\vee}\subset\mathfrak{g}^{\vee}$,
which is canonically identified with $\mathfrak{h}^{*}$, and the roots
$\Delta(\mathfrak{g}^{\vee},\mathfrak{h}^{\vee})$ for $\mathfrak{g}^{\vee}$
coincide with the co-roots $\Delta^{\vee}(\mathfrak{g},\mathfrak{h})$ for
$\mathfrak{g}$. Fix an element
$\gamma\in\mathfrak{h}^{*}\simeq\mathfrak{h}^{\vee}$. Consider the subsystems
of $\Delta^{\vee}$ consisting of _integral_ and _singular_ co-roots

$\Delta^{\vee}_{\gamma}:=\\{\alpha^{\vee}\in\Delta^{\vee}(\mathfrak{g},\mathfrak{h}):\langle\gamma,\alpha^{\vee}\rangle\in\mathbb{Z}\\},\qquad\Delta^{\vee}_{\gamma,0}:=\\{\alpha^{\vee}\in\Delta^{\vee}(\mathfrak{g},\mathfrak{h}):\langle\gamma,\alpha^{\vee}\rangle=0\\}\subset\Delta^{\vee}_{\gamma}$

These root systems define reductive subalgebras $\mathfrak{l}^{\vee}_{\gamma}$
and $\mathfrak{l}^{\vee}_{\gamma,0}$ of $\mathfrak{g}^{\vee}$. Using the
bijection
$\Delta^{\vee}(\mathfrak{g},\mathfrak{h})\simeq\Delta(\mathfrak{g},\mathfrak{h})$
between roots and co-roots, we can produce from $\Delta^{\vee}_{\gamma}$ and
$\Delta^{\vee}_{\gamma,0}$ two subsystems of
$\Delta(\mathfrak{g},\mathfrak{h})$

$\Delta_{\gamma}:=\left(\Delta^{\vee}_{\gamma}\right)^{\vee}\subset\Delta,\qquad\Delta_{\gamma,0}:=\left(\Delta^{\vee}_{\gamma,0}\right)^{\vee}\subset\Delta_{\gamma}.$

These root systems define reductive Lie algebras, denoted by
$\mathfrak{l}_{\gamma}$ and $\mathfrak{l}_{\gamma,0}$. Finally, consider the
nilpotent orbits

$\mathbb{O}^{\vee}_{\gamma}:=\operatorname{Ind}^{L_{\gamma}^{\vee}}_{L_{\gamma,0}^{\vee}}\\{0\\}\subset(\mathfrak{l}_{\gamma}^{\vee})^{*},\qquad\mathbb{O}_{\gamma}:=\mathsf{D}(\mathbb{O}^{\vee}_{\gamma})\subset\mathfrak{l}_{\gamma}^{*},$

where $\mathsf{D}$ is Barbasch-Vogan duality (see [BV85, Appendix]). The
criterion below is a standard consequence of the Barbasch-Vogan algorithm for
the associated variety of a maximal ideal, see [LMM21, Prop 3.3.1] for a
formal proof.

######  Proposition A.0.1.

Suppose $I\subset U(\mathfrak{g})$ is a primitive ideal with infinitesimal
character $\gamma\in\mathfrak{h}^{*}/W$ and associated variety
$\overline{\mathbb{O}}$. Then $I$ is a maximal ideal if and only if

$\operatorname{codim}(\mathbb{O},\mathcal{N})=\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}}).$

###  A.1. Spin groups

Let $\mathfrak{g}=\mathfrak{g}(n)$ be a simple rank-$n$ Lie algebra of type
$B$ or $D$. We start with a technical lemma.

######  Lemma A.1.1.

Let $q=(q_{1},...,q_{m})$ be a partition of $n$. Consider the infinitesimal
character defined in standard coordinates by the formula

$\gamma:=\rho^{+}(h^{\prime}(q\cup q))$

(cf. Definition (4.2.2)). Then

$\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}})=\sum_{i=1}^{m}\dim\mathcal{N}_{\mathrm{GL}(q_{i})}.$

###### Proof.

Let $q^{t}=:r=(r_{1},...,r_{q_{1}})$. Up to permutations

$\gamma=\frac{1}{4}((2q_{1}-1)^{r_{q_{1}}},(2q_{1}-3)^{r_{q_{1}-1}},...,1^{r_{1}})$

where we use superscripts to denote the multiplicities of repeated entries in
$\gamma$. It is easy to see that

$\mathfrak{l}^{\vee}_{\gamma}\simeq\mathfrak{gl}(n),\qquad\mathfrak{l}^{\vee}_{\gamma,0}\simeq\mathfrak{gl}(r_{1})\times\mathfrak{sl}(r_{2})\times...\times\mathfrak{sl}(r_{q_{1}})$

Thus

$\mathbb{O}_{\gamma}^{\vee}=\operatorname{Ind}^{L^{\vee}_{\gamma}}_{L^{\vee}_{\gamma,0}}\\{0\\}=\mathbb{O}_{q}$

where $\mathbb{O}_{q}$ denotes the nilpotent orbit in $\mathfrak{gl}(n)^{*}$
corresponding to $q$. Clearly $\mathfrak{l}_{\gamma}\simeq\mathfrak{gl}(n)$
and

$\mathbb{O}_{\gamma}=d(\mathbb{O}_{\gamma}^{\vee})=\mathbb{O}_{r}$

Note that $\mathbb{O}_{r}$ is induced from the $\\{0\\}$-orbit of the Levi
$\mathfrak{gl}(q_{1})\times\mathfrak{sl}(q_{2})\times...\times\mathfrak{sl}(q_{m})$.
So by (iv) of Proposition 3.3.1

$\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}})=\operatorname{codim}(\\{0\\},\mathcal{N}_{\mathrm{GL}(q_{1})\times...\times\mathrm{GL}(q_{m})})=\sum_{i=1}^{m}\dim\mathcal{N}_{\mathrm{GL}(q_{i})},$

as asserted.

∎

######  Proposition A.1.2.

Let $G$ be a simple rank-$n$ spin-group of type $B$ or $D$ and let
$\widetilde{\mathbb{O}}$ be a $G$-equivariant nilpotent cover. Then
$I_{0}(\widetilde{\mathbb{O}})$ is a maximal ideal.

###### Proof.

Choose a Levi subalgebra

$\mathfrak{m}=\prod_{i}\mathfrak{gl}(a_{i})\times\mathfrak{g}(m)\subset\mathfrak{g},$

and a birationally rigid $M$-equivariant nilpotent cover

$\\{0\\}\times...\times\\{0\\}\times\widetilde{\mathbb{O}}^{\prime}$

such that
$\widetilde{\mathbb{O}}=\mathrm{Bind}^{G}_{M}\\{0\\}\times...\times\\{0\\}\times\widetilde{\mathbb{O}}^{\prime}$.
Here, $m\leq n$ and $a$ is a partition of $n-m$. Write
$\gamma=\gamma_{0}(\widetilde{\mathbb{O}})$ and
$\gamma^{\prime}=\gamma_{0}(\widetilde{\mathbb{O}}^{\prime})$. By Proposition
4.0.3

(A.1.1) $\gamma=(\rho(a),\gamma^{\prime}).$

Let $p$ denote the partition corresponding to $\mathbb{O}^{\prime}$, the orbit
in $\mathfrak{g}(m)$ of which $\widetilde{\mathbb{O}}^{\prime}$ is a cover.
Define $S_{4}(p)$, $p\\#S_{4}(p)$, $x=x((p\\#S_{4}(p))^{t})$,
$y=y((p\\#S_{4}(p))^{t})$, and $z=z((p\\#S_{4}(p))^{t})$ as in Definition
4.2.2. Since $\widetilde{\mathbb{O}}^{\prime}$ is a birationally rigid
$\mathrm{Spin}(m)$-equivariant cover, we can construct a Levi subalgebra

$\mathfrak{l}=\prod_{k\in
S_{4}(p)}\mathfrak{gl}(k)\times\mathfrak{g}(m-|S_{4}(p)|)\subset\mathfrak{g}(m),$

such that

(A.1.2)
$\gamma^{\prime}=(\rho^{+}(h^{\prime}(z^{1/2})),\gamma_{0}(\widetilde{\mathbb{O}}_{p\\#S_{4}(p)})),$

where $\widetilde{\mathbb{O}}_{p\\#S_{4}(p)}$ is the universal
$\mathrm{SO}(m-|S_{4}(p)|)$-equivariant cover of $\mathbb{O}_{p\\#S_{4}(p)}$.
This follows from the proof of Proposition 4.2.6. Substituting (A.1.2) into
(A.1.1), we obtain the following formula for $\gamma$

(A.1.3)
$\gamma=(\rho(a),\rho^{+}(h^{\prime}(z^{1/2})),\gamma_{0}(\widetilde{\mathbb{O}}_{p\\#S_{4}(p)})).$

Form the levi subalgebra of $\mathfrak{g}(n-|S_{4}(p)|)$

$\mathfrak{k}=\prod_{i}\mathfrak{gl}(a_{i})\times\mathfrak{g}(m-|S_{4}(p)|)\subset\mathfrak{g}(n-|S_{4}(p)|),$

and consider the $\mathrm{SO}(n-|S_{4}(p)|)$-equivariant nilpotent cover

$\widetilde{\mathbb{O}}^{\prime\prime}=\mathrm{Bind}^{\mathrm{SO}(n-|S_{4}(p)|)}_{K}\\{0\\}\times...\times\\{0\\}\times\widetilde{\mathbb{O}}_{p\\#S_{4}(p)}.$

