* A list of all pairs $(L,\chi)$ consisting of a Levi subgroup $L \subset G$ and a character $\chi$ of $Z(L)$ such that there is a triple $(L,\OO,\mathcal{E}) \in \mathcal{C}$ and $Z(L)$ acts via (a multiple of) $\chi$ on $\mathcal{E}$. In most cases, $\chi$ is uniquely determined by $L$; in such cases we omit it.

* A description of the relative Weyl groups corresponding to the Levi subgroups in (3).

* A description of the matching (<ref>).

The references are: <cit.> for $SL(N)$, <cit.> for $SO(N)$, <cit.>and $Sp(2n)$, and <cit.> for $Spin(N)$.

For $G=Spin(N)$, we present (1)-(5) in a slightly different manner.
There are natural inclusions

$$\cN_{o,l}^{SO(N)}\to \cN_{o,l}^{Spin(N)}, \quad \overline{\mathfrak C}(SO(N))\to \overline{\mathfrak C}(Spin(N)).$$

With respect to these inclusions there are decompositions

$$\cN_{o,l}^{Spin(N)} =   \mathcal N_{o,l}^{SO(N)}\sqcup \mathcal N_{o,l}^*$$

where $\mathcal{N}_{o,l}^*$ is the set of pairs $(\OO,\mathcal{L}) \in \mathcal{N}_{o,l}^{Spin(n)}$ consisting of a nilpotent orbit $\OO \in \cN_o$ and a $Spin(N)$-equivariant local system $\mathcal{L}$ on $\OO$ which is not $SO(N)$-equivariant, and

$$\overline{\mathfrak C}(Spin(N)) = \overline{\mathfrak C}(SO(N)) \sqcup \overline{\mathfrak C}^*(Spin(N))$$

where $\overline{\mathfrak{C}}^*$ is the set of triples $(J,\mathcal C,\mathcal{E}) \in \overline{\mathfrak C}(Spin(N))$ where $\mathcal{E}$ is not equivariant for the associated Levi subgroup of $SO(N)$. The matching (<ref>) takes $\cN^{SO(N)}_{o,l}$ to $\overline{\mathfrak{C}}(SO(N))$ and $\cN_{o,l}^*$ to $\overline{\mathfrak C}^*(Spin(N))$. In (<ref>) and (<ref>), we describe the correspondence for $\cN_{o,l}^{SO(N)}$, and in (<ref>) and (<ref>), we describe the correspondence for $\cN_{o,l}^* $.

§.§.§ $SL(N)$

* $\cN_{o,l}$ is parameterized by pairs $(\lambda,\rho)$, where $\lambda = (\lambda_1,...,\lambda_k) \in \mathcal{P}(N)$ and $\rho$ is a character of $\ZZ_d$, where $d=\mathrm{gcd}(\lambda_1,...,\lambda_k)$.

* $p_1(\lambda,\rho)=\lambda$.

* $L$ is of type $(N/r)A_{r-1}$ for $r \mid N$, and $\chi$ is a character (trivial on connected center) of order $r$.

* $W_{L}=S_{N/r}$.

* Let $\lambda=(\lambda_1,...,\lambda_k)$, $d=\mathrm{gcd}(\lambda_1,...,\lambda_k)$, and let $\rho$ be a character of $\ZZ_d$ of degree $m$. Then $(\lambda,\rho)$ belongs to a block with to $r=m$. The specific $\chi$ it corresponds to depends on $\rho$, but is not important for our purposes. The corresponding $S_{N/r}$-representation is given by the partition $(\lambda_1/r,...,\lambda_k/r)$ of $N/r$.

§.§.§ $SO(N)$, $N$ odd

Let $n=\lfloor N/2 \rfloor$.

* $\mathcal{N}_{o,l}\leftrightarrow \mathcal Y^{2}_{odd, n}$.

* Let $S\in \mathcal Y^{2}_{odd,n}$.
Then $p_1(S) = \mathcal P(2(X-z)+1 \cup 2(Y-z))$ where $(X,Y) = S^{sp}$.

* $L$ is of type $B_{(r^2-1)/2}$ for $r\ge 0$ odd such that $r^2\le N$.

* $W_L = W_{\frac12(N-r^2)}$.

* Let $S\in \mathcal Y^{2}_{d,n}$ for $d\ge 0$ odd.
Then $S$ belongs to the block corresponding to $r=d$.
The corresponding bipartition is $\mathcal D(S)$.

§.§.§ $SO(N)$, $N$ even

Let $n=\lfloor N/2 \rfloor$.

* $\mathcal{N}_{o,l}\leftrightarrow \mathcal Y^{2}_{even, n}$.

* Let $S\in \mathcal Y^{2}_{even,n}$.
Then $p_1(S) = \mathcal P(2(X-z)+1 \cup 2(Y-z))$ where $(X,Y) = S^{sp}$.

* $L$ is of type $D_{r^2/2}$ for $r\ge 0$ even such that $r^2\le N$.

* $W_L = W_{\frac12(N-r^2)}$ if $r>0$ and $W_n'$ if $r=0$.

* Let $S\in \mathcal Y^{2}_{d,n}$ for $d\ge 0$ even.
Then $S$ belongs to the block corresponding to $r=d$.
The corresponding bipartition is $\mathcal D(S)$.
The label of $\mathcal D(S)$ when $S$ is degenerate is not important for our purposes.

§.§.§ $Sp(2n)$

* $\mathcal{N}_{o,l}\leftrightarrow \mathcal X^{1,1}_{odd, n}$.

* Let $S\in \mathcal X^{1,1}_{odd,n}$.
Then $p_1(S) = \mathcal P(2(X-z) \cup 2(Y-z)+1)$ where $(X,Y) = S^{sp}$.

* $L$ is of type $C_{\frac 12r(r+1)}$ for $r\ge 0$ such that $\frac 12r(r+1)\le n$.

* $W_L = W_{n-\frac 12r(r+1)}$.

* Let $S\in \mathcal X^{1,1}_{d,n}$ for $d$ odd.
Then $S$ belongs to the block corresponding to $r=d-1$ if $d\ge 0$ and $r=-d$ if $d<0$.
The corresponding bipartition is $\mathcal B(S)$ if $d\geq 0$ and the flip of $\mathcal{B}(S)$ otherwise.

§.§.§ $Spin(N)$, $N$ odd

* $\mathcal{N}_{o,l}^* \leftrightarrow \mathcal{P}_{ro}(N)$.

* $p_1(\lambda)=\lambda$.

* $L$ is of type $B_{(d-1)(2d+1)/2} + \frac{1}{4}(N-d(2d-1))A_1$ for $d\in \mathbb Z$, $d \equiv N \mod{4}$ and $d(2d-1) \leq N$.

* $W_L= W_{(N-d(2d-1))/4}$.

* Let $\lambda \in \mathcal{P}_{ro}(N)$. Then $\lambda$ belongs to the block corresponding to $d=d(\lambda)$, cf. (<ref>), and the corresponding bipartition is $\rho(\lambda)$, cf. Proposition <ref>.

§.§.§ $Spin(N)$, $N$ even

* $\mathcal{N}_{o,l}^* \leftrightarrow \mathcal{P}_{ro}(N) \sqcup \mathcal{P}_{ro}(N)$.

* $p_1(\lambda)=\lambda$.

* $L$ is of type $D_{d(2d-1)/2} + \frac{1}{4}(N-d(2d-1))A_1$ for $d\in \mathbb Z$, $d \equiv N \mod{4}$ and $d(2d-1)\leq N$.

* $W_L = W_{(N-d(2d-1))/4}$.

* Let $\lambda \in \mathcal{P}_{ro}(N)$. Then $\lambda$ belongs to the block corresponding to $d=d(\lambda)$, cf. (<ref>), and the corresponding bipartition is $\rho(\lambda)$, cf. Proposition <ref>.

§.§ The Arithmetic-Geometric correspondence

Recall from Section <ref> the definitions of $\bfG, G^\vee, \overline{\mathfrak S}, \overline{\mathfrak U}, \theta$ etc.
Recall from the previous section the definition of the set $\overline{\mathfrak C}(G^\vee)$ and note that there is a natural inclusion $\overline{\mathfrak C}(G^\vee)\to \overline{\mathfrak S}$.
In this section, we will give an explicit description of Lusztig's arithmetic-geometric correspondence (cf. Section <ref>) for the geometric diagrams $\overline{\mathfrak C}(G^\vee)$ for the classical groups.

Below we record

* A list of all pairs $(L^{\vee},\chi)$ consisting of a Levi subgroup $L^{\vee} \subset G^\vee$ and a character $\chi$ of $Z(L^{\vee})$ such that there is a triple $(L^{\vee},\mathcal C^{\vee},\mathcal{E}^\vee) \in \overline{\mathfrak C}(G^{\vee})$ and $Z(L^{\vee})$ acts on the stalks of $\mathcal{E}^\vee$ by (a multiple of) $\chi$. In most cases, $\chi$ is uniquely determined by $L^{\vee}$; in such cases we omit it.

* The type of the affine Weyl group $\mathcal W^\vee(J^\vee,\mathcal C^\vee,\mathcal E^\vee)$.

* A description of $J$ and $\omega$ in $\theta(x) = (J,\bf E,\psi,\omega)$ for each $x\in \overline{\mathfrak C}(G^\vee)$.
In all the cases we consider there is a unique choice of $\bf E$ and so we omit it.

This information is taken from <cit.>.

For $G^{\vee}=\mathrm{Spin}(N)$ we present this information in a slightly different manner. As discussed in Section <ref>, there is a decomposition

$$\overline{\mathfrak C}(Spin(N)) = \overline{\mathfrak C}(SO(N)) \sqcup \overline{\mathfrak C}^*(Spin(N)).$$

In (<ref>) and (<ref>), we describe the restriction of $\theta$ to $\overline{\mathfrak C}(SO(N))$, and in (<ref>) and (<ref>), we describe the restriction of $\theta$ to $\overline{\mathfrak C}^*(Spin(N))$.

§.§.§ $G^\vee=SL(N)$

* $L^\vee$ is of type $(N/r)A_{r-1}$ for $r\mid N$, $\chi$ is of order $r$.

* $\tilde A_{N/r-1}$

* $J$ is $\emptyset$ and $\omega$ is of order $r$.

§.§.§ $G^\vee = SO(N)$, $N$ odd

* $L^\vee$ is of type $B_{\frac 12(r^2-1)}$ for $r\ge 0$ odd and $\frac 12 (r^2-1) \le n$.

* $\tilde C_{\frac12 (N-r^2)}$

* $J$ is $C_{a^2+a}\times C_{a^2+a}$ where $a:=(r-1)/2$. $\omega = 1$.

§.§.§ $G^\vee = SO(N)$, $N$ even

* $L^\vee$ is of type $D_{r^2/2}$ for $r\ge 0$ even and $r^2/2 \le n$.

* $\tilde C_{\frac12 (N-r^2)}$ if $r>0$, $\tilde D_n$ if $r=0$.

* $J$ is $D_{a^2}\times D_{a^2}$ where $a:=r/2$. $\omega = 1$ if $a$ is even and $\omega $ of type I if $a$ is odd.

§.§.§ $G^\vee = Sp(2n)$

* $L^\vee$ is of type $C_{\frac 12r(r+1)}$ for $r\ge 0$ and $\frac 12 r(r+1) \le n$.

* $\tilde C_{n-\frac 12r(r+1)}$ if $r>0$, $\tilde B_n$ if $r=0$..

* $J$ is $D_{a^2}\times B_{a^2+a}$ if r is even and $D_{a^2}\times B_{a^2-a}$ if $r$ is odd where $a:=\lfloor (r+1)/2\rfloor$. $\omega=1$ if $a$ is even and $\omega\ne 1$ if $a$ is odd.

§.§.§ $G^\vee = Spin(N)$, $N$ odd

* $L^\vee$ is of type $B_{(d-1)(2d+1)/2} + \frac{1}{4}(N-d(2d-1))A_1$ for $d \equiv N \mod{4}$ and $d(2d-1) \leq N$.

* $\tilde C_{(N-d(2d-1))/4}$

* $J$ is $C_{(d^2-1)/4}\times A_{d(d-1)/2-1}\times C_{(d^2-1)/4}$, $\omega \ne 1$.

§.§.§ $G^\vee = Spin(N)$, $N$ even

* $L^\vee$ is of type $D_{d(2d-1)/2} + \frac{1}{4}(N-d(2d-1))A_1$ for $d \equiv N \mod{4}$ and $d(2d-1)\leq N$. There are two possible $\chi$.