If we write
$\gamma^{\prime\prime}=\gamma_{0}(\widetilde{\mathbb{O}}^{\prime\prime})$,
then again by Proposition 4.0.3

(A.1.4)
$\gamma^{\prime\prime}=(\rho(a),\gamma_{0}(\widetilde{\mathbb{O}}_{p\\#S_{4}(p)})).$

Combining (A.1.3) and (A.1.4) (and permuting entries) we obtain the following
formula for $\gamma$

$\gamma=(\rho^{+}(h^{\prime}(z^{1/2})),\gamma^{\prime\prime}).$

By [LMM21, Thm 8.5.1], $I_{0}(\widetilde{\mathbb{O}}^{\prime\prime})$ is a
maximal ideal in $U(\mathfrak{g}(n-|S_{4}(p)|)$. So by Proposition A.0.1

$\operatorname{codim}(\mathbb{O}_{\gamma^{\prime\prime}},\mathcal{N}_{L_{\gamma^{\prime\prime}}})=\operatorname{codim}(\mathbb{O}^{\prime\prime},\mathcal{N}_{\mathrm{SO}(n-|S_{4}(p)|)})$

On the other hand, since $\mathbb{O}$ is induced from
$\\{0\\}\times...\times\\{0\\}\times\mathbb{O}^{\prime\prime}\subset\prod_{k\in
S_{4}(p)}\mathfrak{gl}(k)\times\mathfrak{g}(n-|S_{4}(p)|)$, we have by
Proposition 3.3.1(iv)

$\operatorname{codim}(\mathbb{O},\mathcal{N}_{G})=\operatorname{codim}(\mathbb{O}^{\prime\prime},\mathcal{N}_{\mathrm{SO}(n-|S_{4}(p)|)})+\sum_{k\in
S_{4}(p)}\dim\mathcal{N}_{\mathrm{GL}(k)}.$

In view of Proposition A.0.1, it suffices to show that

(A.1.5)
$\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}})=\operatorname{codim}(\mathbb{O}_{\gamma^{\prime\prime}},\mathcal{N}_{L_{\gamma^{\prime\prime}}})+\sum_{k\in
S_{4}(p)}\dim\mathcal{N}_{\mathrm{GL}(k)}.$

Note that $\gamma^{\prime\prime}$ is a tuple in $\frac{1}{2}\mathbb{Z}$, while
$\rho^{+}(h^{\prime}(z^{1/2}))$ is a tuple in
$\frac{1}{4}+\frac{1}{2}\mathbb{Z}$. In particular, the sum or difference of
an entry in $\gamma^{\prime\prime}$ with an entry in
$\rho^{+}(h^{\prime}(z^{1/2}))$ is never contained in $\mathbb{Z}$. So
$\mathfrak{l}^{\vee}_{\gamma}$ splits as a product

$\mathfrak{l}^{\vee}_{\gamma}=\mathfrak{l}^{\vee}_{\gamma^{\prime\prime}}\times\mathfrak{l}^{\vee}_{\rho^{+}(h^{\prime}(z^{1/2}))},$

where $\mathfrak{l}^{\vee}_{\rho^{+}(h^{\prime}(z^{1/2}))}$ is the subalgebra
in $\mathfrak{g}(|S_{4}(p)|)$ corresponding to the integral co-roots for
$\mathfrak{l}^{\vee}_{\rho^{+}(h^{\prime}(z^{1/2}))}$, regarded as an
infinitesimal character for $\mathfrak{g}(|S_{4}(p)|)$. As an immediate
consequence we obtain

$\mathfrak{l}_{\gamma}=\mathfrak{l}_{\gamma^{\prime\prime}}\times\mathfrak{l}_{\rho^{+}(h^{\prime}(z^{1/2}))},\qquad\mathbb{O}_{\gamma}=\mathbb{O}_{\gamma^{\prime\prime}}\times\mathbb{O}_{\rho^{+}(h^{\prime}(z^{1/2}))},$

Since $z^{1/2}=S_{4}(p)\cup S_{4}(p)$, Lemma A.1.1 (applied to $q=S_{4}(p)$)
implies

$\displaystyle\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}})$
$\displaystyle=\operatorname{codim}(\mathbb{O}_{\gamma^{\prime\prime}},\mathcal{N}_{L_{\gamma^{\prime\prime}}})+\operatorname{codim}(\mathbb{O}_{\rho^{+}(h^{\prime}(z^{1/2}))},\mathcal{N}_{L_{\rho^{+}(h^{\prime}(z^{1/2}))}})$
$\displaystyle=\operatorname{codim}(\mathbb{O}_{\gamma^{\prime\prime}},\mathcal{N}_{L_{\gamma^{\prime\prime}}})+\sum_{k\in
S_{4}(p)}\dim\mathcal{N}_{\mathrm{GL}(k)}.$

This proves (A.1.5) and thus completes the proof. ∎

###  A.2. Exceptional groups

######  Proposition A.2.1.

Let $G$ be a simple simply connected group of exceptional type and let
$\widetilde{\mathbb{O}}$ be a $G$-equivariant nilpotent cover. Then
$I_{0}(\widetilde{\mathbb{O}})$ is a maximal ideal.

The proof will come after a lemma.

######  Lemma A.2.2.

Let $\mathbb{O}$ be the nilpotent orbit $E_{7}(a_{4})$ in $E_{7}$ and let
$\widetilde{\mathbb{O}}$ be a 2-fold cover of $\mathbb{O}$. Then
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}])$ contains a
codimension 2 leaf.

###### Proof.

Assume the opposite. We note that
$\pi_{1}(\mathbb{O})\simeq\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, and
$\pi_{1}(\widetilde{\mathbb{O}})\simeq\mathbb{Z}_{2}$. There are three
codimension $2$ orbits in $\overline{\mathbb{O}}$, namely
$\mathbb{O}_{1}=A_{6}$, $\mathbb{O}_{2}=D_{5}+A_{1}$ and
$\mathbb{O}_{3}=D_{6}(a_{1})$. For each of these orbits the corresponding
singularity $\Sigma_{j}$ is of type $A_{1}$. Let $\widehat{\mathbb{O}}$ be the
universal cover of $\mathbb{O}$. The preimage of each $\Sigma_{i}$ in
$\widehat{X}$ is the union of two copies of $\mathbb{C}^{2}$. Let
$K\subset\pi_{1}(\mathbb{O})$ be such that
$\pi_{1}(\mathbb{O})\simeq\pi_{1}(\widetilde{\mathbb{O}})\oplus K$, and let
$\widecheck{\mathbb{O}}$ be the corresponding cover of $\mathbb{O}$. Since the
map $\pi_{1}(\mathbb{O})\stackrel{{\scriptstyle}}{{\to}}S_{2}$ permuting the
components of the preimage of $\Sigma_{i}$ has kernel
$\pi_{1}(\widetilde{\mathbb{O}})$, it follows that the preimage of
$\Sigma_{i}$ in $\widecheck{X}$ is the union of two copies of $\Sigma_{i}$ for
each $i=1,2,3$. Therefore, $\dim\mathfrak{P}^{\widecheck{X}}\geq 3$. Using
[DE09, Tables], we see that there is a single pair of Levi $L$ of corank $3$
and a nilpotent orbit $\mathbb{O}_{L}\subset\mathfrak{l}^{*}$, such that
$\mathbb{O}$ is induced from $\mathbb{O}_{L}$, namely
$(L(A_{2}+2A_{1};2,3,5,6),\\{0\\})$. Thus, there is at most one cover
birationally induced from a corank $3$ Levi. However, we have
$\dim\mathfrak{P}^{X}=3$, and $\dim\mathfrak{P}^{\widecheck{X}}\geq 3$, so we
get a contradiction. ∎

###### Proof of Proposition A.2.1.

Let $L\subset G$ be a Levi subgroup of $G$. The universal cover
$\widetilde{L}$ of $L$ is a product
$T\times\widetilde{L}_{1}\times...\times\widetilde{L}_{t}$, where $T$ is a
torus and $\widetilde{L}_{1},...,\widetilde{L}_{t}$ are simple simply
connected groups. For each $i$, choose a birationally rigid
$\widetilde{L}_{i}$-equivariant nilpotent cover $\widetilde{\mathbb{O}}_{i}$
and regard $\gamma_{0}(\widetilde{\mathbb{O}}_{i})$ as an infinitesimal
character for $\mathfrak{g}$ via the natural embedding
$\mathfrak{l}_{i}\subset\mathfrak{l}$. Consider the infinitesimal character

(A.2.1)
$\gamma=\sum_{i=1}^{t}\gamma_{0}(\widetilde{\mathbb{O}}_{i})\in\mathfrak{h}^{*}/W$

Any infinitesimal character which arises in this way will be called _pseudo-
unipotent_. By Proposition 4.0.3, every unipotent infinitesimal character is
pseudo-unipotent, but there are many others.

In the tables below, we list all pseudo-unipotent infinitesimal character for
simple exceptional groups. We will explain the procedure and give an example:

  * •

List all standard Levi subgroups $L\subset G$.

  * •

For each $L$, determine the simple factors $\mathfrak{l}_{i}$ of
$\mathfrak{l}$. This is evident from the Dynkin diagram.

  * •

For each simple factor $\mathfrak{l}_{i}$ compute the fundamental weights for
$\mathfrak{l}_{i}$ in terms of fundamental weights for $\mathfrak{g}$.

  * •

For each simple factor $\mathfrak{l}_{i}$ of $\mathfrak{l}$, list, in
fundamental weight coordinates, all infinitesimal characters
$\gamma_{0}(\widetilde{\mathbb{O}}_{i})$ attached to birationally rigid covers
for the simply connected group $\widetilde{L}_{i}$. If $\mathfrak{l}_{i}$ is
exceptional, we use the tables in Sections 4.3.1 and 4.3.2. If
$\mathfrak{l}_{i}$ is classical, we use [LMM21, Prop 8.2.8].

  * •

For each simple factor $\mathfrak{l}_{i}$ and
$\gamma_{0}(\widetilde{\mathbb{O}}_{i})$, write the infinitesimal character
$\gamma_{0}(\widetilde{\mathbb{O}}_{i})$ in terms of fundamental weights for
$G$.

  * •

Compute $\gamma=\sum_{i=1}^{t}\gamma_{0}(\widetilde{\mathbb{O}}_{i})$.

######  Example 1.

Let $G=E_{8}$ and choose $L=L(D_{4}+A_{2};2,3,4,5,7,8)$. Let
$\mathfrak{l}_{1}$ denote the $D_{4}$ factor and let $\mathfrak{l}_{2}$ denote
the $A_{2}$ factor. In terms of fundamental weights for $\mathfrak{g}$, the
fundamental weights for $\mathfrak{l}_{1}$ are

$\frac{1}{2}(-1,2,0,0,0,-1,0,0),\frac{1}{2}(-2,0,2,0,0,-1,0,0),(-1,0,0,1,0,-1,0,0),\frac{1}{2}(-1,0,0,0,2,-2,0,0),$

where the third weight corresponds to the central node. The fundamental
weights for $\mathfrak{l}_{2}$ are

$\frac{1}{3}(0,0,0,0,0,-2,3,0),\frac{1}{3}(0,0,0,0,0,-1,0,3).$

Choose $\widetilde{\mathbb{O}}_{1}$ to be the minimal nilpotent orbit in
$\mathfrak{l}_{1}$, corresponding to the partition $(2^{2},1^{4})$. By [LMM21,
Prop 8.2.8], $\gamma_{0}(\widetilde{\mathbb{O}}_{1})$ is the sum of the three
fundamental weights corresponding to the non-central nodes, i.e.

$\displaystyle\gamma_{0}(\widetilde{\mathbb{O}}_{1})$
$\displaystyle=\frac{1}{2}(-1,2,0,0,0,-1,0,0)+\frac{1}{2}(-2,0,2,0,0,-1,0,0)+\frac{1}{2}(-1,0,0,0,2,-2,0,0)$
$\displaystyle=(-2,1,1,0,1,-2,0,0)$

Choose $\widetilde{\mathbb{O}}_{2}$ to be the universal cover of the principal
nilpotent orbit in $\mathfrak{l}_{2}$. By [LMM21, Prop 8.2.8],
$\gamma_{0}(\widetilde{\mathbb{O}}_{2})$ is $\rho(\mathfrak{l}_{2})/3$, i.e.

$\gamma_{0}(\widetilde{\mathbb{O}}_{2})=\frac{1}{3}(\frac{1}{3}(0,0,0,0,0,-2,3,0)+\frac{1}{3}(0,0,0,0,0,-1,0,3))=\frac{1}{3}(0,0,0,0,0,-1,1,1).$

So

$\gamma=\gamma_{0}(\widetilde{\mathbb{O}}_{1})+\gamma_{0}(\widetilde{\mathbb{O}}_{2})=\frac{1}{3}(-6,3,3,0,3,-7,1,1).$

This is $W$-conjguate to the dominant weight

$\gamma=\frac{1}{3}(0,1,0,0,0,0,2,0).$

If $\gamma$ is a pseudo-unipotent infinitesimal character corresponding to the
data $(L\subset G,\widetilde{\mathbb{O}}_{1},...,\widetilde{\mathbb{O}}_{t})$,
define the integer

$n(\gamma):=\sum_{i=1}^{t}\operatorname{codim}(\mathbb{O}_{i},\mathcal{N}_{\mathfrak{l}_{i}})$

For each pseudo-unipotent infinitesimal character, we will record $n(\gamma)$
in the tables below. Suppose $\gamma$ is the infinitesimal character of a
unipotent ideal $I_{0}(\widetilde{\mathbb{O}})$. Then $\mathbb{O}$ is induced
from $\mathbb{O}_{1}\times...\times\mathbb{O}_{t}\subset\mathfrak{l}^{*}$ and

$n(\gamma)=\operatorname{codim}(\mathbb{O},\mathcal{N}_{G}).$

So if

(A.2.2)
$n(\gamma)=\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}}),$

then

$\operatorname{codim}(\mathbb{O},\mathcal{N}_{G})=\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}}),$

which implies that $I_{0}(\widetilde{\mathbb{O}})$ is maximal by Proposition
A.0.1. Thus, it suffices to show that (A.2.2) holds for every unipotent
$\gamma$.

In fact, we compute
$\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}})$ for every
_pseudo-unipotent_ $\gamma$ in the tables below. In types $G_{2}$, $F_{4}$,
$E_{6}$, and $E_{7}$, the atlas command
‘$\mathrm{GK\\_dim\\_maximal\\_ideal}(G,\gamma)$’ is helpful, but one can also
compute by hand, as illustrated below.

######  Example 2.

Consider the pseudo-unipotent infinitesimal character computed in the example
above

$\gamma=\frac{1}{3}(0,1,0,0,0,0,2,0).$

The following is a simple system for $\Delta^{\vee}_{\gamma}$

$\alpha_{1}^{\vee},\alpha_{3}^{\vee},\alpha_{4}^{\vee},\alpha_{5}^{\vee},\alpha_{6}^{\vee},\alpha_{8}^{\vee},\alpha_{2}^{\vee}+\alpha_{4}^{\vee}+\alpha_{5}^{\vee}+\alpha_{6}^{\vee}+\alpha_{7}^{\vee}.$

And below is a simple system for $\Delta^{\vee}_{0}$

$\alpha_{1}^{\vee},\alpha_{3}^{\vee},\alpha_{4}^{\vee},\alpha_{5}^{\vee},\alpha_{6}^{\vee},\alpha_{8}^{\vee}$

Thus, $\mathfrak{l}^{\vee}_{\gamma}$ is of type $E_{7}$ and
$\mathfrak{l}^{\vee}_{\gamma,0}$ is a Levi of type $A_{5}+A_{1}$ in
$\mathfrak{l}^{\vee}_{\gamma}$ (the embedding
$\mathfrak{l}_{\gamma,0}^{\vee}\subset\mathfrak{l}_{\gamma}^{\vee}$
matters—$\mathfrak{l}_{\gamma,0}^{\vee}$ can be regarded as the standard Levi
in $E_{7}$ obtained by deleting the node adjacent to the end point of the
short leg of the Dynkin diagram). Now $\mathbb{O}^{\vee}_{\gamma}$ is the
Richardson orbit in $\mathfrak{l}^{\vee}_{\gamma}$ induced from the
$\\{0\\}$-orbit in $\mathfrak{l}^{\vee}_{\gamma,0}$. Hence by [DE09, Sec 4],
$\mathbb{O}_{\gamma}^{\vee}=D_{4}(a_{1})$. By [Car93, Sec 13.4],
$\mathbb{O}_{\gamma}=\mathsf{D}(\mathbb{O}_{\gamma}^{\vee})=E_{7}(a_{5})\subset
E_{7}$. And by [CM93, Sec 8.4],
$\operatorname{codim}(E_{7}(a_{5}),\mathcal{N}_{E_{7}})=14$.