* $\tilde C_{(N-d(2d-1))/4}$

* $J$ is $D_{d^2/4}\times A_{d(d-1)/2-1}\times D_{d^2/4}$, $\omega$ of type II.

§.§ Faithful nilpotent orbits

Recall the notation $\bfG, G^\vee, \overline{\mathfrak S}, \overline{\mathfrak U}, \theta$ from Section <ref>, $\cN_{o,l},(\cN_{o,l})_x,W_x,\overline{\mathfrak C}$ from Section <ref>, $d_A, \mathcal K_I, \overline{\mathbb L}$ from Section <ref>, and $\unip, \unip_{\bfL,\bf E}, \WF(\phi)$ from Section <ref>.

* If $x=(J^\vee,\mathcal C^\vee,\mathcal E^\vee)\in \overline{\mathfrak S}$, and $\theta(x) = (K,\bf E,\psi,\omega)$, write $\mathcal W_x$ for the affine Weyl group $\mathcal W(K,\omega,\bf E)$ and $S_x$ for the Coxeter generators for $\mathcal W_x$.
Recall from Section <ref> that $S_x\leftrightarrow (I - K)/\omega$. Write $W_x$ for the finite part of $\mathcal W_x$ (when $x\in \overline{\mathfrak C}(G^\vee)$ this agrees with the definition of $W_x$ given in Section <ref>).
Write $\pi:\mathcal W_x\to W_x$ for the projection map and $\mathcal I:(I-K)/\omega \xrightarrow{\sim} S_x$ for the bijection.

* Let $\mathcal J_x = \{J \mid K\subseteq J\subsetneq I, \ \omega J = J \}$.

* Write $I_{0,K}$ for the element of $\mathcal J_x$ with $\mathcal I(I_{0,K}-K)$ equal to the Coxeter generators  of $W_x$.

* Let $\mathcal F_x = \{(J,\phi) \mid J\in \mathcal J_x, \ \phi \text{ a family of }\unip(\bfL_{c^\omega(J)}(\mathbb F_q)) \}$.

* For $J\in \mathcal J_x$ write $\mathcal W_{x,J}$ for $\langle y\mid y\in \mathcal I((J-K)/\omega)\rangle$ and $W_{x,J}$ for $\pi(\mathcal W_{x,J})$.

* Let $J\in \mathcal J_x$.
Then $\bfL_{c^\omega(K)}(\mathbb F_q)$ is a $\bfL_{c^\omega(J)}(\mathbb F_q)$-split Levi subgroup of $\bfL_{c^\omega(J)}(\mathbb F_q)$ affording the cuspidal representation $\bf E$, and $\mathcal H(\bfP_{c^\omega(J)(\mf o)},\bf E)$ (c.f. Section <ref>) has underlying coxeter system $\mathcal W_{x,J} \cong W_{x,J}$.
Thus $\mathrm{Irr}(W_{x,J})$ (resp. $\mathrm{Irr}(\mathcal W_{x,J})$) parameterizes the irreducible representations in the Harish-Chandra series of $\bfL_{c^\omega(J)}(\mathbb F_q)$ determined by the cuspidal data $(\bfL_{c^\omega(K)},\bf E)$.
Let $\mathcal U_{x,J}:\mathrm{Irr}(W_{x,J})\to \unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q))$ (resp. $\tilde{\mathcal U}_{x,J}:\mathrm{Irr}(\mathcal W_{x,J})\to \unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q))$) denote this parameterisation.
We also write $\mathcal U_{x,J}$ for the composition

$$\mathrm{Irr}(W_{x,J})\xrightarrow{\sim} \unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q)) \hookrightarrow \unip(\bfL_{c^\omega(J)}(\mathbb F_q)).$$

Let $x=(J^\vee,\mathcal C^\vee,\mathcal E^\vee)\in \overline{\mathfrak C}(G^\vee)$, $\theta(x) = (K,\bf E,\psi,\omega)$ and $\OO^{\vee} \in (\cN_o^\vee)_x$. We say that $\OO^{\vee}$ is $x$-faithful if there exists a pair $(J, \phi) \in \mathcal{F}_{x}$ such that

(i) $\overline{\mathbb L}(J,\WF(\phi,\CC)) = d_A(\OO^{\vee},1)$,

and for every irreducible representation $\rho$ of $A(\OO^{\vee})$ such that $(\OO,\rho)\in (\cN_{o,l}^\vee)_x$,

(ii) there is a representation $F\in \mathrm{Irr}(W_{x,J})$ such that $\mathscr U_{x,J}(F) \in \phi$ and

$$\Hom_{W_{x,J}}(F,E(\OO^\vee, \rho)|_{W_{x,J}}) \neq 0;$$

(iii) for every $J'\in \mathcal J_x$ and $F'\in \mathrm{Irr}(W_{x,J'})$, if

$$\Hom_{W_{x,J'}}(F',E(\OO^\vee, \rho)|_{W_{x,J'}}) \neq 0$$

\begin{equation}
\label{eq:bound}
\overline{\mathbb L}(J',\WF(\mathcal U_{x,J'}(F'),\CC)) \le d_A(\OO^{\vee},1).
\end{equation}

Note that Equation <ref> is equivalent to the condition

$$d_S(J',\WF(\mathcal U_{x,J'}(F'),\CC)) \ge \OO^\vee.$$

It suffices to check (iii) in Definition <ref> for $J'$ maximal in $\mathcal J_x$ with respect to inclusion.

Suppose condition (iii) of Definition <ref> holds for all $J'$ maximal.
Let $J'\in \mathcal J_x$ and $F'\in \mathrm{Irr}(W_{x,J'})$ be such that

$$\Hom_{W_{x,J'}}(F',E(\OO^\vee, \rho)|_{W_{x,J'}}) \neq 0.$$

Let $J''\in \mathcal J_x$ be maximal among elements of $\mathcal J_x$ containing $J'$.
Then there exists an $F''\in \mathrm{Irr}(W_{x,J''})$ such that

$$\Hom_{W_{x,J''}}(F'',E(\OO^\vee, \rho)|_{W_{x,J''}}) \neq 0$$

$$\Hom_{W_{x,J'}}(F',F''|_{W_{x,J'}}) \neq 0.$$

By assumption

$$d_S(J'',\WF(\mathcal U_{x,J''}(F''),\CC)) \ge \OO^\vee.$$

Thus it suffices to prove

$$d_S(J',\WF(\mathcal U_{x,J'}(F'),\CC)) \ge d_S(J'',\WF(\mathcal U_{x,J''}(F''),\CC)).$$

By <cit.> and the remarks in the proof we have

$$d_S(J',\WF(\mathcal U_{x,J'}(F'),\CC)) = d_S(J'',\mathrm{Sat}_{L_{J'}}^{L_{J''}}\WF(\mathcal U_{x,J'}(F'),\CC)).$$

Thus it suffices to prove that for $F'\in \mathrm{Irr}(W_{x,J'}), F''\in \mathrm{Irr}(W_{x,J''})$

\begin{equation}
\label{eq:suff}
\Hom_{W_{x,J'}}(F',F''|_{W_{x,J'}}) \neq 0\implies \mathrm{Sat}_{L_{J'}}^{L_{J''}}\WF(\mathcal U_{x,J'}(F'),\CC) \le \WF(\mathcal U_{x,J''}(F'')).
\end{equation}

By <cit.>,

$$\mathcal U_{x,J''}\left(\mathrm{Ind}_{W_{x,J'}}^{W_{x,J''}}F'\right) = \mathrm{Ind}_{L_{J'}}^{L_{J''}}\mathcal U_{x,J'}(F')$$

where $\mathrm{Ind}$ applied to unipotent characters means Harish-Chandra induction.
Thus by Frobenius reciprocity we have that

\begin{equation}
\label{eq:part1}
\Hom_{W_{x,J'}}(F',F''|_{W_{x,J'}}) \neq 0\Leftrightarrow \mathcal U_{x,J''}(F'')\subseteq \mathrm{Ind}_{L_{J'}}^{L_{J''}}\mathcal U_{x,J'}(F').
\end{equation}

By <cit.> (and noting that we may replace the Deligne-Lusztig induction in the lemma by Harish-Chandra induction since $\bfL_{c^\omega(J')}$ is a $\bfL_{c^\omega(J'')}$-split Levi of $\bfL_{c^\omega(J'')}$ by <cit.>) we have that

\begin{equation}
\label{eq:part2}
\mathcal U_{x,J''}(F'')\subseteq \mathrm{Ind}_{L_{J'}}^{L_{J''}}\mathcal U_{x,J'}(F') \implies \mathrm{Sat}_{L_{J'}}^{L_{J''}}\WF(\mathcal U_{x,J'}(F'),\CC) \le \WF(\mathcal U_{x,J''}(F'')).
\end{equation}

Stringing together Equation <ref> and Equation <ref> gives Equation <ref> as required.

Suppose $\mathbf{G}$ is a simple group of adjoint type and let $\OO^{\vee} \in \cN_o^{\vee}$. Then for every $x \in \overline{\mathfrak C}(G^{\vee})$ such that $\OO^\vee\in (\cN_o)_x$, $\OO^{\vee}$ is $x$-faithful.

We consider separately the cases of classical and exceptional groups.
If $\mathbf{G}$ is classical, the statement is proved in Section <ref>. If $\mathbf{G}$ is exceptional, it is proved in Section <ref>.

§.§ Proof of faithfulness in classical types

§.§.§ $G^\vee = SL(N)$

Let $n=N-1$.
Let $x\in \overline{\mathfrak C}(G^\vee)$ correspond to $\chi \in \mathrm{Irr}(\mathbb Z_N)$ of order $r\mid N$.
Then $\theta(x) = (\emptyset,1,\psi,\chi)$ and $\mathcal W_x$ is of type $\tilde A_{k-1}$ where $k = N/r$.
A nilpotent orbit $\OO^\vee\in \cN_o^\vee \leftrightarrow \lambda = (\lambda_1\le\lambda_2\le \dots \le \lambda_e)$ lies in $(\cN_o^\vee)_x$ iff $r\mid \lambda_i$ for all $i$.

We wish to show that $\lambda$ is $x$-faithful.
Let $J = rA_{k-1}$ and $\mu = (\lambda_1/r\le \lambda_2/r\le \dots \le \lambda_e/r)$. Then $\bfL_{c^\chi(J)}$ is isomorphic to $A_{k-1}(q)$ and so the families are parameterised by $\mathcal P(k)$.
Let $\phi$ be the family parameterised by the partition $\mu$.
Then $\bar {\mathbb L}(J,\WF(\phi,\CC)) = \mu^t \cup \mu^t\cup \cdots \cup\mu^t$ where there are $r$ terms in this expression.
But $\mu^t\cup\mu^t\cup\cdots\cup \mu^t = (\mu + \mu + \cdots + \mu)^t = \lambda^t = d(\OO^\vee,1)$.
Thus condition (i) holds.
Condition (ii) holds because $W_{x,J} = W_x$ and $E(\OO^\vee,\rho) = \mu$.
Finally condition (iii) holds by using Lemma <ref> to reduce to the case where $J\in \mathcal J_x$ is maximal and noting that for such $J$, $W_{x,J} = W_x$.
Condition (iii) then holds as a consequence of condition (ii) (the only possible $F'$ that can arise is $F' = F$).