Comparing $\operatorname{codim}(\mathbb{O}_{\gamma},\mathcal{N}_{L_{\gamma}})$
and $n(\gamma)$ for every pseudo-unipotent $\gamma$, we find that (A.2.2)
holds in all but one case (highlighted in red in the tables below). The case
in question is

(A.2.3) $G=E_{8},\qquad\gamma=\frac{1}{4}(0,1,0,0,1,0,0,1)$

This infinitesimal character corresponds to the following choices:

$L=L(E_{7}),\qquad\widetilde{\mathbb{O}}_{1}=\text{universal cover of
}E_{7}(a_{4})\subset\mathcal{N}_{L}$

Indeed, in fundamental weight coordinates for $\mathfrak{l}$

$\gamma_{0}(\widetilde{\mathbb{O}}_{1})=\frac{1}{4}(1,2,0,0,1,0,1),$

see Table LABEL:table:covers:E7. Converting into fundamental weight
coordinates for $\mathfrak{g}$, we obtain the weight

(A.2.4)
$\gamma_{0}(\widetilde{\mathbb{O}}_{1})=\frac{1}{4}(1,2,0,0,1,0,1,-8).$

The dominant weight in (A.2.3) is obtained from (A.2.4) by conjugating by $W$.
Note that we have

$n(\gamma)=\operatorname{codim}(E_{7}(a_{4}),\mathcal{N}_{E_{7}})=10,$

whereas

$\operatorname{codim}(\mathbb{O}_{\gamma},L_{\gamma})=\operatorname{codim}(\mathbb{O}_{(4,3,1)}\times\\{0\\},\mathcal{N}_{\mathfrak{sl}(2)\times\mathfrak{sl}(8)})=12$

To show that $\gamma$ is _not_ unipotent, we must show that the orbit
$\mathbb{O}_{L}=E_{7}(a_{4})\subset\mathfrak{l}^{*}$ does not admit an
$L$-equivariant cover in the same equivalence class as the universal cover of
$\mathbb{O}_{L}$. An atlas computation shows that
$\pi_{1}^{L}(\mathbb{O}_{L})\simeq\mathbb{Z}_{2}$. Write
$\widetilde{\mathbb{O}}_{L}$ for the nontrivial $L$-equivariant cover of
$\mathbb{O}_{L}$. By Lemma A.2.2,
$\operatorname{Spec}(\mathbb{C}[\widetilde{\mathbb{O}}_{L}])$ contains a
codimension 2 leaf. On the other hand, by Corollary 3.9.4,
$\operatorname{Spec}(\mathbb{C}[\widehat{\mathbb{O}}_{L}])$ does not.
Consequently, $[\widetilde{\mathbb{O}}_{L}]\neq[\widehat{\mathbb{O}}_{L}]$, as
desired. This completes the proof. ∎