§.§.§ $SO(N)$, $N$ odd

Let $n=\lfloor N/2\rfloor$, $G^\vee = Spin(N)$.
Fix $\OO^\vee\in \cN^\vee_o$. We show that $\OO^\vee$ is $x$-faithful for all $x\in \overline{\mathfrak C}(SO(N))$ such that $\OO^\vee\in (\cN_o^\vee)_x$ by exhibiting a single pair $(J,\phi)$ that works for all such blocks.
Let $\lambda = (\lambda_1\le\lambda_2\le \dots \le \lambda_e)$ be the orthogonal partition of $N$ corresponding to $\OO^\vee$.
Define $\mu(\lambda) \subseteq \lambda^t$ as follows

$$m_{\mu(\lambda)}(x) = \begin{cases}
2 &\mbox{if } x \text{ is even and } m_{\lambda^t}(x) \text{ is}\\
&\mbox{even and nonzero}\\
1 & \mbox{if } x \text{ is even and } m_{\lambda^t}(x) \text{ is}\\
& \mbox{odd}\\
0 &\mbox{otherwise} \end{cases}$$

and define $\nu(\lambda) = d_{BV}(\lambda)\setminus \mu(\lambda)$.
By <cit.> $\mu(\lambda)$ and $\nu(\lambda)$ are both special symplectic partitions.
We set $J$ to be the set $C_m\times C_{n-m}$ where $m = |\mu|/2$, and $\phi$ to be the family characterised by $\WF(\phi,\CC) = (\mu(\lambda),\nu(\lambda))$.
Since $J = C_m\times C_{n-m}$, the family $\phi$ decomposes as a product $\phi_1\times \phi_2$ where $\phi_1$ is a family of $C_m$, $\phi_2$ is a family of $C_{n-m}$ and

$$\WF(\phi_1) = \mu(\lambda), \quad \WF(\phi_2) = \nu(\lambda).$$

It follows from <cit.> that condition (i) holds for this choice of $(J,\phi)$.
For condition (ii), let $x\in \overline{\mathfrak C}(SO(N))$ and suppose $(\OO^\vee,\rho)\in (\mathcal N_{o,l})_x$.
As described in Section <ref> (3) let $r$ be the odd integer corresponding to $x$.
Then $\theta(x) = (C_{a^2+a}\times C_{a^2+a},\bf E,\psi,1)$, $a = (r-1)/2$, $\bf E$ is a uniquely determined cuspidal unipotent representation, and $\mathcal W_x$ is of type $\tilde C_{\frac 12(r^2-1)}$.
By Section <ref>, $\unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q)) \leftrightarrow \mathcal Y^1_{r,m}\times\mathcal Y^1_{r,n-m}$.
Note that $K\subseteq J$ (and hence $J\in \mathcal J_x$) if and only if $\mathcal Y^1_{r,m}\times\mathcal Y^1_{r,n-m}$ is non-empty.
Under the bijection in Section <ref> (1), $(\OO^\vee,\rho)$ corresponds to a $S\in \mathcal Y^2_{r,n}$ with $p_1(S) = \lambda$.
By <cit.>, $S$ has the following description in terms of $S^{sp} = (X,Y)$.
Let $X=(X_1,\dots,X_c), Y = (Y_1,\dots,Y_c)$ be the refinements of $X$ and $Y$ (see paragraph preceeding <cit.>).
Then $S=(A,B)$ where $A=(A_1,\dots,A_c),B=(B_1,\dots,B_c)$ and $\{A_i,B_i\} = \{X_i,Y_i\}$ for all $i$.
Let $S^{1,sp}= (X^1,Y^1)\in \mathcal Y_{1,m}^1, S^{2,sp}=(X^2,Y^2)\in \mathcal Y_{1,n-m}^1$ be the special symbols in the families $\phi_1$ and $\phi_2$ respectively.
Taking sufficiently large representatives in the relevant shift equivalence classes we have by condition (i) and <cit.> that $S^{sp} = S^{1,sp} + S^{2,sp}$.
Let $S^{i,sp}=(X^i,Y^i)=((X_1^i,\dots,X_c^i),(Y_1^i,\dots,Y_c^i))$ be the decomposition into subsequences along the same indices as the decomposition of $X,Y$.
Define $S^i = ((A_1^i,\dots,A_c^i),(B_1^i,\dots,B_c^i))$ where

\begin{equation}
(A_j^i,B_j^i) = \begin{cases}
(X_j^i,Y_j^i) & \mbox{ if } (A_j,B_j)=(X_j,Y_j) \\
(Y_j^i,X_j^i) & \mbox{ if } (A_j,B_j)=(Y_j,X_j).
\end{cases}
\end{equation}

By construction $S^{i,sp}\sim S^i$ and $S = S^1+S^2$.
Moreover, by <cit.> and <cit.> we have that $S^1\in \mathcal Y^1_{r,m}, S^2\in \mathcal Y^1_{r,n-m}$.
Thus $\unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q)) \ne \emptyset$, so $J\in \mathcal J_x$, and

\begin{equation}
\label{eq:sumSOodd}
\mathcal D(S) = \mathcal D(S^1) + \mathcal D(S^2).
\end{equation}

Note that $\mathcal D(S)$ parameterises an irreducible representation of $W_x$ and $(\mathcal D(S^1),\mathcal D(S^2))$ parameterises an irreducible representation of $W_{x,J}$.
By the Littlewood-Richardson rule (c.f. the proof of <cit.>) Equation <ref> implies that

$$\Hom_{W_{x,J}}(F,E(\OO^\vee, \rho)|_{W_{x,J}}) \neq 0.$$

Finally for condition (iii), let $S\in \mathcal Y^2_{r,n}$ of defect $r\ge 1$ correspond to $(\OO^\vee,\rho)$ as above and suppose $J\in \mathcal J_x$ is maximal and $F\in \mathrm{Irr}(W_{x,J'})$ satisfies

$$\Hom_{W_{x,J}}(F,E(\OO^\vee, \rho)|_{W_{x,J}}) \neq 0.$$

Since $J$ is maximal it is of the form $C_m\times C_{n-m}$ and $W_{x,J}$ is of the form $W_k\times W_{l-k}$.
Let the pair of bipartitions $(B^1,B^2)$ correspond to $F$.
Let $(S^1,S^2)$ be the corresponding symbols of defect $r$.
By the Littlewood-Richardson rule (cf. <cit.>) we have that $S^1+S^2\ge S$.
We wish to show that

$$d_S(J,\WF(\mathcal U_{x,J}(F),\CC)) \ge \OO^\vee.$$

By <cit.> it suffices to show that the inequality holds with respect to the dominance order for the respective s-symbols viewed as non-decreasing sequences i.e. partitions.
The s-symbol for $E(\OO^\vee,1)$ is just $S^{sp}$.
Let us compute the s-symbol for the left hand side.
Let $\lambda_1 = \mathcal P(2S_1^{1,sp}\cup 2S_2^{1,sp}+1)$ and $\lambda_2 = \mathcal P(2S_2^{2,sp}\cup 2S_2^{2,sp}+1)$ (these are both special symplectic partitions).

$$\WF(\mathcal U_{x,J}(F)) = \lambda_1^t\times \lambda_2^t.$$

Thus by <cit.>

$$d_S(J,\WF(\mathcal U_{x,J}(F),\CC)) = j_{W_J}^{W^\vee}(E(\lambda_1 ,1)\otimes E(\lambda_2,1)).$$

where $j$ is $j$-induction, see <cit.>. By <cit.>, $W_J$ is the parahoric of $W$ (the Weyl group of $\bfG$) determined by $J$, and the s-symbol for the $j$-induced representation is

\begin{equation}
\label{eq:ssymb}
\end{equation}

Since $S^1+S^2\ge S$, by Lemma <ref> we have that $S^{1,sp}+S^{2,sp}\ge S$.
This is the required inequality.

§.§.§ $SO(N)$, $N$ even

Suppose $\lambda$ is not one of the edge cases listed in <cit.>.
Then the proof is identical to the $N$ odd case except for a few minor technical differences which we note.

* The partition $\mu(\lambda)$ is defined as

$$m_{\mu(\lambda)}(x) = \begin{cases}
2 &\mbox{if } x \text{ is odd and } m_{\lambda^t}(x) \text{ is}\\
&\mbox{even and nonzero}\\
1 & \mbox{if } x \text{ is odd and } m_{\lambda^t}(x) \text{ is}\\
& \mbox{odd}\\
0 &\mbox{otherwise}. \end{cases}$$

* $J = D_m\times D_{n-m}$.

* $r$ is even. $\omega = 1$ if $4\mid r$, and $\omega$ is type I if $4\mid r-2$. $K=D_{a^2}\times D_{a^2}$ where $a = r/2$.
Despite these differences, we still have $\unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q)) \leftrightarrow \mathcal Y^1_{r,m}\times\mathcal Y^1_{r,n-m}$.

* The rest of the argument follows through identically, except for the case $r=0$ (see Equation <ref> for why this case is different) which is treated in <cit.>.

Now suppose $\lambda$ is one of the edge cases listed in <cit.>.
Recall that for these $\lambda$, $\mu(\lambda) = (1^2)$ which is a partition of $2$.
The issue with these edge cases can be traced down to the fact that $m = 1$, but $D_1 \times D_{n-1}$ is not a subgraph of $\tilde D_n$ if we interpret $D_1$ as $A_1$.
If we interpret $D_1$ as the empty graph however, then as long as we interpret things correctly, the same argument as the one given for $\lambda$ not an edge case works.
We briefly give the details.

Suppose $\lambda$ is one of said edge cases.
Then $\lambda$ is special, $r$ takes values in $\{0,2\}$, $\mu(\lambda) = (1^2)$ and $\nu(\lambda) = d_{BV}(\lambda)\setminus \mu(\lambda)$.
For both $r=0,2$ we have that $K = \emptyset$ (since <cit.> uses the convention $D_1 = \emptyset$).
Thus we can take $J$ to be $D_{n-1}$.
Take $\phi$ to be the family characterised by $\WF(\phi,\CC) = \nu(\lambda)$.
Since $d_{BV}(\lambda) = \nu(\lambda) \cup \mu(\lambda) = \nu(\lambda) \cup (1^2)$, we have that $\nu(\lambda)$ saturates to $d_{BV}(\lambda)$ in $D_n$ and so since $\lambda$ is special we get that

$$\bar {\mathbb L}(D_{n-1},\nu(\lambda)) = \bar {\mathbb L}(D_n,d_{BV}(\lambda)) = d_A(\OO^\vee,1).$$

This establishes property (i).
For property (ii) we can construct as usual the a-symbols $S_1,S_2$ so that Equation <ref> holds.
Supposing $r\ne 0$ (the $r=0$ case has already been treated in <cit.>), then $W_x = W_{l}$ where $l=\frac12(N-r^2)$, and $W_{x,J} = W_{l-1}$.
Since $W_{l-1}\le W_1\times W_{l-1}\le W_l$ and $\mathcal D(S^2) \subseteq \mathcal D(S^1)\otimes \mathcal D(S^2)|_{W_{l-1}}$, we get from Equation <ref> that

$$\Hom_{W_{x,J}}(F,E(\OO^\vee, \rho)|_{W_{x,J}}) \neq 0$$

as required.
The proof for condition (iii) requires no modifications.

§.§.§ $Sp(2n)$

Suppose $\lambda$ is not one of the edge cases listed in <cit.>.
The proof for this case is similar to the $SO(N)$, $N$ odd case with a few (more substantial) technical differences which we note.

* $\lambda$ is a symplectic partition of $2n$. The partition $\mu(\lambda)$ is defined as

$$m_{\mu(\lambda)}(x) = \begin{cases}
2 &\mbox{if } x \text{ is odd and } m_{\lambda^t}(x) \text{ is}\\
&\mbox{even and nonzero}\\
1 & \mbox{if } x \text{ is odd and } m_{\lambda^t}(x) \text{ is}\\
& \mbox{odd}\\
0 &\mbox{otherwise}. \end{cases}$$

* $J = D_m\times B_{n-m}$.

* $r$ has no parity restriction. $K=D_{a^2}\times B_{a^2+a}$ if $r$ is even and $K = D_{a^2}\times B_{a^2-a}$ if $r$ is odd where $a = \lfloor (r+1)/2\rfloor$.
$\omega = 1$ if $a$ is even, and $\omega = -1$ if $a$ is odd.

* $S\in \mathcal X^{1,1}_{d,n}$ where $d=r+1$ if $r$ is even and $d=-r$ if $r$ is odd.

* We have $\unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q)) \leftrightarrow \mathcal Y^1_{d-1,m}\times\mathcal Y^1_{d,n-m}$.

* $S^{1,sp}$ lies in $\mathcal Y^1_{0,m}$.
The additive identity is $S^{sp} = (S^{1,sp})^! + S^{2,sp}$.

* $(S^1)^!$ is constructed from $(S^{1,sp})^!$. We have $S^1\in \mathcal X_{d-1,m}^{1,0}, (S^1)^! \in \mathcal X_{d,m}^{0,1}, S^2\in \mathcal X_{d,n-m}^{1,0}$ and $S = (S^1)^!+S^2$. The $\bullet^!$ map reflects similarity classes and so $S^1 \sim S^{1,sp}$.

* The rest of the argument follows through identically.

If $\lambda$ is one of the edge cases listed in <cit.> then if we interpret $D_1$ as the empty graph, the same argument as in Section <ref> follows through.

§.§.§ $Spin(N)$, $N$ odd

Let $n=\lfloor N/2\rfloor$, $G^\vee = Spin(N)$.
Fix $(\OO^\vee,\rho)\in \cN_{o,l}^*$ and let $\lambda = (\lambda_1^{a_1},\lambda_2^{a_2},\dots,\lambda_k^{a_k})\in \cN_{ro}$ for $\lambda_1<\cdots<\lambda_k$ and $a_i\ge 1, i=1,\dots,k$ be the corresponding partition under the bijection in Section <ref> (1).
Define $\mu(\lambda) \subseteq \lambda^t$ as follows

$$m_{\mu(\lambda)}(x) = \begin{cases}
2 &\mbox{if } x \text{ is even and } 4\mid m_{\lambda^t}(x) \\
&\mbox{and } m_{\lambda^t}(x) \ne 0\\
1 & \mbox{if } x \text{ is even and } m_{\lambda^t}(x) \text{ is}\\
& \mbox{odd}\\
0 &\mbox{otherwise.} \end{cases}$$

Recall from Corollary <ref> that we can write $d_{BV}(\lambda) = \underline{d_{BV}(\lambda)}\cup \underline{d_{BV}(\lambda)}$.

$\mu(\lambda)\subseteq \underline{d_{BV}(\lambda)}$.

Arguing as in the proof of Lemma <ref>, we can reduce to the case when $\lambda$ consists of only odd parts, i.e. $\lambda=(\lambda_1<\lambda_2<...<\lambda_k)$. Recall that

$$\lambda^t = (k^{\overline{\lambda}_1},(k-1)^{\overline{\lambda}_2},...,1^{\overline{\lambda}_k})$$

$$d_{BV}(\lambda) = (k^{\overline{\lambda}_1+\delta_1},(k-1)^{\overline{\lambda}_2+\delta_2},...,1^{\overline{\lambda}_k}+\delta_k)$$

where $\delta=(-1,2,-2,...,2,-2)$. If $x$ is even, then $m_{\lambda^t}(x) \geq 2$. So $m_{d_{BV}(\lambda)}(x) = m_{\lambda^t}(x)+2 \geq 4$. Hence, $m_{\underline{d_{BV}(\lambda)}}(x) = \frac{1}{2}m_{d_{BV}(\lambda)}(x) \geq 2$.

Define $\nu(\lambda) = \underline{d_{BV}(\lambda)}\ \setminus \mu(\lambda)$.

Let $x\in \overline{\mathfrak C}^{*}(Spin(N))$ be such that $(\OO,\rho)\in (\cN_{o,l}^*)_x$.
$x$ is uniquely determined and corresponds to the block parameterised by $d = d(\lambda)$ in the description of blocks given in <ref> (3).
We have $\theta(x) = (C_{(d^2-1)/4}\times A_{d(d-1)/2-1}\times C_{(d^2-1)/4},\bf E,\psi,\omega)$ where $\omega = -1$, and the affine Weyl group is of type $\tilde C$.
Note that since $N$ is odd and $d(\lambda)\equiv N\pmod 2$ we have that $d$ is also odd.
We set $J$ to be the $\omega$-stable set $C_m\times A_{n-2m-1}\times C_m$ where $m = |\mu|/2$.
The reductive quotient has type $C_m(q)\times \hphantom{ }^2A_{n-2m-1}(q^2)$.
By Section <ref>,

\begin{align*}
%\unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q)) &\leftrightarrow \mathcal Y^1_{d,m}\times\mathrm{Irr}(W_{\frac12(n-2m-\frac12r(r+1))})
\mathrm{Irr}(W_{m-(d^2-1)/4})\times\mathrm{Irr}(W_{\frac12(n-2m-\frac12r(r+1))}) &\leftrightarrow \unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q))
\end{align*}

where $r = d-1$ if $d>0$ and $r=-d$ if $d\le 0$.
It will be useful to note that

$$\unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q)) \subseteq \unip(C_m(q))\times \unip(\hphantom{ }^2A_{n-2m-1}).$$

and that $\mathcal U_{x,J}$ is a product of maps

$$\mathrm{Irr}(W_{m-(d^2-1)/4})\to \unip(C_m(q)), \quad \mathrm{Irr}(W_{\frac12(n-2m-\frac12r(r+1))})\to \unip(\hphantom{ }^2A_{n-2m-1}).$$

We call these maps $\mathcal U_{x,J,1}$ and $\mathcal U_{x,J,2}$ respectively.

We set $\phi$ to be the family characterised by $\WF(\phi,\CC) = (\mu(\lambda),\nu(\lambda))$.
Note the family $\phi$ decomposes as a product $\phi_1\times \phi_2$ where $\phi_1$ is a family of $C_m(q)$, $\phi_2$ is a family of $^2A_{n-2m-1}(q^2)$ and

$$\WF(\phi_1) = \mu(\lambda), \quad \WF(\phi_2) = \nu(\lambda).$$

The following Lemma establishes condition (i) for the pair $(J,\phi)$.

$\bar{\mathbb L}(J,\WF(\phi,\CC)) = d_A(\OO^\vee,1)$.

We have

\begin{align*}
\bar{\mathbb L}(J,\WF(\phi,\CC)) &= \bar{\mathbb L}(C_m\times A_{n-2m-1}\times C_m,(\mu(\lambda),\nu(\lambda),\mu(\lambda))) \\
&=\bar{\mathbb L}(C_m\times C_{n-m},(\mu(\lambda),\nu(\lambda)\cup\nu(\lambda)\cup\mu(\lambda))) \\
&=\bar{\mathbb L}(C_m\times C_{n-m},(\mu(\lambda),d_{BV}(\lambda)\setminus \mu(\lambda)) \\
\end{align*}

where the last equality follows from essentially the same argument given in <cit.>.

Recall the definition of $\rho$ from Section <ref> and let $B = (B_1,B_2) = \rho(\lambda)$.
Note that there is a case distinction involving a flip depending on if $d(\lambda)>0$ or $\le 0$.
Let us assume for now that $d(\lambda)>0$.
We will discuss the case $d(\lambda)\le0$ separately at the end.

To verify condition (ii) it suffices to find $B^1\in \mathcal B(m-(d^2-1)/4), B^2\in \mathcal B(\frac12(n-2m-\frac12r(r+1)))$ such that

$$B = B^1+B^2$$

(c.f. the proof of <cit.>) and $\mathcal U_{x,J,1}(B^1)\in \phi_1,\mathcal U_{x,J,2}(B^2) \in \phi_2$.
Note that since the families in $^2A_{n-2m-1}$ are all singletons, the condition $\WF(\mathcal U_{x,J,2}(B^2)) = \nu(\lambda)$ in fact completely determines $B^2$ provided such a $B^2$ exists.
For this reason it is covenient to determine $B^2$ first.

Recall the definitions of $f, \delta, \bar\lambda_i,A_i$ from the end of Section <ref>.
Let $(a,b]$ denote the set $\{a+1,\dots,b\}$ when $a<b$ and $\emptyset$ otherwise. Let $(a,b)$ denote the set $\{a+1,\dots,b-1\}$ if $a+1<b$ and $\emptyset$ otherwise.
Let $[P]$ denote the function that takes value $1$ if $P$ is true and $0$ if $P$ is false.
Define $\xi_i = [4\mid \bar\lambda_i], \zeta_i = [2\nmid \bar\lambda_i]$ and $\mathcal S = \bigcup_{i=1}^l(f(2i-1),f(2i)]$.
Since $a_i=1$ whenever $\lambda_i$ is odd, we have that $A_i$ is even if and only if $i\in \mathcal S$.
Therefore, we can write $\mu$ as $(A_i^{\zeta_i+2\xi_i})_{i\in \mathcal S}$ (this is a decreasing sequence in $i$).
It follows that

$$\nu = (A_i^{\frac12(\bar\lambda_i+\delta_i)-[i\in \mathcal S](\zeta_i+2\xi_i)}).$$

A direct computation then shows that

$$\nu^t = (\eta_1^{a_1},\eta_2^{a_2},\dots,\eta_k^{a_k})$$

where $\eta_1\le\eta_2\le\cdots\le \eta_k$,

$$\eta_i = \frac12(\lambda_i+[i\in \mathcal O_0]-[i\in \mathcal O_1])-E_i, \quad E_i = \sum_{j\le i, j\in \mathcal S}\zeta_i+2\xi_i$$

$$\mathcal O_i = \{f(j) \mid 1\le j\le 2l,\ j \equiv i\pmod 2\}.$$

By Section <ref> (2), the partition $\nu^t$ parameterises $\mathcal U_{x,J,2}(B^2)$ as en element of $\unip(\hphantom{ }^2A_{n-2m-1}(q^2))$.
Let us now compute $B_2 = (B_1^2,B_2^2)$ using the algorithm in Section <ref> (6).

Let $z_i = \lfloor \lambda_i/2 \rfloor$ so $z_i$ is even if $r_i = 0,1$ and is odd if $r_i = 2,3$.
In terms of $z_i$ we have

$$\eta_i = z_i + [i\in \mathcal O_0] - E_i.$$

The $\beta$-set $Z$ for $\nu^t$ of length $\#\lambda$ consists of blocks of sequences of the form

$$z_i+[i\in \mathcal O_0]-E_i+C_i+(0,1,\dots,a_i-1)$$

where $C_i = \sum_{j<i}a_j$.
Note that

\begin{align*}
&z_i + [i\in \mathcal O_0]-E_i+C_i \\
\equiv &[r_i=2,3] + [i\in \mathcal S, \lambda_i \text{ is odd}] + [i\in \mathcal S, \lambda_i \text{ is even}] + [i\in \mathcal S] \pmod 2 \\
\equiv &[r_i = 2,3] \pmod 2.
\end{align*}

Thus the $\beta$-set for $B_1^2$, which we denote $S_1^2$, consists of blocks of sequences of the form

\begin{equation}
\begin{cases}
q_i+\frac12 \Delta_i+(0,1,\dots,a_i/2-1) & \mbox{ if $r_i=0$} \\
q_i+\frac12\Delta_i+(0) & \mbox{ if $r_i=1$} \\
q_i+\frac12 \Delta_i+ 1 +(0,1,\dots,a_i/2-1) & \mbox{ if $r_i=2$} \\
\emptyset & \mbox{ if $r_i=3$}
\end{cases}
\end{equation}

and the $\beta$-set for $B_2^2$, which we denote $S_2^2$, consists of blocks of sequences of the form

\begin{equation}
\begin{cases}
q_i+\frac12\Delta_i+(0,1,\dots,a_i/2-1) & \mbox{ if $r_i=0$} \\
\emptyset & \mbox{ if $r_i=1$} \\
q_i+\frac12\Delta_i+(0,1,\dots,a_i/2-1) & \mbox{ if $r_i=2$} \\
q_i+\frac12\Delta_i+(0) & \mbox{ if $r_i=3$.}
\end{cases}
\end{equation}

where $\Delta_i = [i\in \mathcal O_0]+C_i-E_i$ and $1\le i\le k$.
The partitions $B_1^2,B_2^2$ can then be obtained by applying $\mathcal P$ to the $\beta$-sets.
Note that $|S_1^2|-|S_2^2|=d$.

Let us now produce a bipartition $B_1$ satisfying $\mathcal U_{x,J,1}(B_1) \in \phi_1$ which, we will see, satisfies $B= B^1+B^2$.
First suppose such a $B^1$ exists.
Let $S = \mathcal D^{-1}(B^1)\in \mathcal Y_{d,m}^1$.
Using Section <ref> (3), we can compute $S^{sp}$ from the condition $\WF(\mathcal U_{x,J,1}(B^1)) = \mu(\lambda)$.
A direct computation shows that

$$\mu^t = (E_1^{a_1},E_2^{a_2},\dots,E_k^{a_k}).$$

Taking a $\beta$-set of length $\#\lambda$ one computes that $S^{sp} = (X,Y)$ where $X$ consists of blocks of sequences of the form

\begin{equation}
\begin{cases}
\emptyset & \mbox{ if $i\in \mathcal O_0$} \\
(C_i+E_i)/2 + (0) & \mbox{ if $i\in \mathcal O_1$} \\
(C_i+E_i)/2 + (0,1,\dots,a_i/2-1) & \mbox{ if $\lambda_i$ is even}
\end{cases}
\end{equation}

and $Y$ consists of blocks of sequences of the form

\begin{equation}
\begin{cases}
(C_i+E_i-1)/2 + (0) & \mbox{ if $i\in \mathcal O_0$} \\
\emptyset & \mbox{ if $i\in \mathcal O_1$} \\
(C_i+E_i)/2 + (0,1,\dots,a_i/2-1) & \mbox{ if $\lambda_i$ is even.}
\end{cases}
\end{equation}