Table 19. Pseudo-unipotent infinitesimal characters: type $G_{2}$ central char | codim | central char | codim | central char | codim | central char | codim
---|---|---|---|---|---|---|---
$(0,0)$ | $0$ | $\frac{1}{2}(1,0)$ | $2$ | $\frac{1}{2}(0,1)$ | $2$ | $(1,0)$ | $2$
$\frac{1}{2}(1,1)$ | $4$ | $\frac{1}{3}(3,1)$ | $6$ | $(1,1)$ | $12$ |  |
Table 20. Pseudo-unipotent infinitesimal characters: type $F_{4}$ central char | codim | central char | codim | central char | codim | central char | codim
---|---|---|---|---|---|---|---
$(0,0,0,0)$ | $0$ | $\frac{1}{3}(1,0,0,0)$ | $0$ | $\frac{1}{3}(0,0,0,1)$ | $0$ | $\frac{1}{4}(1,0,0,0)$ | $0$
$\frac{1}{4}(0,0,1,0)$ | $0$ | $\frac{1}{4}(0,0,0,1)$ | $0$ | $\frac{1}{12}(1,0,3,0)$ | $0$ | $\frac{1}{12}(0,1,2,0)$ | $0$
$\frac{1}{2}(1,0,0,0)$ | $2$ | $\frac{1}{2}(0,0,0,1)$ | $2$ | $\frac{1}{4}(1,0,1,0)$ | $2$ | $\frac{1}{4}(0,1,0,0)$ | $2$
$\frac{1}{4}(1,0,0,2)$ | $2$ | $\frac{1}{6}(0,1,1,0)$ | $2$ | $\frac{1}{6}(1,0,2,0)$ | $2$ | $\frac{1}{2}(1,0,0,1)$ | $4$
$\frac{1}{2}(0,0,1,0)$ | $4$ | $(0,0,0,1)$ | $6$ | $(1,0,0,0)$ | $6$ | $\frac{1}{4}(3,0,1,0)$ | $6$
$\frac{1}{4}(1,0,2,0)$ | $6$ | $\frac{1}{4}(0,1,0,2)$ | $6$ | $(0,0,1,0)$ | $8$ | $\frac{1}{2}(1,0,0,2)$ | $8$
$\frac{1}{2}(0,1,0,0)$ | $8$ | $\frac{1}{2}(0,1,0,1)$ | $8$ | $\frac{1}{2}(1,0,1,0)$ | $8$ | $\frac{1}{2}(1,1,0,0)$ | $10$
$\frac{1}{2}(0,1,0,2)$ | $12$ | $\frac{1}{3}(1,1,1,1)$ | $12$ | $\frac{1}{4}(1,0,2,4)$ | $12$ | $\frac{1}{4}(1,1,2,2)$ | $14$
$\frac{1}{2}(2,1,0,1)$ | $18$ | $(0,0,1,1)$ | $18$ | $(1,0,1,0)$ | $20$ | $(1,0,1,1)$ | $26$
$\frac{1}{2}(1,1,2,2)$ | $32$ | $(1,1,1,1)$ | $48$ |  |  |  |
Table 21. Pseudo-unipotent infinitesimal characters: type $E_{6}$ central char | codim | central char | codim | central char | codim
---|---|---|---|---|---
$(0,0,0,0,0,0)$ | $0$ | $\frac{1}{3}(0,1,0,0,0,0)$ | $0$ | $\frac{1}{3}(1,0,0,0,0,1)$ | $0$
$\frac{1}{4}(0,1,0,0,0,0)$ | $0$ | $\frac{1}{4}(1,0,0,0,0,1)$ | $0$ | $\frac{1}{4}(0,0,0,1,0,0)$ | $0$
$\frac{1}{5}(1,1,0,0,0,1)$ | $0$ | $\frac{1}{8}(1,0,1,0,1,1)$ | $0$ | $\frac{1}{8}(1,2,0,0,0,1)$ | $0$
$\frac{1}{12}(1,0,0,3,0,1)$ | $0$ | $\frac{1}{12}(2,1,1,0,1,2)$ | $0$ | $\frac{1}{12}(3,1,0,0,0,3)$ | $0$
$\frac{1}{12}(0,0,1,2,1,0)$ | $0$ | $\frac{1}{20}(1,0,3,1,3,1)$ | $0$ | $\frac{1}{2}(0,1,0,0,0,0)$ | $2$
$\frac{1}{4}(0,0,1,0,1,0)$ | $2$ | $\frac{1}{4}(1,1,0,0,0,1)$ | $2$ | $\frac{1}{6}(2,1,0,0,0,2)$ | $2$
$\frac{1}{6}(0,0,1,1,1,0)$ | $2$ | $\frac{1}{8}(2,0,1,0,1,2)$ | $2$ | $\frac{1}{10}(2,0,1,1,1,2)$ | $2$
$\frac{1}{12}(1,0,2,1,2,1)$ | $2$ | $\frac{1}{2}(1,0,0,0,0,1)$ | $4$ | $\frac{1}{4}(1,2,0,0,0,1)$ | $4$
$\frac{1}{4}(1,0,0,1,0,1)$ | $4$ | $\frac{1}{4}(0,1,1,0,1,0)$ | $4$ | $\frac{1}{6}(1,0,0,2,0,1)$ | $4$
$(0,1,0,0,0,0)$ | $6$ | $\frac{1}{2}(0,0,0,1,0,0)$ | $6$ | $\frac{1}{3}(0,1,1,0,1,0)$ | $6$
$\frac{1}{3}(1,2,0,0,0,1)$ | $6$ | $\frac{1}{4}(1,3,0,0,0,1)$ | $6$ | $\frac{1}{4}(0,2,1,0,1,0)$ | $6$
$\frac{1}{4}(1,0,1,0,1,1)$ | $6$ | $\frac{1}{6}(2,1,1,0,1,2)$ | $6$ | $\frac{1}{12}(1,5,3,0,3,1)$ | $6$
$\frac{1}{2}(1,1,0,0,0,1)$ | $8$ | $\frac{1}{2}(0,1,0,1,0,0)$ | $8$ | $\frac{1}{4}(1,1,1,0,1,1)$ | $8$
$\frac{1}{6}(1,1,2,0,2,1)$ | $6$ | $\frac{1}{6}(1,2,1,1,1,1)$ | $6$ | $\frac{1}{2}(0,0,1,0,1,0)$ | $10$
$\frac{1}{4}(1,2,1,0,1,1)$ | $10$ | $(1,0,0,0,0,1)$ | $12$ | $\frac{1}{2}(1,2,0,0,0,1)$ | $12$
$\frac{1}{4}(1,2,0,2,0,1)$ | $12$ | $\frac{1}{4}(1,3,1,0,1,1)$ | $12$ | $\frac{1}{4}(3,0,0,1,0,3)$ | $12$
$\frac{1}{4}(2,1,1,0,1,2)$ | $12$ | $(0,0,0,1,0,0)$ | $14$ | $\frac{1}{2}(0,1,1,0,1,0)$ | $14$
$\frac{1}{2}(1,0,0,1,0,1)$ | $14$ | $\frac{1}{2}(1,1,0,1,0,1)$ | $16$ | $\frac{1}{3}(1,1,1,1,1,1)$ | $18$
$(1,1,0,0,0,1)$ | $20$ | $\frac{1}{4}(3,3,1,0,1,3)$ | $20$ | $\frac{1}{2}(1,1,1,0,1,1)$ | $22$
$(0,1,0,1,0,0)$ | $24$ | $\frac{1}{3}(1,3,1,1,1,1)$ | $24$ | $\frac{1}{4}(1,4,1,2,1,1)$ | $24$
$\frac{1}{2}(1,2,1,0,1,1)$ | $26$ | $(1,0,0,1,0,1)$ | $30$ | $\frac{1}{2}(2,1,1,0,1,2)$ | $30$
$\frac{1}{2}(1,1,1,1,1,1)$ | $32$ | $(1,1,0,1,0,1)$ | $40$ | $(1,1,1,0,1,1)$ | $50$
$(1,1,1,1,1,1)$ | $72$ |  |  |  |
Table 22. Pseudo-unipotent infinitesimal characters: type $E_{7}$ central char | codim | central char | codim | central char | codim
---|---|---|---|---|---
$(0,0,0,0,0,0,0)$ | $0$ | $\frac{1}{2}(0,0,0,0,0,0,1)$ | $0$ | $\frac{1}{3}(1,0,0,0,0,0,0)$ | $0$
$\frac{1}{3}(0,0,0,0,0,1,0)$ | $0$ | $\frac{1}{4}(1,0,0,0,0,0,0)$ | $0$ | $\frac{1}{4}(0,0,1,0,0,0,0)$ | $0$
$\frac{1}{4}(0,1,0,0,0,0,1)$ | $0$ | $\frac{1}{5}(1,0,0,0,0,1,0)$ | $0$ | $\frac{1}{6}(0,2,0,0,0,0,1)$ | $0$
$\frac{1}{6}(1,0,0,0,0,1,1)$ | $0$ | $\frac{1}{7}(0,0,0,1,0,1,0)$ | $0$ | $\frac{1}{8}(1,0,0,0,0,1,2)$ | $0$
$\frac{1}{8}(0,0,0,1,0,1,0)$ | $0$ | $\frac{1}{8}(0,0,1,0,1,0,1)$ | $0$ | $\frac{1}{8}(2,0,0,0,0,1,0)$ | $0$
$\frac{1}{12}(0,0,1,1,1,0,1)$ | $0$ | $\frac{1}{12}(0,0,2,1,0,0,0)$ | $0$ | $\frac{1}{12}(1,0,0,1,0,2,0)$ | $0$
$\frac{1}{12}(0,0,3,0,0,1,0)$ | $0$ | $\frac{1}{12}(1,0,0,0,0,3,0)$ | $0$ | $\frac{1}{15}(0,0,2,1,0,2,0)$ | $0$
$\frac{1}{20}(0,0,1,3,0,1,0)$ | $0$ | $\frac{1}{24}(0,2,3,0,3,0,1)$ | $0$ | $\frac{1}{24}(0,0,2,1,0,5,0)$ | $0$
$\frac{1}{2}(1,0,0,0,0,0,0)$ | $2$ | $\frac{1}{2}(0,1,0,0,0,0,0)$ | $2$ | $\frac{1}{4}(1,0,0,0,0,0,2)$ | $2$