Thus we know that if such a $S$ exists, it must have entries coming from the above special symbol.
We now define such an $S$.
Let $S=(S_1^1,S_2^1)$ where $S_1^1$ consists of blocks of sequences of the form

\begin{equation}
\begin{cases}
(C_i+E_i-[i\in \mathcal O_0])/2 + (0) & \mbox{ if $r_i=1$} \\
\emptyset & \mbox{ if $r_i=3$} \\
(C_i+E_i)/2 + (0,1,\dots,a_i/2-1) & \mbox{ if $r_i=0,2$}
\end{cases}
\end{equation}

and $S_2^1$ consists of blocks of sequences of the form

\begin{equation}
\begin{cases}
\emptyset & \mbox{ if $r_i=1$} \\
(C_i+E_i-[i\in \mathcal O_0])/2 + (0) & \mbox{ if $r_i=3$} \\
(C_i+E_i)/2 + (0,1,\dots,a_i/2-1) & \mbox{ if $r_i=0,2$.}
\end{cases}
\end{equation}

Note as a sanity check that $C_i+E_i \equiv [i\in \mathcal S] + [i\in \mathcal S,\lambda_i \text{ even}] \equiv [i\in \mathcal O_0] \pmod 2$ since $\mathcal O_0 = \{i\in \mathcal S\mid \text{$\lambda_i$ is odd}\}$ and so all the entries are integers.
It is clear that $S$ has the same entries as $S^{sp}$ and has defect $d$.
To show that $S$ lies in $\mathcal Y_{d,m}^1$ it suffices to check that $X',Y'$ are both strictly increasing.
Clearly within each block the sequence is strictly increasing so we only need to compare endpoints of blocks.
We prove this for $S_1^1$. The proof for $S_2^1$ is analogous.
Let $o_i = \lfloor (C_i+E_i)/2\rfloor = (C_i+E_i-[i\in\mathcal O_0])/2$, $e_i=(C_i+E_i)/2$ and $e_i' = (C_{i+1}+E_i)/2-1$.
Here $e_i$ and $e_i'$ are the entries at the start and at the end of the $i$th block when $\lambda_i$ is even.
Let $i$ and $j$ index consecutive blocks of $S_1^1$ (so $i<j$ and $r_l=3$ for all $i<l<j$).
There are a number of cases to consider:

* $r_{i}=1,r_j=1$ $\implies$ $o_{i}<o_j$,

* $r_{i}=1,r_j\in\{0,2\}$ $\implies$ $o_{i}<e_j$,

* $r_i\in\{0,2\},r_j=1$ $\implies$ $e_{i}'<o_j$,

* $r_i\in\{0,2\},r_j\in\{0,2\}$ $\implies$ $e_{i}'<e_j$.

We prove each case as follows:

* Note that $C_j = C_i + (j-i)$ and so $o_i<o_j$ certainly holds if $j-i\ge 2$.
Thus it remains to verify the case when $j=i+1$.
Suppose $j=i+1$. Then $C_j=C_i+1$.
Moreover, since $r_{j-1}=r_i=r_j$ we have $\xi_j = 1$ and so $E_j=E_i+2[j\in \mathcal S]$.

$$o_j = o_i + (1+2[j\in \mathcal S]+[j-1\in \mathcal O_0]-[j\in \mathcal O_0])/2 = o_i+1>o_i$$

as required.

* By a similar reduction to the one performed in the previous case we may assume that $j=i+1$.
Then $C_j = C_i+1$ and

$$E_j = E_i + [j\in \mathcal S] = E_i + 1-[j-1\in \mathcal O_0].$$

$$e_j = (C_j+E_j)/2 = (C_i+E_i-[i\in \mathcal O_0])/2+1>o_i$$

as required.

* Suppose $j-i\ge 2$. Then $C_j = C_{i+1}+(j-i-1)\ge C_{i+1}+1$ and $E_j \ge E_i$ so

$$o_j = (C_j+E_j-[j\in \mathcal O_0])/2 \ge (C_{i+1}+E_i+1-[j\in \mathcal O_0])/2 > e_i'.$$

If $j=i+1$ then $E_j = E_i+[j\in \mathcal O_0]$ so $o_j = e_i'+1 > e_i'$ as required.

* Since $j>i$ we have $C_j\ge C_{i+1}$ and $E_j\ge E_i$ and so

$$e_i' \le (C_j+E_j)/2-1 < e_j.$$

It thus now remains to check that $\mathcal P(S_y^1) + \mathcal P(S_y^2) = B_y$ for $y=1,2$.
For this we prove the equivalent statement that

\begin{equation}
\label{eq:sum}
S_y^1+S_y^2 = B_y + 2\cdot(0,1,2,\dots).
\end{equation}

Let $\lambda^i$ denote the partition $(\lambda_1^{a_1},\lambda_2^{a_2},\dots,\lambda_{i-1}^{a_{i-1}})$.
We note that by Proposition <ref>, $B_1$ consists of blocks of the form

\begin{equation}
\begin{cases}
q_i-d(\lambda^i) + (0,0,\dots,0) & \mbox{ if $r_i=0$} \\
q_i-d(\lambda^i) + (0) & \mbox{ if $r_i=1$} \\
q_i-d(\lambda^i)+1+ (0,0,\dots,0) & \mbox{ if $r_i=2$} \\
\emptyset & \mbox{ if $r_i=3$}
\end{cases}
\end{equation}

where in the $r_i = 0,2$ cases the blocks have length $a_i/2$, and $B_2$ consists of blocks of sequences of the form

\begin{equation}
\begin{cases}
q_i+d(\lambda^i)+(0,0,\dots,0) & \mbox{ if $r_i=0$} \\
\emptyset & \mbox{ if $r_i=1$} \\
q_i+d(\lambda^i) + (0,0,\dots,0) & \mbox{ if $r_i=2$} \\
q_i+d(\lambda^i) + (0) & \mbox{ if $r_i=3$}
\end{cases}
\end{equation}

where in the $r_i = 0,2$ cases the blocks have length $a_i/2$.
We prove Equation <ref> for $y=1$. The case $y=2$ is analogous.
Note that the block structures of $S_i^1,S_i^2$, and $B_i$ all coincide (by which we mean they have the same number of blocks and the blocks have the same size) and so we prove the equality on an arbitrary block.
For this it will be necessary to decompose the sequence $(0,1,\dots)$ into blocks of the same size.
We have for $y=1$, the sequence $(0,1,\dots)$ decomposes into the blocks

\begin{equation}
\begin{cases}
F_i + (0,1,\dots,a_i/2-1) & \mbox{ if $r_i=0$} \\
F_i + (0) & \mbox{ if $r_i=1$} \\
F_i + (0,1,\dots,a_i/2-1) & \mbox{ if $r_i=2$} \\
\emptyset & \mbox{ if $r_i=3$}
\end{cases}
\end{equation}

where $F_i = \sum_{j<i}([2\mid \lambda_j]a_j/2+[r(\lambda_j) = 1])$.
Unpacking Equation <ref> for a block with $r_i = 0,2$ we see that we must show that

\begin{align*}
&q_i+\frac12\Delta_i+[r_i=2]+(0,1,\dots,a_i/2-1)+(C_i+E_i)/2+(0,1,\dots,a_i/2-1) \\
= &q_i-d(\lambda^i)+[r_i=2]+2F_i+2(0,1,\dots,a_i/2-1)
\end{align*}

After cancelling equal terms we get that we must show

\begin{equation}
\label{eq:red}
\frac12\Delta_i + (C_i+E_i)/2 = 2F_i-d(\lambda^i).
\end{equation}

Now the left hand side simplifies to $C_i$, whereas the right hand side simplifies to

\begin{align*}
&\sum_{j<i}\left([2\mid a_j]a_j+2[r(\lambda_j)=1]\right)-\sum_{j<i}\left([r(\lambda_j)=1]-[r(\lambda_j)=3]\right) \\
= &\sum_{j<i} \left([2\mid a_j]a_j + [r(\lambda_j)=1] + [r(\lambda_j)=3] \right) = C_i.
\end{align*}

Thus we have the equality that we desire. For blocks with $r_i = 1$ we must show that

\begin{align*}
&q_i+\frac12\Delta_i+(0) + (C_i+E_i-[i\in \mathcal O_0])/2 + (0) \\
= &q_i-d(\lambda^i)+2F_i+2(0).
\end{align*}

Cancelling equal terms yields exactly Equation <ref> which we have just established.
Finally, the $r_i = 3$ case is vacuous.
This concludes our proof that condition (ii) holds when $d(\lambda) >0$.
The case $d(\lambda)\le 0$ is analogous after making the following modifications

* In terms of $B_1,B_2$ as defined above, $B = (B_2,B_1)$;

* In terms of $B_1^2,B_2^2$ as defined above, $B_2 = (B_2^2,B_1^2)$;

* In terms of $S_1^1,S_2^1$ as defined above, we define $S = (S_2^1,S_1^1)$.

Since all the terms are just flipped the proof follows through identically.

Finally we show that condition (iii) holds.
We first need a preparatory lemma.
For $r\in \{0,1,2,3\}$ and $a\ge 0$, define $\gamma(r,a)$ to be the sequence of length $a$ of the form

\begin{equation}
\begin{cases}
(0,1,0,1,\dots) & \mbox{if } r=0,2 \\
(0,0,0,0,\dots) & \mbox{if } r=1,3.
\end{cases}
\end{equation}

\begin{align*}
\gamma(\lambda) &= (\gamma(r_1,a_1),\gamma(r_2,a_2),\dots,\gamma(r_k,a_k)).
\end{align*}

Let $\lambda\in \cN_{ro}$.
Let $\Lambda\in \mathcal Y^2_{\bullet,n}$ be the symbol with $\mathcal D(\Lambda) = E(\lambda,1)$.

$$Z^*(\Lambda) =  2q(\lambda)+z+\epsilon(\lambda)-2\delta(\lambda).$$

Write $\lambda = (\lambda_1^{a_1},\dots,\lambda_k^{a_k})$ with $\lambda_1<\lambda_2<\dots<\lambda_k$.
Let $1\le n_1<\cdots<n_l\le k$ denote the indicies for which $\lambda_i$ is odd. Let $n_0 = 0$, $\lambda_0=0$, and $a_0 = 0$.
We can break $\lambda$ up into subsequences of the form $(\lambda_{n_i}^{a_{n_i}},\dots, \lambda_{n_{i+1}-1}^{a_{n_{i+1}-1}})$.
Note that for $i>0$, $a_{n_i} = 1$.
Let $A_i = \sum_{j<i}a_j$.
Then $\lambda + z$ is block-wise of the form

\begin{equation}
\label{eq:lplusz}
A_{n_i} + (\lambda_{n_i}^{a_{n_i}},\dots, \lambda_{n_{i+1}-1}^{a_{n_{i+1}-1}}) + (0,1,\dots).
\end{equation}

Since $A_{n_i} \equiv i-1 \pmod 2$ for $i>0$ and $A_{n_0} = 0$, considering this sequence in Equation <ref> modulo $2$ we see that it is of the form $i+(0,0,1,0,1,\dots) \pmod 2$ for $i>0$ and $(0,1,0,1,\dots) \pmod 2$ for $i=0$.
Thus the a-symbol consists of blocks of the form

\lambda_{n_i}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots \\
\lambda_{n_i}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots
\end{pmatrix}+\begin{pmatrix}
0, & 1, & \dots \\
0, & 1, & \dots
\end{pmatrix}$$

for $i=0$,

$$A_{n_i}/2 + \begin{pmatrix}
\lfloor\lambda_{n_i}/2\rfloor,&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots \\
&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots
\end{pmatrix}+\begin{pmatrix}
0, & 0, & 1,& \dots \\
& 1, & 2,& \dots
\end{pmatrix}$$

for $i>0$ and odd, and

$$(A_{n_i}-1)/2 + \begin{pmatrix}
&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2 & \dots \\
\lfloor\lambda_{n_i}/2\rfloor,&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots
\end{pmatrix}+\begin{pmatrix}
& 1, & 2,& \dots \\
1,& 1, & 2,& \dots
\end{pmatrix}$$

for $i>0$ and even.
The s-symbol thus consists of blocks of the form

\lambda_{n_i}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots \\
\lambda_{n_i}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots
\end{pmatrix}+\begin{pmatrix}
0, & 2, & \dots \\
0, & 2, & \dots
\end{pmatrix}$$

for $i=0$,

$$A_{n_i} + \begin{pmatrix}
\lfloor\lambda_{n_i}/2\rfloor,&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots \\
&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots
\end{pmatrix}+\begin{pmatrix}
0, & 1, & 3,& 5,& \dots \\
& 1, & 3,& 5,& \dots
\end{pmatrix}$$

for $i>0$ and odd, and

$$A_{n_i} + \begin{pmatrix}
&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2 & \dots \\
\lfloor\lambda_{n_i}/2\rfloor,&\lambda_{n_i+1}/2,& \dots, & \lambda_{n_{i+1}-1}/2, & \dots
\end{pmatrix}+\begin{pmatrix}
& 1, & 3,& 5,& \dots \\
0,& 1, & 3,& 5,& \dots
\end{pmatrix}$$

for $i>0$ and even.
Thus as a non-decreasing sequence, the s-symbol is equal to

$$\lfloor\lambda/2\rfloor + z - \gamma(\lambda).$$

But $\lfloor \lambda/2\rfloor = 2\lfloor \lambda/4\rfloor + [r=2,3]$.
Since $[r=2,3] - \gamma(\lambda) = \epsilon(\lambda)-2\delta(\lambda)$ we get that

$$\lfloor\lambda/2\rfloor + z - \gamma(\lambda) = 2q(\lambda)+z+\epsilon(\lambda)-2\delta(\lambda).$$

We handle the case $d(\lambda)>0$ first.
Suppose $J\in \mathcal J_x$ is maximal and $F\in \mathrm{Irr}(W_{x,J})$ is such that

\begin{equation}
\label{eq:res}
\Hom_{W_{x,J}}(F, E(\OO^\vee,\rho)|_{W_{x,J}})\ne0.
\end{equation}

Since $J$ is maximal $W_{x,J} = W_{i}\times W_{l-i}$ for $l = (N-d(2d-1))/4$ and some $0\le i \le l$.
Let $(B^1,B^2)$ be the pair of bipartitions parameterising $F$ (so $B^1$ is a bipartition of $i$, and $B^2$ is a bipartition of $l-i$).
We wish to show that

$$d_S(J,\WF(\mathcal U_{x,J}(F),\CC)) \ge \OO^\vee.$$

By <cit.> it suffices to show that the inequality holds for the respective s-symbols viewed as non-decreasing sequences.
By Lemma <ref> the multiset of entries of the s-symbol for $E(\OO^\vee,1)$ viewed as a non-decreasing sequence is

$$2q(\lambda) + z + \epsilon(\lambda) -2\delta(\lambda).$$

Let us compute the s-symbol attached to $d_S(J,\WF(\mathcal U_{x,J}(F),\CC))$.
Let $S^i \in \mathcal X_{d(\lambda)}^{1,0}$ be the symbol associated to $B^i$.
Write $S^i = (S_1^i,S_2^i)$.
Let $\lambda_1 = \mathcal P(2S_1^{1,sp}\cup 2S_2^{1,sp}+1)$ (a special symplectic partition) and $\lambda_2 = \mathcal P(2S_1^2\cup 2S_2^2+1)$.

$$\WF(\mathcal U_{x,J,1})(B^1) = \lambda_1^t, \quad \WF(\mathcal U_{x,J,2}(B^2)) = \lambda_2^t.$$

Thus by <cit.>

$$d_S(J,\WF(\mathcal U_{x,J}(F),\CC)) = j_{W_J}^{W^\vee}(E(\lambda_1,1)\otimes E(\lambda_2,1)\otimes E(\lambda_1,1))$$

where $W_J$ is the parahoric of $W$ (the Weyl group of $\bfG$) determined by $J$.
By <cit.>, $Z^*$ of the s-symbol for the $j$-induced representation is

\begin{equation}
\label{eq:ssymb}
\end{equation}

Thus we must show that

\begin{equation}
\label{eq:ineq}
2S^{1,sp}+\widetilde{S^{2}}-z\ge 2q(\lambda) + z + \epsilon(\lambda) -2\delta(\lambda).
\end{equation}

For this we first observe that by the Littlewood-Richardson rule (c.f. <cit.>), Equation <ref> implies that $B^1+B^2 \ge B$ where $B = \rho(\lambda)$.
Let $q=q(\lambda),\epsilon=\epsilon(\lambda),\delta=\delta(\lambda)$ and $\Lambda$ be the symbol with linear presentation $(q+z-\delta,\epsilon)$.
By Proposition <ref>, $\Lambda = \rho(B) + 2(z,z)$.

$$B^1+B^2 \ge B \Rightarrow S^1+S^2 \ge \Lambda.$$

Since $\epsilon$ has exactly $d(\lambda)$ many $1$'s, there is a unique sequence $S^{i,r}$ such that $(S^{i,r},\epsilon)$ is a linear presentation for $S^i$.
Write $\Lambda^r$ for $q+z-\delta$.
We thus get that

$$S^{1,r}+S^{2,r} - \Lambda^r \ge 0.$$

By Lemma <ref> we have that

$$S^{2,sp}+\epsilon^*(S^2)\ge S^{2,r}+\epsilon.$$

In particular

$$S^{2,sp}+\epsilon^*(S^2) - S^{2,r} \ge \epsilon.$$

Finally we have by Lemma <ref> that

$$2S^{1,sp}+\widetilde{S^2}-z = 2(S^{1,sp}+S^{2,sp})+\epsilon^*(S^2)-z$$

\begin{align*}
&2S^{1,sp}+\widetilde{S^2}-z \\
= &2(S^{1,sp}+S^{2,sp})+\epsilon^*(S^2)-z \\
= &2(S^{1,sp}+S^{2,sp}-S^{1,r}-S^{2,r}+S^{1,r}+S^{2,r}-\Lambda^r+\Lambda^r)+\epsilon^*(S^2)-z \\
= &2(S^{1,sp}-S^{1,r}) + (S^{2,sp}-S^{2,r}) + (S^{2,sp}+\epsilon^*(S^2)-S^{2,r}) + 2(S^{1,r}+S^{2,r}-\Lambda) + 2q + z - 2\delta \\
\ge &2\cdot 0 + 0 + \epsilon + 2\cdot 0 + 2q + z - 2\delta \\
= &2q+z + \epsilon - 2\delta.
\end{align*}

where we use Lemma <ref> to deduce that $S^{i,sp}-S^{i,r}\ge0$.
This proves Equation <ref> as required.
For the case $d(\lambda)\le 0$ we note that $\Lambda$ and $S^2$ get replaced by their flips, $\Lambda^f$ and $(S^2)^f$.
The linear presentation for $\Lambda^f$ is just $(q+z-\delta,1-\epsilon)$.
Also $\widetilde{(S^2)^f} = 2S^{2,sp}+1-\epsilon^*(S^2)$.
In particular

$$S^{2,sp} + 1-\epsilon^*(S^2)-S^{2,r} \ge 1-\epsilon$$

where $S^{2,r}$ is now the linear presentation with respect to to $1-\epsilon$, and so the rest of the argument now follows through identically.

§.§.§ $Spin(N)$, $N$ even

When $N$ is even, $\mathcal{N}_{o,l}^*$ consists of two copies of $\mathcal{P}_{ro}(N)$ instead of one.
However the associated Hecke algebras are the same for both copies and so we treat both cases the same.
Otherwise the proof is essentially the same as the $N$ odd case and all the relevant lemmas have been proved in sufficient generality for the even case.
We do not repeat the proof, but mention the main technical differences.

* The partition $\mu(\lambda)$ is defined as

$$m_{\mu(\lambda)}(x) = \begin{cases}
2 &\mbox{if } x \text{ is odd and } 4\mid m_{\lambda^t}(x) \\
&\mbox{and } m_{\lambda^t}(x) \ne 0\\
1 & \mbox{if } x \text{ is odd and } m_{\lambda^t}(x) \text{ is}\\
& \mbox{odd}\\
0 &\mbox{otherwise.} \end{cases}$$

* $J = D_m\times A_{n-2m-1} \times D_m$.

* $d$ is even. $\omega$ is type II. $K=D_{d^2}\times A_{d(d-1)/2-1}\times D_{d^2}$.

§.§ Proof of faithfulness in exceptional types

Let $G^\vee$ be a complex exceptional group and $x\in \overline{\mathfrak C}(G^\vee)$.
Let $\theta(x) = (K,\bf E,\psi,\omega)$.
The block $x$ takes one of the following three forms:

* $L^\vee=T$ and $\mathcal E^\vee$ is the trivial local system;

* $L^\vee=G^\vee$ and $\mathcal E^\vee$ is a cuspidal local system;

* $L^\vee \ne G^\vee,T^\vee$.

The first case is covered in <cit.>, and the second case in <cit.>.
We now cover the third case.
There are very few possibilities for this case: we must have $G^\vee$ is either of type $E_6$ or $E_7$.
If $G^\vee = E_6$ there are two possibilities for $x$.
In both cases however the relative Weyl group is of type $G_2$ and $\mathcal J_x$ is the same set for both.
We can treat these two cases simultaneously.
If $G^\vee = E_7$ there is only one choice for $x$ and the relative Weyl group is of type $F_4$.
We use GAP to exhibit explict pairs $(J,\phi)\in \mathcal F_x$ that realise faithfulness.
We summarise the results in the next paragraph.

We treat the cases $E_6,E_7$ simultaneously.
Suppose $\OO^\vee \in (\cN_o^\vee)_x$.
If $\OO^\vee$ appears in Table <ref> or Table <ref> take the indicated $(J,\phi)$ (recall that $\WF(\phi,\CC)$ completely characterises $\phi$).
If $\OO^\vee$ does not appear then $\OO^\vee$ has the following properly: $\OO^\vee$ is special and $d(\OO^\vee)$ intersects $\mathbb L_{c^\omega(I_{0,K})}(\overline{\mathbb F_q})$ in a single $F_\omega$-stable special orbit which we denote $\OO_0$.
We take $(J,\phi)$ to be such that $J=I_{0,K}$ and $\WF(\phi,\CC) = \OO_0$.

$\OO^\vee$     $J$     Type of $\mathbf L_{c^\omega(J)}(\mathbb F_q)$     $\mathrm{WF}(\phi,\CC)$

$A_5$     [extended,labels=×,×,,,×,,×] E6     $A_1\times A_1$     $(2)\times (2)$

$2A_2+A_1$     [extended,labels=×,×,×,×,,×,×] E6     $A_2$     $(3)$

$(J,\phi)$ for $G^\vee = E_6$

$\OO^\vee$     $J$     Type of $\mathbf L_{c^\omega(J)}(\mathbb F_q)$     $\mathrm{WF}(\phi,\CC)$

$D_6$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(2^4)$

$E_7(a_4)$     [extended,labels=×,,×,×,×,×,,×] E7     $A_1\times \hphantom{ }^2D_4$     $(2)\times (3^21^2)$

$D_6(a_2)$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(42^2)$

$D_5(a_1)+A_1$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(431)$

$A_5+A_1$     [extended,labels=×,×,×,,×,,×,×] E7     $A_2\times A_2$     $(3)\times(3)$

$D_4+A_1$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(4^2)$

$A_3+A_2+A_1$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(53)$

$A_3+2A_1$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(62)$

$A_2+3A_1$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(71)$

$4A_1$     [extended,labels=×,×,,×,×,×,×,×] E7     $^2A_7$     $(8)$

$(J,\phi)$ for $G^\vee = E_7$

We make a brief remark on how to interpret the table.
Take for example $G^\vee = E_7, \OO^\vee = E_7(a_4)$.
$J$ takes the form

$$\dynkin[extended,labels={\times,,\times,\times,\times,\times,,\times}] E7.$$

The reductive quotient $\mathbf L_{c^\omega(J)}(\overline{\mathbb F_q})$ is of type $A_1\times D_4\times A_1$ and comes with a Frobenius $F_\omega$ that identifies the two $A_1$ factors and folds the $D_4$ factor.
Thus the $\mathbb F_q$ points $\mathbf L_{c^\omega(J)}(\mathbb F_q)$ has type $A_1\times \hphantom{ }^2D_4$.
When we say $\WF(\phi,\CC) = (2)\times (3^21^2)$ we are listing the geometric orbit in $A_1\times D_4$.
The actual geometric orbit in $\mathbf L_{c^\omega(J)}(\overline{\mathbb F_q})$ is $(2)\times (3^21^2)\times (2)$.

We remark that there appears to be a mistake in table for the generalised Springer correspondence for $E_7$ in <cit.> (and hence in GAP).
It is listed that

$$E(E_7(a_4),\epsilon'') = \chi_{8,3}, \quad E(D_5+A_1,-1) = \chi_{2,3}.$$

However, it should be

$$E(E_7(a_4),\epsilon'') = \chi_{2,3}, \quad E(D_5+A_1,-1) = \chi_{8,3}.$$

This was verified in <cit.>, where the $W(F_4)$-structure of the simple tempered modules with real infinitesimal character (and as a byproduct the generalized Springer correspondence) was computed for the graded affine Hecke algebra attached to the cuspidal local system on the principal nilpotent orbit in the Levi subgroup $(3A_1)'\subset E_7$.