$\frac{1}{4}(0,0,0,1,0,0,0)$ | $2$ | $\frac{1}{4}(1,0,0,0,0,1,0)$ | $2$ | $\frac{1}{4}(0,0,0,0,1,0,1)$ | $2$
$\frac{1}{6}(0,0,1,1,0,0,0)$ | $2$ | $\frac{1}{6}(1,0,0,0,0,2,0)$ | $2$ | $\frac{1}{6}(0,1,1,0,0,1,0)$ | $2$
$\frac{1}{8}(0,0,0,1,0,2,0)$ | $2$ | $\frac{1}{8}(1,0,0,0,0,2,2)$ | $2$ | $\frac{1}{8}(0,0,1,1,0,0,2)$ | $2$
$\frac{1}{8}(0,2,1,0,0,1,0)$ | $2$ | $\frac{1}{10}(0,0,1,1,0,2,0)$ | $2$ | $\frac{1}{12}(0,1,0,0,3,0,2)$ | $2$
$\frac{1}{12}(0,0,1,2,0,1,0)$ | $2$ | $\frac{1}{24}(0,0,5,1,2,0,4)$ | $2$ | $\frac{1}{2}(0,0,0,0,0,1,0)$ | $4$
$\frac{1}{2}(1,0,0,0,0,0,1)$ | $4$ | $\frac{1}{4}(2,0,0,0,0,1,0)$ | $4$ | $\frac{1}{4}(0,0,1,0,0,0,2)$ | $4$
$\frac{1}{4}(1,0,0,0,1,0,1)$ | $4$ | $\frac{1}{4}(0,1,0,0,1,0,0)$ | $4$ | $\frac{1}{4}(0,0,0,0,0,1,2)$ | $4$
$\frac{1}{4}(1,0,0,1,0,0,0)$ | $4$ | $\frac{1}{4}(0,0,1,0,0,1,0)$ | $4$ | $\frac{1}{6}(1,0,0,0,2,0,1)$ | $4$
$\frac{1}{6}(0,0,2,0,0,1,0)$ | $4$ | $\frac{1}{8}(0,2,3,0,2,0,0)$ | $4$ | $\frac{1}{12}(2,0,0,3,0,1,0)$ | $4$
$\frac{1}{12}(0,1,3,0,0,0,5)$ | $4$ | $(0,0,0,0,0,0,1)$ | $6$ | $(1,0,0,0,0,0,0)$ | $6$
$\frac{1}{2}(0,0,0,0,1,0,0)$ | $6$ | $\frac{1}{2}(1,1,0,0,0,0,0)$ | $6$ | $\frac{1}{2}(0,0,1,0,0,0,0)$ | $6$
$\frac{1}{3}(1,0,0,0,0,1,1)$ | $6$ | $\frac{1}{3}(0,1,0,0,0,0,2)$ | $6$ | $\frac{1}{3}(1,0,0,1,0,0,0)$ | $6$
$\frac{1}{3}(2,0,0,0,0,1,0)$ | $6$ | $\frac{1}{4}(2,0,0,1,0,0,0)$ | $6$ | $\frac{1}{4}(3,0,0,0,0,1,0)$ | $6$
$\frac{1}{4}(1,0,0,0,0,1,2)$ | $6$ | $\frac{1}{4}(0,0,0,1,0,1,0)$ | $6$ | $\frac{1}{4}(0,0,0,1,0,0,2)$ | $6$
$\frac{1}{4}(0,1,1,0,0,0,1)$ | $6$ | $\frac{1}{4}(0,1,0,0,0,0,3)$ | $6$ | $\frac{1}{4}(2,0,0,1,0,0,0)$ | $6$
$\frac{1}{5}(2,0,0,1,0,1,0)$ | $6$ | $\frac{1}{6}(1,0,0,1,0,2,0)$ | $6$ | $\frac{1}{8}(4,0,0,1,0,2,0)$ | $6$
$\frac{1}{8}(3,2,0,1,0,1,0)$ | $6$ | $\frac{1}{12}(0,0,1,2,1,0,5)$ | $6$ | $\frac{1}{12}(5,0,0,3,0,1,0)$ | $6$
$\frac{1}{12}(1,0,0,3,0,1,4)$ | $6$ | $\frac{1}{2}(1,0,0,0,0,1,0)$ | $8$ | $\frac{1}{2}(0,1,0,0,0,0,1)$ | $8$
$\frac{1}{2}(1,0,1,0,0,0,0)$ | $8$ | $\frac{1}{4}(2,1,1,0,0,0,1)$ | $8$ | $\frac{1}{4}(1,2,0,0,0,1,0)$ | $8$
$\frac{1}{4}(1,0,0,1,0,1,0)$ | $8$ | $\frac{1}{4}(0,0,1,0,1,0,1)$ | $8$ | $\frac{1}{6}(3,0,1,1,0,1,0)$ | $8$
$\frac{1}{6}(0,0,1,1,1,0,1)$ | $8$ | $\frac{1}{6}(1,0,0,2,0,1,0)$ | $8$ | $\frac{1}{8}(1,2,0,2,0,1,0)$ | $8$
$\frac{1}{2}(0,1,0,0,0,1,0)$ | $10$ | $\frac{1}{2}(1,1,0,0,0,0,1)$ | $10$ | $\frac{1}{2}(0,0,0,1,0,0,0)$ | $10$
$\frac{1}{2}(1,0,0,0,1,0,0)$ | $10$ | $\frac{1}{4}(2,0,0,1,0,1,0)$ | $10$ | $\frac{1}{4}(1,2,0,0,1,0,1)$ | $10$
$\frac{1}{4}(0,0,0,1,0,2,0)$ | $10$ | $\frac{1}{4}(0,2,0,1,0,0,0)$ | $10$ | $(0,0,0,0,0,1,0)$ | $12$
$(0,1,0,0,0,0,0)$ | $12$ | $\frac{1}{2}(2,0,0,0,0,1,0)$ | $12$ | $\frac{1}{2}(1,0,0,0,0,1,1)$ | $12$
$\frac{1}{4}(4,0,0,0,0,1,2)$ | $12$ | $\frac{1}{4}(2,0,2,0,0,1,0)$ | $12$ | $\frac{1}{4}(2,1,1,0,1,0,0)$ | $12$
$\frac{1}{4}(0,2,1,0,0,1,0)$ | $12$ | $\frac{1}{4}(1,0,0,1,1,0,1)$ | $12$ | $\frac{1}{4}(1,0,0,1,0,1,2)$ | $12$
$\frac{1}{4}(0,0,1,0,0,3,0)$ | $12$ | $\frac{1}{4}(3,0,0,1,0,1,0)$ | $12$ | $\frac{1}{4}(0,1,1,0,1,0,2)$ | $12$
$\frac{1}{4}(1,0,0,1,0,2,0)$ | $12$ | $\frac{1}{4}(3,0,0,1,0,0,2)$ | $12$ | $\frac{1}{6}(4,0,0,2,0,1,0)$ | $12$
$\frac{1}{12}(8,0,0,3,1,0,5)$ | $12$ | $(1,0,0,0,0,0,1)$ | $14$ | $(0,0,1,0,0,0,0)$ | $14$
$\frac{1}{2}(1,0,0,1,0,0,0)$ | $14$ | $\frac{1}{2}(0,0,1,0,0,1,0)$ | $14$ | $\frac{1}{2}(0,0,1,1,0,1,1)$ | $14$
$\frac{1}{2}(0,0,0,1,0,0,1)$ | $14$ | $\frac{1}{2}(0,0,1,0,1,0,0)$ | $14$ | $\frac{1}{3}(2,0,0,0,1,0,2)$ | $14$
$\frac{1}{4}(2,0,0,1,1,0,1)$ | $14$ | $\frac{1}{4}(2,2,0,0,1,0,1)$ | $14$ | $\frac{1}{4}(0,0,2,1,0,1,0)$ | $14$
$\frac{1}{4}(3,0,0,0,1,0,3)$ | $14$ | $\frac{1}{4}(0,1,3,0,0,0,1)$ | $14$ | $\frac{1}{2}(1,0,0,0,1,0,1)$ | $16$
$\frac{1}{2}(0,1,1,0,0,0,1)$ | $16$ | $\frac{1}{2}(1,0,1,0,0,1,0)$ | $16$ | $\frac{1}{4}(2,1,1,0,1,1,0)$ | $16$
$\frac{1}{2}(0,0,0,1,0,1,0)$ | $18$ | $\frac{1}{2}(1,0,0,1,0,0,1)$ | $18$ | $\frac{1}{2}(1,1,0,0,1,0,0)$ | $18$
$\frac{1}{3}(1,0,1,1,0,1,0)$ | $18$ | $\frac{1}{4}(2,1,1,0,1,0,2)$ | $18$ | $\frac{1}{4}(0,0,0,1,2,1,0)$ | $18$
$\frac{1}{6}(2,1,2,1,1,1,1)$ | $18$ | $(1,0,0,0,0,1,0)$ | $20$ | $(0,0,0,0,1,0,0)$ | $20$
$\frac{1}{3}(2,0,0,1,0,2,0)$ | $20$ | $\frac{1}{4}(3,0,0,1,0,3,0)$ | $20$ | $\frac{1}{2}(1,0,0,1,0,1,0)$ | $22$
$(1,0,1,0,0,0,0)$ | $24$ | $\frac{1}{2}(2,0,1,0,1,0,0)$ | $24$ | $\frac{1}{3}(3,0,1,1,0,1,0)$ | $24$
$\frac{1}{4}(1,1,0,2,1,1,3)$ | $24$ | $\frac{1}{4}(2,1,1,0,1,2,2)$ | $24$ | $\frac{1}{4}(4,0,2,1,0,1,0)$ | $24$
$(0,0,0,1,0,0,0)$ | $26$ | $\frac{1}{2}(2,1,1,0,0,0,1)$ | $26$ | $\frac{1}{2}(1,0,0,1,0,1,1)$ | $26$
$\frac{1}{2}(2,0,0,1,0,1,0)$ | $26$ | $\frac{1}{4}(4,1,1,0,2,0,1)$ | $26$ | $\frac{1}{4}(4,0,0,1,2,1,0)$ | $26$
$(1,0,0,0,1,0,0)$ | $28$ | $\frac{1}{2}(1,1,1,0,0,1,1)$ | $28$ | $(1,0,0,0,0,1,1)$ | $30$
$(0,0,1,0,0,1,0)$ | $30$ | $\frac{1}{2}(1,0,0,1,0,2,0)$ | $30$ | $\frac{1}{4}(3,0,0,1,0,3,4)$ | $30$
$(0,0,0,1,0,0,1)$ | $32$ | $\frac{1}{2}(1,0,1,1,0,1,0)$ | $32$ | $\frac{1}{2}(1,0,0,1,0,1,2)$ | $32$
$\frac{1}{2}(1,1,1,0,1,0,1)$ | $32$ | $\frac{1}{2}(0,1,1,0,1,0,2)$ | $32$ | $\frac{1}{2}(1,1,0,1,0,1,1)$ | $34$
$\frac{1}{3}(1,1,1,1,1,1,1)$ | $36$ | $(1,0,1,0,0,1,0)$ | $40$ | $\frac{1}{2}(2,0,1,1,0,1,1)$ | $40$