§ MAIN RESULTS

Our main result is a general formula for the (canonical unramified) wavefront set of an irreducible unipotent representation with real infinitesimal character.

For this we first need an analogue of <cit.>.
Let $X$ be a unipotent representation of $\bfG^\omega(\sfk)$ and let $x = \mathsf{Gcusp}(X) = (J^\vee,\mathcal C^\vee,\mathcal E^\vee)$ and $\theta(x)
= \mathsf{Acusp}(X)=(K,\mathbf E,\psi,\omega)$ (c.f. Section <ref> and Section <ref>]).
Recall from Section <ref> that we can associate to $X$ a module $m_{\bf E}(X)$ of the Hecke algebra $\mathcal H = \mathcal H(\bfG^\omega(\sfk),\bf E)$.
Our first order of business is to describe the structure of $X^{\bfU_c(\mf o)}$ in terms of restrictions of $m_{\bf E}(X)$ to subalgebras of $\mathcal H$.

Recall from Section <ref> the definition of the local wavefront sets $^K\WF_c(X)$ for $c\in \mathcal B(\bfG^\omega,\sfk)$.

We have

$$^K\WF(X) = \max_{J}\hphantom{ }^K\WF_{c^\omega(J)}(X)$$

where $J$ ranges over $\omega$-stable subsets of $I$ containing $K$.

By <cit.>, if $X^{\bfU_{c^\omega(J)}(\mf o)} \ne 0$ then there exists a $\sigma\in \Omega$ such that $\sigma(J)\supseteq K$.
But since $^K\WF_{c^\omega(J)}$ is constant on $\Omega$-orbits the result follows from the analogue of <cit.> for canonical unramified wavefront sets (this is essentially the content of <cit.>).

Recall from Section <ref> the definition of $\mathcal J_x, \tilde{\mathcal U}_{x,J}$, $\mathcal W_x,W_x,\mathcal W_{x,J}$ and $W_{x,J}$.
Recall from Section <ref> the definition of the $\mathcal W_x$ representation $\widetilde \sigma_0(\theta^*(\mathsf m_{\mathbf E}(V))$.
Write $E$ for this representation.

We have

$$^K\WF(X) = \max\{\overline{\mathbb L}(J,\WF(\tilde{\mathcal U}_{x,J}(F),\CC)) \mid J\in \mathcal J_x,F\in \mathrm{Irr}(\mathcal W_{x,J}), \Hom_{\mathcal W^\vee_{J}}(F, E) \ne 0\}.$$

Let $J\in \mathcal J_x$ .
By <cit.> the representation $X^{\bfU_{c^\omega(J)}(\mf o)}$ of $\bfL_{c^\omega(J)}(\mf o)$ is a direct sum of representations in $\unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q))$.
Moreover $(\theta^*\circ m_{\bf E})_{q\to 1}$ furnishes a bijection between $\unip_{\bfL_{c^\omega(K)}(\mathbb F_q),\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q))$ and $\mathrm{Irr}(\mathcal W_{J})$ with inverse $\tilde{\mathcal U}_{x,J}$.
The result then follows from Equation <ref> and Lemma <ref>.

Let $X=X(s,n,\rho)$ be an irreducible unipotent $\mathbf G^\omega(\mathsf k)$-representation with $s\in T^\vee_\mathbb R$. Write $\OO^{\vee}_X = G^{\vee}n \subset \mathfrak{g}^{\vee}$. Then $^K\WF(\AZ(X))$, $\hphantom{ }^{\bar{\sfk}}\WF(\AZ(X))$ are singletons, and

\begin{align*}
^K\WF(\mathsf{AZ}(X)) &= d_A(\OO^{\vee}_X,1)\\
\hphantom{ }^{\bar{\sfk}}\WF(\AZ(X)) &= d(\OO^{\vee}_X).
\end{align*}

Arguing as in the proof of <cit.>, we can reduce to the case when $\mathbf{G}$ is simple and adjoint.

By Equation (<ref>) and the identity

\begin{equation}
\pr_1(d_A(\OO^\vee_X,1)) = d(\OO^\vee_X)
\end{equation}

the formula for $\hphantom{ }^{\bar{\sfk}}\WF(\AZ(X))$ follows from the formula for $^K\WF(\AZ(X))$. So it suffices to show that $^K\WF(\AZ(X)) = d_A(\OO^{\vee}_X,1)$.

By inverting the character identities (<ref>), we see that there is an equality in $R(\mathbf{G}^\omega(\sfk))$

\[X = Y(s,n,\rho) + \sum_{n \in \partial (G^{\vee}n')} m_{s,n',\rho'}Y(s,n',\rho')
\]

for standard modules $Y(s,n',\rho')$ and $m_{s,n',\rho'} \in \ZZ$.
Let $x = \mathsf{Gcusp}(X(s,n,\rho)) = (J^\vee,\mathcal C^\vee,\mathcal E^\vee)$ and $\theta(x) = \mathsf{Acusp}(X(s,n,\rho))=(K,\mathbf E,\psi,\omega)$.
By Remark <ref> $x$ lies in $\overline{\mathfrak C}(G^\vee)$.

If $J\in \mathcal J_x$ and $\tau\in \unip_{\bfL_{c^\omega(K)},\bf E}(\bfL_{c^\omega(J)}(\mathbb F_q))$, let $\tau|_{q\to 1}$ be the corresponding irreducible $\mathcal W_{x,J}$-representation (this was denoted $\theta^*(m_{\mathbf E}(\tau))_{q\to 1}$ in section <ref>). By Corollary <ref>,

\begin{equation}\label{eq:restr-1}
\begin{aligned}
\Hom_{\mathbf P_{c^\omega(J)}(\mathfrak o)}(\tau,Y(s,n',\rho'))&=\Hom_{W_{x,J}}(\tau_{q\to 1}, Y_{\mathbf H}(s,n',\rho')|_{W_{x}}),\\
%&=\Hom_{\mathcal W^\vee_{{J'}^\vee}}(\theta^*(m_{\mathbf E}(\tau))_{q\to 1}, H_\bullet (\mathcal P_{n},\dot{\mathcal F}^\vee)^\rho)   ,
\end{aligned}
\end{equation}

where $Y_{\mathbf H}(s,n,\rho)$ is the standard module for the graded affine Hecke algebra $\mathbf H(G^\vee,G^\vee_{J^\vee},\mathcal C^\vee,\mathcal F^\vee)$. To simplify the notation, let us denote by $\Res_{\bfL_{c^\omega(J)}(\mathbb F_q)}(X)$ the character of the parahoric restriction of an admissible representation $V$. Then we can rewrite (<ref>) in the following form:

\begin{equation}
\Res_{\bfL_{c^\omega(J)}(\mathbb F_q)}(Y(s,n',\rho'))=\Res^{W_x}_{W_{x,J}}( Y_{\mathbf H}(s,n',\rho')),
\end{equation}

by which we understand that the multiplicity of $\tau$ in the left hand side equals the multiplicity of $\tau_{q\to 1}$ in the right hand side.

By Proposition <ref>, we have the following identity of $W_x$-characters:

\[
Y_\bH(s,n',\rho')|_{W_x}=\sum_{\tilde\rho\in \widehat{A(n')}_{\CL}} m(\rho',\tilde\rho) \mu(n',\tilde\rho) + \sum_{(n'',\tilde\rho'')} m_{(n',\tilde\rho),(n'',\tilde\rho')} \mu(n'',\tilde\rho'),
\]

\[m(\rho',\tilde\rho)=\dim \Hom_{A(s,n')}[\rho',\tilde\rho|_{A(s,n')}],\]
$m_{(n',\tilde\rho),(n'',\tilde\rho')}\in \mathbb Z_{\ge 0}$, $\tilde\rho'\in \widehat{A(n'')}_{\CL}$, and $n''$ ranges over a set of representatives of $G^\vee$-orbits in $\mathcal N^\vee$ such that $n'\in\partial {G^\vee n''}$.

This means that

\begin{equation}
\Res_{\bfL_{c^\omega(J)}(\mathbb F_q)}(X)=\sum_{\tilde\rho\in \widehat{A(n)}_{\CL}} m(\rho,\tilde\rho) \mu(n,\tilde\rho)|_{W_{x,J}} + \sum_{\substack{\mu'\in\Irr(W_x)\\n\in \partial\OO^\vee(\mu',\mathbb C)}} m_{\mu'}\mu'|_{W_{x,J}},
\end{equation}

for some integers $m_{\mu'}$. Consequently (see Remark <ref>),

\begin{equation}\label{e:tensor-sign}
\Res_{\bfL_{c^\omega(J)}(\mathbb F_q)}(\AZ(X))=\sum_{\tilde\rho\in \widehat{A(n)}_{\CL}} m(\rho,\tilde\rho) ~(\mu(n,\tilde\rho)\otimes\mathrm{sgn})|_{W_{x,J}} + \sum_{\substack{\mu'\in\Irr(W_x)\\n\in \partial\OO^\vee(\mu',\mathbb C)}} m_{\mu'}~(\mu'\otimes\mathrm{sgn})|_{W_{x,J}},
\end{equation}

Notice that our normalization is such that $\mu(n,\tilde\rho)\otimes\mathrm{sgn}$ is precisely the irreducible generalized Springer representation attached to the pair $(n,\tilde\rho)$ i.e. $E(\OO^\vee_n,\tilde\rho)$.

\begin{equation}
\Res_{\bfL_{c^\omega(J)}(\mathbb F_q)}(\AZ(X))=\sum_{\tilde\rho\in \widehat{A(n)}_{\CL}} m(\rho,\tilde\rho) E(\OO^\vee_X,\tilde\rho)|_{W_{x,J}} + \sum_{\substack{\OO^\vee < \OO'^\vee,\\ (\OO'^\vee,\tilde\rho')\in (\cN_{o,l})_x}} m_{\OO'^\vee,\tilde\rho'} E(\OO^\vee_{1},\tilde\rho')|_{W_{x,J}}.
\end{equation}

Let $F$ be an irreducible constituent of $\Res_{\bfL_{c^\omega(J)}(\mathbb F_q)}(\AZ(X))$.
Then $\Hom_{W_{x,J}}(F,E(\OO'^\vee,\tilde \rho'))\ne 0$ for some $\OO'^\vee\ge \OO^\vee_X$.
By condition (iii) of faithfulness, we have

$$\overline{\mathbb L}(J,\WF(\mathcal U_{x,J}(F),\CC)) \le d_A(\OO'^\vee,1)\le d_A(\OO^\vee_X,1).$$

Thus by Lemma <ref> $^K\WF(X)\le d_A(\OO^\vee_X,1)$.
To establish equality note that there exists at least one $\tilde\rho$ such that $m(\rho,\tilde\rho)\neq 0$.
By conditions (i) and (ii) of faithfulness there exists a $J$ and $F\in \mathrm{Irr}(W_{x,J})$ such that $\Hom_{W_{x,J}}(F,E(\OO^\vee_X,\tilde\rho))\ne 0$ and

$$\overline{\mathbb L}(J,\WF(\mathcal U_{x,J}(F),\CC)) = d_A(\OO^\vee_X,1).$$

$$\Hom_{W_{x,J}}(F,E(\OO^\vee_X,\tilde\rho))\ne 0 \implies \Hom_{W_{x,J}}(F,\Res_{\bfL_{c^\omega(J)}(\mathbb F_q)}(\AZ(X)))\ne 0$$

since $m(\rho,\tilde\rho)\neq 0$ and so by Lemma <ref> we get $^K\WF(\AZ(X)) = d_A(\OO^\vee_X,1)$ as desired.

In the above proof, we have used implicitly (see (<ref>)) that the restrictions of $\AZ(X)$ are obtained from those of $X$ by tensoring with $\mathrm{sgn}$. In the Iwahori-spherical case, we presented the details in <cit.>. In the general case, the argument is identical using Kato's duality for affine Hecke algebras with unequal parameters <cit.> (see also <cit.> for graded Hecke algebras), and the known compatibility between the parabolic induction (respectively, Jacquet restriction) functors for $\bfG^\omega(\mathsf k)$-representations  in a Bernstein component and the parabolic induction (respectively, restriction) functors for the modules of the corresponding Hecke algebras (see <cit.>).

Now let $\OO^{\vee} \subset \mathfrak{g}^{\vee}$ be a nilpotent orbit. Choose an $\mathfrak{sl}(2)$-triple $(e^{\vee},f^{\vee},h^{\vee})$ with $e^{\vee} \in \OO^{\vee}$.