$\frac{1}{4}(4,1,3,0,1,3,0)$ | $40$ | $\frac{1}{4}(4,1,3,0,1,2,2)$ | $40$ | $(0,0,0,1,0,1,0)$ | $42$
$\frac{1}{2}(2,1,1,0,1,1,0)$ | $42$ | $\frac{1}{2}(1,1,1,0,1,1,1)$ | $42$ | $\frac{1}{2}(2,1,1,0,1,0,2)$ | $42$
$(1,0,0,1,0,0,1)$ | $44$ | $(1,0,0,1,0,1,0)$ | $50$ | $\frac{1}{2}(1,1,1,1,1,1,1)$ | $56$
$(1,0,0,1,0,1,1)$ | $60$ | $\frac{1}{2}(2,1,1,0,1,2,2)$ | $60$ | $\frac{1}{2}(1,1,1,1,1,1,2)$ | $62$
$(1,0,1,1,0,1,0)$ | $72$ | $\frac{1}{2}(2,1,2,1,1,1,1)$ | $72$ | $(1,1,1,0,1,0,1)$ | $74$
$(1,1,1,0,1,1,1)$ | $92$ | $(1,1,1,1,1,1,1)$ | $126$ |  |
Table 23. Pseudo-unipotent infinitesimal characters: type $E_{8}$ central char | codim | central char | codim | central char | codim
---|---|---|---|---|---
$(0,0,0,0,0,0,0,0)$ | $0$ | $\frac{1}{3}(0,0,0,0,0,0,0,1)$ | $0$ | $\frac{1}{3}(1,0,0,0,0,0,0,0)$ | $0$
$\frac{1}{4}(0,0,0,0,0,0,0,1)$ | $0$ | $\frac{1}{4}(1,0,0,0,0,0,0,0)$ | $0$ | $\frac{1}{4}(0,0,0,0,0,0,1,0)$ | $0$
$\frac{1}{4}(0,1,0,0,0,0,0,0)$ | $0$ | $\frac{1}{5}(1,0,0,0,0,0,0,1)$ | $0$ | $\frac{1}{7}(1,0,0,0,0,1,0,0)$ | $0$
$\frac{1}{8}(1,0,0,0,0,0,0,2)$ | $0$ | $\frac{1}{8}(0,0,1,0,0,0,1,0)$ | $0$ | $\frac{1}{8}(1,0,0,0,0,1,0,0)$ | $0$
$\frac{1}{8}(1,0,0,0,1,0,0,0)$ | $0$ | $\frac{1}{12}(2,0,0,0,0,1,0,1)$ | $0$ | $\frac{1}{12}(0,2,0,0,1,0,0,0)$ | $0$
$\frac{1}{12}(1,0,0,0,0,0,3,0)$ | $0$ | $\frac{1}{12}(3,0,0,0,0,0,0,1)$ | $0$ | $\frac{1}{12}(0,2,1,0,0,0,0,0)$ | $0$
$\frac{1}{12}(0,0,1,0,0,1,1,0)$ | $0$ | $\frac{1}{12}(0,0,0,0,0,1,2,0)$ | $0$ | $\frac{1}{15}(2,0,0,0,0,1,2,0)$ | $0$
$\frac{1}{16}(1,0,0,1,0,1,1,0)$ | $0$ | $\frac{1}{20}(0,1,3,0,0,0,1,0)$ | $0$ | $\frac{1}{20}(1,0,0,0,0,3,1,0)$ | $0$
$\frac{1}{24}(5,0,0,0,0,1,2,0)$ | $0$ | $\frac{1}{24}(0,0,1,0,2,0,3,0)$ | $0$ | $\frac{1}{28}(0,0,3,1,0,0,3,0)$ | $0$
$\frac{1}{40}(2,0,0,1,4,0,3,0)$ | $0$ | $\frac{1}{60}(0,3,4,0,5,0,3,0)$ | $0$ | $\frac{1}{2}(0,0,0,0,0,0,0,1)$ | $2$
$\frac{1}{4}(1,0,0,0,0,0,0,1)$ | $2$ | $\frac{1}{4}(0,0,1,0,0,0,0,0)$ | $2$ | $\frac{1}{4}(0,0,0,0,0,1,0,0)$ | $2$
$\frac{1}{6}(2,0,0,0,0,0,0,1)$ | $2$ | $\frac{1}{6}(0,0,0,0,0,1,1,0)$ | $2$ | $\frac{1}{8}(0,0,0,1,0,0,1,0)$ | $2$
$\frac{1}{8}(2,0,0,0,0,1,0,0)$ | $2$ | $\frac{1}{10}(2,0,0,0,0,1,1,0)$ | $2$ | $\frac{1}{12}(1,0,0,0,0,2,1,0)$ | $2$
$\frac{1}{12}(0,0,2,0,1,0,0,0)$ | $2$ | $\frac{1}{12}(1,0,0,1,0,0,2,0)$ | $2$ | $\frac{1}{12}(0,1,2,0,0,0,1,0)$ | $2$
$\frac{1}{14}(0,0,0,1,1,0,2,0)$ | $2$ | $\frac{1}{20}(0,0,1,1,2,0,1,0)$ | $2$ | $\frac{1}{24}(0,0,2,1,0,0,5,0)$ | $2$
$\frac{1}{30}(0,1,3,0,2,0,4,0)$ | $2$ | $\frac{1}{2}(1,0,0,0,0,0,0,0)$ | $4$ | $\frac{1}{4}(1,0,0,0,0,0,0,2)$ | $4$
$\frac{1}{4}(0,0,0,0,1,0,0,0)$ | $4$ | $\frac{1}{4}(0,0,0,0,0,1,0,1)$ | $4$ | $\frac{1}{4}(1,0,0,0,0,0,1,0)$ | $4$
$\frac{1}{4}(0,0,1,0,0,0,0,1)$ | $4$ | $\frac{1}{6}(1,0,0,0,0,0,2,0)$ | $4$ | $\frac{1}{6}(0,1,0,0,1,0,0,0)$ | $4$
$\frac{1}{8}(0,0,0,0,1,0,2,0)$ | $4$ | $\frac{1}{8}(0,0,0,1,0,1,0,1)$ | $4$ | $\frac{1}{10}(0,1,0,0,1,0,2,0)$ | $4$
$\frac{1}{12}(0,1,0,0,2,0,1,0)$ | $4$ | $\frac{1}{12}(1,0,0,0,0,3,0,2)$ | $4$ | $\frac{1}{12}(0,0,2,1,0,0,0,2)$ | $4$
$\frac{1}{20}(0,0,1,3,0,1,0,2)$ | $4$ | $(0,0,0,0,0,0,0,1)$ | $6$ | $\frac{1}{2}(0,0,0,0,0,0,1,0)$ | $6$
$\frac{1}{3}(1,0,0,0,0,0,0,2)$ | $6$ | $\frac{1}{3}(0,0,0,0,0,1,0,1)$ | $6$ | $\frac{1}{4}(1,0,0,0,0,0,0,3)$ | $6$
$\frac{1}{4}(1,0,0,0,0,1,0,0)$ | $6$ | $\frac{1}{4}(0,0,1,0,0,0,0,2)$ | $6$ | $\frac{1}{4}(0,0,0,1,0,0,0,0)$ | $6$
$\frac{1}{4}(0,0,0,0,0,1,0,2)$ | $6$ | $\frac{1}{4}(0,1,0,0,0,0,1,0)$ | $6$ | $\frac{1}{5}(1,0,0,0,0,1,0,2)$ | $6$
$\frac{1}{6}(0,2,0,0,0,0,1,0)$ | $6$ | $\frac{1}{6}(2,0,0,0,0,1,0,1)$ | $6$ | $\frac{1}{8}(2,0,0,0,0,1,0,4)$ | $6$
$\frac{1}{8}(0,0,0,1,0,1,0,3)$ | $6$ | $\frac{1}{12}(0,0,2,1,0,0,0,5)$ | $6$ | $\frac{1}{12}(1,0,0,0,0,3,0,5)$ | $6$
$\frac{1}{12}(0,0,3,0,0,1,0,4)$ | $6$ | $\frac{1}{12}(1,0,0,2,0,1,0,1)$ | $6$ | $\frac{1}{12}(0,0,2,1,0,0,0,5)$ | $6$
$\frac{1}{12}(0,0,1,2,0,0,1,0)$ | $6$ | $\frac{1}{20}(0,0,1,3,0,1,0,7)$ | $6$ | $\frac{1}{2}(1,0,0,0,0,0,0,1)$ | $8$
$\frac{1}{2}(0,1,0,0,0,0,0,0)$ | $8$ | $\frac{1}{2}(0,0,0,0,0,0,1,1)$ | $8$ | $\frac{1}{4}(1,0,0,0,0,1,0,1)$ | $8$
$\frac{1}{4}(0,0,0,1,0,0,0,1)$ | $8$ | $\frac{1}{4}(0,0,1,0,0,0,1,0)$ | $8$ | $\frac{1}{4}(0,1,0,0,0,0,1,2)$ | $8$
$\frac{1}{4}(1,0,0,0,1,0,0,0)$ | $8$ | $\frac{1}{6}(0,2,0,0,0,0,1,3)$ | $8$ | $\frac{1}{6}(0,0,1,0,0,1,1,0)$ | $8$
$\frac{1}{6}(1,0,0,0,0,1,1,3)$ | $8$ | $\frac{1}{6}(1,0,0,0,0,2,0,1)$ | $8$ | $\frac{1}{6}(0,0,1,1,0,0,0,1)$ | $8$
$\frac{1}{8}(0,0,0,1,0,2,0,1)$ | $8$ | $\frac{1}{8}(1,0,0,1,0,1,1,0)$ | $8$ | $\frac{1}{10}(0,0,1,1,0,2,0,1)$ | $8$
$\frac{1}{12}(0,0,1,2,0,1,0,2)$ | $8$ | $\frac{1}{12}(0,0,1,1,1,0,1,6)$ | $8$ | $\frac{1}{2}(0,1,0,0,0,0,0,1)$ | $10$
$\frac{1}{2}(0,0,0,0,0,1,0,0)$ | $10$ | $\frac{1}{4}(1,0,0,0,0,1,0,2)$ | $10$ | $\frac{1}{4}(0,0,1,0,0,1,0,0)$ | $10$
$\frac{1}{4}(0,1,0,0,1,0,0,1)$ | $10$ | $\frac{1}{4}(0,0,0,1,0,0,0,2)$ | $10$ | $\frac{1}{4}(2,0,0,0,0,1,0,0)$ | $10$
$\frac{1}{4}(1,0,0,1,0,0,0,0)$ | $10$ | $\frac{1}{6}(0,0,2,0,0,1,0,0)$ | $10$ | $\frac{1}{6}(0,1,1,0,0,1,0,2)$ | $10$
$\frac{1}{12}(0,0,1,2,0,1,0,5)$ | $10$ | $(1,0,0,0,0,0,0,0)$ | $12$ | $\frac{1}{2}(0,0,1,0,0,0,0,0)$ | $12$
$\frac{1}{2}(1,0,0,0,0,0,0,2)$ | $12$ | $\frac{1}{4}(0,0,0,1,0,0,1,0)$ | $12$ | $\frac{1}{4}(0,0,1,0,0,1,0,1)$ | $12$
$\frac{1}{4}(2,0,0,0,1,0,0,0)$ | $12$ | $\frac{1}{4}(1,0,0,0,0,1,0,3)$ | $12$ | $\frac{1}{4}(0,0,0,0,1,0,1,2)$ | $12$
$\frac{1}{4}(1,0,0,0,0,0,2,2)$ | $12$ | $\frac{1}{4}(0,0,0,1,0,0,0,3)$ | $12$ | $\frac{1}{4}(2,0,0,0,0,1,0,1)$ | $12$
$\frac{1}{4}(1,0,0,1,0,0,0,1)$ | $12$ | $\frac{1}{4}(3,0,0,0,0,0,1,0)$ | $12$ | $\frac{1}{4}(0,1,1,0,0,0,1,0)$ | $12$
$\frac{1}{6}(1,0,0,0,0,2,0,4)$ | $12$ | $\frac{1}{8}(0,0,0,1,0,2,0,5)$ | $12$ | $\frac{1}{10}(0,0,1,1,0,2,0,6)$ | $12$
$\frac{1}{12}(0,0,1,2,0,1,0,8)$ | $12$ | $\frac{1}{12}(0,1,0,0,3,0,2,6)$ | $12$ | $(0,0,0,0,0,0,1,0)$ | $14$
$\frac{1}{2}(0,0,0,0,0,1,0,1)$ | $14$ | $\frac{1}{2}(1,1,0,0,0,0,0,0)$ | $14$ | $\frac{1}{2}(1,0,0,0,0,0,1,0)$ | $14$
$\frac{1}{3}(0,1,0,0,0,0,2,0)$ | $14$ | $\frac{1}{4}(0,0,1,0,0,1,0,2)$ | $14$ | $\frac{1}{4}(1,0,0,0,1,0,1,0)$ | $14$
$\frac{1}{4}(0,1,0,0,0,0,3,0)$ | $14$ | $\frac{1}{4}(0,0,0,1,0,0,2,0)$ | $14$ | $\frac{1}{4}(0,0,0,1,0,1,0,0)$ | $14$
$\frac{1}{4}(1,0,0,0,0,1,2,0)$ | $14$ | $\frac{1}{6}(0,0,2,0,0,1,0,3)$ | $14$ | $\frac{1}{2}(0,0,1,0,0,0,0,1)$ | $16$
$\frac{1}{2}(0,1,0,0,0,0,1,0)$ | $16$ | $\frac{1}{2}(1,0,0,0,0,0,1,1)$ | $16$ | $\frac{1}{2}(0,0,0,0,1,0,0,0)$ | $16$
$\frac{1}{4}(0,0,0,1,0,1,0,1)$ | $16$ | $\frac{1}{4}(1,0,0,1,0,0,1,0)$ | $16$ | $\frac{1}{4}(1,0,0,0,1,0,1,2)$ | $16$
$\frac{1}{4}(0,0,0,1,0,1,0,1)$ | $16$ | $\frac{1}{6}(1,0,0,0,2,0,1,3)$ | $16$ | $\frac{1}{2}(1,0,0,0,0,1,0,0)$ | $18$
$\frac{1}{2}(0,0,0,0,1,0,0,1)$ | $18$ | $\frac{1}{3}(1,0,0,0,0,1,1,1)$ | $18$ | $\frac{1}{4}(1,0,0,1,0,1,0,0)$ | $18$
$\frac{1}{4}(1,2,0,0,0,1,0,0)$ | $18$ | $\frac{1}{4}(0,0,0,1,0,1,0,2)$ | $18$ | $\frac{1}{4}(0,1,1,0,0,0,1,2)$ | $18$
$\frac{1}{12}(1,0,0,3,0,1,4,4)$ | $18$ | $\frac{1}{12}(2,1,1,0,1,2,4,4)$ | $18$ | $(1,0,0,0,0,0,0,1)$ | $20$
$(0,1,0,0,0,0,0,0)$ | $20$ | $\frac{1}{2}(0,0,0,1,0,0,0,0)$ | $20$ | $\frac{1}{3}(2,0,0,0,0,1,0,2)$ | $20$
$\frac{1}{3}(1,1,0,0,0,1,0,1)$ | $20$ | $\frac{1}{4}(1,0,0,1,0,1,0,1)$ | $20$ | $\frac{1}{4}(2,0,0,1,0,0,0,3)$ | $20$
$\frac{1}{4}(3,0,0,0,0,1,0,3)$ | $20$ | $\frac{1}{6}(0,0,1,1,1,0,1,2)$ | $20$ | $\frac{1}{8}(4,0,0,1,0,2,0,5)$ | $20$
$\frac{1}{12}(5,0,0,3,0,1,0,8)$ | $20$ | $\frac{1}{2}(1,0,0,0,0,1,0,1)$ | $22$ | $\frac{1}{2}(0,1,0,0,0,1,0,0)$ | $22$
$\frac{1}{4}(1,0,0,1,0,1,0,2)$ | $22$ | $\frac{1}{6}(1,0,0,2,0,1,0,3)$ | $22$ | $(0,0,0,0,0,0,1,1)$ | $24$
$\frac{1}{2}(0,0,0,1,0,0,0,1)$ | $24$ | $\frac{1}{2}(1,0,0,0,1,0,0,0)$ | $24$ | $\frac{1}{3}(1,0,0,0,0,1,1,3)$ | $24$
$\frac{1}{3}(0,1,0,0,0,0,2,3)$ | $24$ | $\frac{1}{4}(2,1,1,0,0,0,1,2)$ | $24$ | $\frac{1}{4}(1,0,0,1,0,1,1,0)$ | $24$
$\frac{1}{4}(0,0,0,1,0,2,0,1)$ | $24$ | $\frac{1}{4}(0,1,0,0,0,0,3,4)$ | $24$ | $\frac{1}{4}(0,0,0,1,0,0,2,4)$ | $24$
$\frac{1}{4}(1,0,0,0,0,1,2,4)$ | $24$ | $\frac{1}{12}(0,0,1,2,1,0,5,12)$ | $24$ | $\frac{1}{12}(1,0,0,3,0,1,4,12)$ | $24$
$(0,0,0,0,0,1,0,0)$ | $26$ | $\frac{1}{2}(0,1,0,0,0,0,1,2)$ | $26$ | $\frac{1}{2}(1,0,0,0,0,1,0,2)$ | $26$
$\frac{1}{4}(0,0,1,0,1,0,1,4)$ | $26$ | $\frac{1}{4}(0,0,1,0,0,3,0,0)$ | $26$ | $\frac{1}{4}(2,0,0,1,0,1,0,2)$ | $26$
$\frac{1}{4}(1,0,0,1,0,1,0,4)$ | $26$ | $\frac{1}{4}(1,2,0,0,0,1,0,4)$ | $26$ | $\frac{1}{6}(1,0,0,2,0,1,0,6)$ | $26$
$\frac{1}{6}(0,0,1,1,1,0,1,6)$ | $26$ | $\frac{1}{2}(0,0,0,1,0,0,0,2)$ | $28$ | $\frac{1}{2}(0,0,1,0,0,1,0,0)$ | $28$
$\frac{1}{2}(1,1,0,0,0,0,1,1)$ | $28$ | $(1,0,0,0,0,0,1,0)$ | $30$ | $(0,1,0,0,0,0,0,1)$ | $30$
$\frac{1}{2}(2,0,0,0,0,1,0,1)$ | $30$ | $\frac{1}{4}(2,0,0,1,0,1,2,0)$ | $30$ | $\frac{1}{4}(0,2,1,0,0,1,0,3)$ | $30$
$\frac{1}{4}(3,0,0,1,0,1,0,2)$ | $30$ | $\frac{1}{4}(3,0,0,0,1,0,3,0)$ | $30$ | $\frac{1}{4}(1,0,0,1,1,0,1,2)$ | $30$
$(0,0,0,0,1,0,0,0)$ | $32$ | $\frac{1}{2}(1,0,0,0,1,0,1,0)$ | $32$ | $\frac{1}{2}(1,0,0,1,0,0,0,1)$ | $32$
$\frac{1}{2}(0,1,1,0,0,0,1,0)$ | $32$ | $\frac{1}{2}(0,0,1,0,0,1,0,1)$ | $32$ | $\frac{1}{2}(1,0,0,0,0,1,1,1)$ | $32$
$\frac{1}{2}(0,1,1,0,0,0,1,0)$ | $32$ | $\frac{1}{2}(0,0,0,1,0,1,0,0)$ | $32$ | $\frac{1}{2}(0,0,0,1,0,0,1,0)$ | $32$
$\frac{1}{4}(0,1,1,0,1,0,2,2)$ | $32$ | $\frac{1}{4}(1,0,0,1,0,1,2,2)$ | $32$ | $\frac{1}{2}(0,0,1,0,1,0,0,1)$ | $34$
$\frac{1}{2}(1,1,0,0,0,1,0,1)$ | $34$ | $\frac{1}{2}(0,0,0,1,0,0,1,1)$ | $34$ | $\frac{1}{2}(1,0,0,1,0,0,1,0)$ | $36$
$\frac{1}{3}(0,1,1,0,1,0,1,1)$ | $36$ | $\frac{1}{4}(2,1,1,0,1,1,0,1)$ | $36$ | $\frac{1}{6}(2,1,1,0,1,2,2,2)$ | $36$
$\frac{1}{4}(1,1,1,0,1,1,1,1)$ | $38$ | $\frac{1}{6}(2,2,1,1,1,1,1,1)$ | $38$ | $(1,0,0,0,0,0,1,1)$ | $40$
$\frac{1}{3}(2,0,0,0,1,0,2,3)$ | $40$ | $\frac{1}{4}(3,0,0,0,1,0,3,4)$ | $40$ | $\frac{1}{4}(2,1,1,0,0,0,3,4)$ | $40$
$\frac{1}{4}(2,0,0,1,0,1,2,4)$ | $40$ | $\frac{1}{5}(1,1,1,1,1,1,1,1)$ | $40$ | $(1,0,0,0,0,1,0,0)$ | $42$
$\frac{1}{2}(1,0,0,1,0,0,1,1)$ | $42$ | $\frac{1}{2}(1,0,0,0,1,0,1,2)$ | $42$ | $\frac{1}{2}(0,1,1,0,0,0,1,2)$ | $42$
$\frac{1}{2}(0,1,1,0,0,0,1,2)$ | $42$ | $\frac{1}{4}(2,1,1,0,1,0,2,2)$ | $42$ | $\frac{1}{4}(3,0,0,1,0,3,0,0)$ | $42$
$\frac{1}{2}(0,0,0,1,0,1,0,2)$ | $44$ | $\frac{1}{2}(1,0,0,1,0,1,0,0)$ | $44$ | $\frac{1}{2}(1,1,0,0,1,0,0,2)$ | $44$
$(0,0,0,0,1,0,0,1)$ | $46$ | $\frac{1}{2}(1,0,0,1,0,1,0,1)$ | $48$ | $(1,0,0,0,0,1,0,1)$ | $50$
$\frac{1}{4}(3,0,0,1,0,3,0,4)$ | $50$ | $\frac{1}{2}(1,0,0,1,0,1,0,2)$ | $52$ | $\frac{1}{4}(1,1,1,1,1,1,1,1)$ | $52$