Let $X=X(\frac{1}{2}h^{\vee},n,\rho)$ be an irreducible unipotent $\mathbf{G}^{\omega}(\sfk)$-representation. Then

\begin{align}\label{eq:WFsetbound}
\begin{split}
d_A(\OO^{\vee}, 1) &\leq_A \hphantom{ } ^K\WF(X)\\
d(\OO^{\vee}) &\leq \hphantom{ }^{\bar{\sfk}}\WF(X)
\end{split}
\end{align}

The second inequality follows from the first by applying $\mathrm{pr}_1$ to both sides. So it suffices to show that $d_A(\OO^{\vee}, 1) \leq_A \hphantom{ } ^K\WF(X)$.

Let $X'=\AZ(X)$. Note that $X' = X(\frac{1}{2}h^{\vee},n',\rho')$, for some $n'$ and $\rho'$. By <cit.>, we have $\OO^{\vee}_{X'} = G^{\vee}n' \leq \OO^{\vee}$. In particular, $(\OO^\vee_{X'},1)\le_A(\OO^\vee,1)$ and so $d_A(\OO^\vee,1)\le_A d_A(\OO^\vee_{X'},1)$ since $d_A$ is order-reversing. But by Theorem <ref>, $d_A(\OO^\vee_{X'},1) = \hphantom{ }^K\WF(X)$. This completes the proof.

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# Data-Link: High Fidelity Manufacturing Datasets for Model2Real Transfer
under Industrial Settings

Sunny Katyara, Mohammad Mujtahid, Court Edmondson

###### Abstract

High-fidelity datasets play a pivotal role in imbuing simulators with realism,
enabling the benchmarking of various state-of-the-art deep inference models.
These models are particularly instrumental in tasks such as semantic
segmentation, classification, and localization. This study showcases the
efficacy of a customized manufacturing dataset comprising 60 classes in the
creation of a high-fidelity digital twin of a robotic manipulation
environment. By leveraging the concept of transfer learning, different 6D pose
estimation models are trained within the simulated environment using domain
randomization and subsequently tested on real-world objects to assess domain
adaptation. To ascertain the effectiveness and realism of the created data-
set, pose accuracy and mean absolute error (MAE) metrics are reported to
quantify the model2real gap.

##  I Introduction

We are living in a world where vast amounts of data hold the power to
revolutionize industries and shape the future of manufacturing. Datasets play
a pivotal role in virtually every field within today’s digital world, enabling
data-driven decision-making. In the manufacturing industry, data-sets assume a
critical position, offering invaluable insights to enhance product quality,
optimize production processes, streamline supply chains, and achieve
heightened operational efficiencies. However, creation of data-sets represents
a laborious and time-consuming endeavour, necessitating the acquisition of
high-quality, consistent, scalable, and adaptable data for process automation
and optimization, particularly in the realms of robotic grasping and
manipulation, assumes utmost significance.

The process of annotating 6D poses in data-sets for robotic grasping and
manipulation represents a labour-intensive endeavour, surpassing the
challenges encountered in 2D image labelling. To mitigate this challenge, a
viable solution lies in the utilization of synthetic data, which offers
meticulously annotated samples at a low cost for training pose estimation
models [1][2]. However, the substantial disparities between synthetic (source)
and real (target) models result in sub-optimal performance. To bridge this
gap, a promising approach emerges, combining domain randomization and photo-
realistic synthetic data [3][4], aiming to address the domain shift between
the source and target domains. In our study, we adopt the real-sim-real
transfer method as a means of domain adaptation to overcome sensor noise and
realism issues.

While certain generic data-sets, such as YCB Videos [5], MVTech AD [6], and
Dex-Net 2.0 [7], have been employed for training models in semantic
segmentation, classification, and localization, their limitations become
apparent in terms of restricted object variety and the absence of real-world
manufacturing context. Consequently, our research proposes the creation of an
extensive, high-fidelity data-set encompassing a range of 3D objects commonly
employed within the manufacturing industry. As illustrated in Fig. 1, we
discretize the captured 3D object, acquired using a high-resolution camera,
into descriptive components, comprising texture, material, shape, and inertial
dynamics. This process is facilitated by a customized neural network known as
Disc-Net, which accepts RGBD data and CAD models of the object of interest as
inputs, enabling the extraction of desired object features. The proposed Disc-
Net architecture incorporates two distinct neural network components: style
extraction, encompassing texture and material, while the other focuses on
shape and inertial parameters. These extracted features are subsequently
transferred to synthetic models within the digital twin environment, where
they are annotated using a standard bounding box annotator [8]. The resulting
synthetic annotated data-set is then utilized for training various pose
estimation networks including PoseCNN [5], PVNet [9], and DOPE [10] and
assessing their performance within real-world settings, thereby providing a
benchmark for evaluating the realism and effectiveness of generated data-set
for sim2real transfer.

Figure 1: Data-Link pipeline for creating a high-fidelity dataset to bridge
the domain gap for moder2real transfer under industrial conditions.

In summary, this research makes following contributions:

  * •

We propose a robust pipeline that facilitates extraction of desired rendering
and physics features from real objects, enabling their seamless transfer to
synthetic models. This approach allows for low-cost augmentation and creation
of a synthetic data-set with comprehensive 6D annotations, operating under the
paradigm of domain adaptation.

  * •

• Use high-fidelity synthetic dataset to train different PoseNets within
designed digital twin under domain randomization and eventually evaluate their
performance on real world setup using pose accuracy and mean absolute error
(MAE) metrics to quantify domain gap.

Figure 2: Performance evaluation of PoseNets trained using synthetic data and
tested on real-world object instances (a) Pose accuracy is higher than 90%,
(b) MAE is less than 5% threshold.

##  II Methodology and Discussion

The primary objective of this research is to leverage 6D pose estimation
models trained on low-cost synthetic data for real-world settings, eliminating
the need for laborious fine-tuning or expensive model retraining. The proposed
methodology encompasses the extraction of style features, including texture
and material properties, using StyleNet. StyleNet is an autoencoder
architecture with an additional layer dedicated to semantic understanding of
texture and material properties derived from latent space representation. In
addition, shape and inertial parameters of 3D objects of interest are
extracted using PhysNet, which combines a modified PointNet [11] with a
customized NeRF architecture [12]. These two networks are integrated into a
composite model architecture named Disc-Net, enabling simultaneous inference.
The StyleNet layer takes RGB data as input, while the PhysNet layer processes
PointCloud data and CAD models of candidate objects, yielding descriptive
descriptors encompassing material, texture, shape, and inertial properties
from the composite model (DiscNet).

Using the extracted descriptors from the real world, virtual objects are
developed within the Unity 3D engine to establish a digital twin environment
for manipulation scenarios, serving as a domain adaptation technique with a
specific focus on bin-picking tasks. To ensure clarity and comprehensiveness,
we have modelled 60 distinct manufacturing object classes, ranging from motors
and gears to wrenches and Allen keys. These models form the basis for
generating a synthetic 6D annotated dataset, incorporating domain
randomizations. The randomizations encompass variations not only in rendering-
related factors, such as lighting, object positions, and camera locations, but
also in physics parameters, including mass, friction, inertias, and other
relevant factors. The dataset consists of 3000 training and 1800 test images,
each with dimensions of 640 $\times$ 480 pixels. These images are utilized to
train three distinct PoseNets: PoseCNN, PVNet, and DOPE. Operating on RGB
images as input, these PoseNets estimate 6D poses of objects of interest
within the scene, leveraging 2D to 3D correspondences. The estimated poses
from these models are subsequently utilized by the Moveit API for grasp
planning, with evaluation conducted in both virtual and real setups.

The experimental findings indicate that despite being trained on synthetic
data, all the networks exhibit comparable performance levels during inference
for various objects within both the digital twin and real setups. This
observation is substantiated by the outcomes presented in Fig. 2, illustrating
the pose accuracy and MAE values. However, it is important to note that the
grasp planner requires fine-tuning when adapting to real setups. This
necessity arises due to unmodeled dynamics, including object slip, external
disturbances, stochasticity, and inherent limitations in the planner’s
adaptability.

##  III Conclusion

This research proposed a novel methodology for developing high-fidelity
manufacturing data-set using domain adaptation and domain randomization
strategies. The methodology leveraged digital twins of manipulation scenarios
to eliminate the need for laborious manual 6D annotations. Our approach
encompassed not only the extraction of style features but also the
incorporation of inertial dynamics, aiming to achieve a heightened level of
realism and model plausibility in rendering and object interactions. The
evaluation results demonstrated that our method effectively bridges the domain
gap, as evidenced by lower values of MAE and higher pose accuracy rates over
sim2real tests. Importantly findings indicated that the proposed approach not
only demonstrated superior performance for larger objects but also exhibits
notable adaptability to smaller components, even when employing low-cost
camera sensors.

This research, however, addresses the reality gap in perceptual pipelines by
enabling them to adapt to diverse environments without the need for fine-
tuning. Nonetheless, it is important to note that with domain changes, the
motion planners also require tuning. In future endeavors, our objective is to
extend this work by developing a control pipeline that harnesses the
capabilities of reinforcement learning algorithms. The overarching goal is to
devise manipulation sequences that are both robust and optimal, thereby
facilitating enhanced adaptability and efficiency in manipulation tasks [13].

## Acknowledgments

Authors would like to extend their sincere gratitude to the Industrial
Steering Board (ISB) at IMR, Ireland, for generously providing the funding
required to undertake this significant research endeavor. This study aims to
address the gap between AI-driven academic research outcomes and the
prevailing traditional industrial issues.

## References

  * [1] Hinterstoisser S., Olivier P., Hauke H., Marek M., and Martin B.. ”An annotation saved is an annotation earned: Using fully synthetic training for object detection.” In Proceedings of the IEEE/CVF international conference on computer vision workshops, pp. 0-0. 2019.
  * [2] Yang Z., Xin Y., and Yi Y., ”Dsc-posenet: Learning 6dof object pose estimation via dual-scale consistency.” In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 3907-3916. 2021.
  * [3] Tremblay J., Thang T., Balakumar S., Yu X., Dieter F., and Stan B., ”Deep object pose estimation for semantic robotic grasping of household objects.” arXiv preprint arXiv:1809.10790 (2018).
  * [4] Alghonaim R., and Edward J., ”Benchmarking domain randomisation for visual sim-to-real transfer.” In 2021 IEEE International Conference on Robotics and Automation (ICRA), pp. 12802-12808. IEEE, 2021.
  * [5] Xiang Y., Tanner S., Venkatraman N., and Dieter F.. ”Posecnn: A convolutional neural network for 6d object pose estimation in cluttered scenes.” arXiv preprint arXiv:1711.00199 (2017).
  * [6] Bergmann P., Michael F., David S., and Carsten S., ”MVTec AD–A comprehensive real-world dataset for unsupervised anomaly detection.” In Proceedings of the IEEE/CVF conference on computer vision and pattern
  * [7] Mahler, J., Jacky L., Sherdil N., Michael L., Richard D., Xinyu L., Juan Aparicio O., and Ken G., ”Dex-net 2.0: Deep learning to plan robust grasps with synthetic point clouds and analytic grasp metrics.” arXiv preprint arXiv:1703.09312 (2017).
  * [8] Manhardt F., Manuel N., Sven M., Luca M., and Nassir N.. ”Cps: Class-level 6d pose and shape estimation from monocular images.” arXiv preprint arXiv:2003.05848 3, no. 8 (2020).
  * [9] Peng S., Yuan L., Qixing H., Xiaowei Z., and Hujun B.. ”Pvnet: Pixel-wise voting network for 6dof pose estimation.” In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 4561-4570. 2019.
  * [10] Tremblay J., Thang T., Balakumar S., Yu X., Dieter F., and Stan B., ”Deep object pose estimation for semantic robotic grasping of household objects.” arXiv preprint arXiv:1809.10790 (2018).
  * [11] Qi Charles R., Hao S., Kaichun M., and Leonidas J. G.. ”Pointnet: Deep learning on point sets for 3d classification and segmentation.” In Proceedings of the IEEE conference on computer vision and pattern recognition, pp. 652-660. 2017.
  * [12] Le C., Simon, H. Yu, Michelle G., Taylor H., Ruohan G., Jiajun W., Zachary M., and Mac S.. ”Differentiable physics simulation of dynamics-augmented neural objects.” IEEE Robotics and Automation Letters 8, no. 5 (2023): 2780-2787.
  * [13] Sandha, Sandeep S., Luis G., Bharathan B., Fatima A., and Mani S., ”Sim2real transfer for deep reinforcement learning with stochastic state transition delays.” In Conference on Robot Learning, pp. 1066-1083. PMLR, 2021.


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\end{thebibliography}

\end{document}
\endinput