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The relevant information is in the article of Hesselink: Polarizations in the classical groups.

Fix a parabolic $P$ and a Richardson element $u$. Let $N(u,P)$ be the the number of conjugates of $P$ that contain $u$ (the number of polarizations of $u$). In Hesselink's notation, this is $N_1(P)$.

EDIT: I realized I had misread Hesselink; $N(P)$ is not the same for all polarizations.

Theorem (Fu): A nilpotent orbit closure $\bar{O}$ has a symplectic resolution if and only if $O$ is normal and Richardson with a polarization such that $N(P)=1$.

I misunderstood what was going on in Hesselink's tables. You should look for entries that only have one conjugacy class of polarizations which have $N_1=2$ (this is stronger than what you need, but all of Hesselink's examples have this form). It looks as though the first bad examples in each series are the Richardson orbits for the stabilizer of a line in $Sp(6)$ (denoted $C_2$), the stabilizer of a 4-space in $SO(9)$ (denoted $A_3$), and the stabilizer of a 5-space in $SO(12)$ (denoted $A_4$; that one I'm less confident I got right). If I understand correctly, inducing these up should give bad examples of higher rank in these series.

The relevant information is in the article of Hesselink: Polarizations in the classical groups (Wayback Machine).

Fix a parabolic $P$ and a Richardson element $u$. Let $N(u,P)$ be the the number of conjugates of $P$ that contain $u$ (the number of polarizations of $u$). In Hesselink's notation, this is $N_1(P)$.

EDIT: I realized I had misread Hesselink; $N(P)$ is not the same for all polarizations.

Theorem (Fu): A nilpotent orbit closure $\bar{O}$ has a symplectic resolution if and only if $O$ is normal and Richardson with a polarization such that $N(P)=1$.

I misunderstood what was going on in Hesselink's tables. You should look for entries that only have one conjugacy class of polarizations which have $N_1=2$ (this is stronger than what you need, but all of Hesselink's examples have this form). It looks as though the first bad examples in each series are the Richardson orbits for the stabilizer of a line in $Sp(6)$ (denoted $C_2$), the stabilizer of a 4-space in $SO(9)$ (denoted $A_3$), and the stabilizer of a 5-space in $SO(12)$ (denoted $A_4$; that one I'm less confident I got right). If I understand correctly, inducing these up should give bad examples of higher rank in these series.

The revelant information is in the article of Hesselink: Polarizations in the classical groups.

Fix a parabolic $P$ and a Richardson element $u$. Let $N(u,P)$ be the the number of conjugates of $P$ that contain $u$ (the number of polarizations of $u$). In Hesselink's notation, this is $N_1(P)$.

EDIT: I realized I had misread Hesslink; $N(P)$ is not the same for all polarizations.

Theorem (Fu): A nilpotent orbit closure $\bar{O}$ has a symplectic resolution if and only if $O$ is normal and Richardson with a polarization such that $N(P)=1$.

I misunderstood what was going on in Hesselink's tables. You should look for entries that only have one conjugacy class of polarizations which have $N_1=2$ (this is stronger than what you need, but all of Hesselink's examples have this form). It looks as though the first bad examples in each series are the Richardson orbits for the stabilizer of a line in $Sp(6)$ (denoted $C_2$), the stabilizer of a 4-space in $SO(9)$ (denoted $A_3$), and the stabilizer of a 5-space in $SO(12)$ (denoted $A_4$; that one I'm less confident I got right). If I understand correctly, inducing these up should give bad examples of higher rank in these series.

The relevant information is in the article of Hesselink: Polarizations in the classical groups.

Fix a parabolic $P$ and a Richardson element $u$. Let $N(u,P)$ be the the number of conjugates of $P$ that contain $u$ (the number of polarizations of $u$). In Hesselink's notation, this is $N_1(P)$.

EDIT: I realized I had misread Hesselink; $N(P)$ is not the same for all polarizations.

Theorem (Fu): A nilpotent orbit closure $\bar{O}$ has a symplectic resolution if and only if $O$ is normal and Richardson with a polarization such that $N(P)=1$.

I misunderstood what was going on in Hesselink's tables. You should look for entries that only have one conjugacy class of polarizations which have $N_1=2$ (this is stronger than what you need, but all of Hesselink's examples have this form). It looks as though the first bad examples in each series are the Richardson orbits for the stabilizer of a line in $Sp(6)$ (denoted $C_2$), the stabilizer of a 4-space in $SO(9)$ (denoted $A_3$), and the stabilizer of a 5-space in $SO(12)$ (denoted $A_4$; that one I'm less confident I got right). If I understand correctly, inducing these up should give bad examples of higher rank in these series.

The revelant information is in the article of Hesselink: Polarizations in the classical groups.

Fix a parabolic $P$ and a Richardson element $u$. Let $N(u,P)$ be the the number of conjugates of $P$ that contain $u$ (the number of polarizations of $u$). In Hesselink's notation, this is $N_1(P)$.

EDIT: I realized I had misread Hesslink; $N(P)$ is not the same for all polarizations.

Theorem (Fu): A nilpotent orbit closure $\bar{O}$ has a symplectic resolution if and only if $O$ is normal and Richardson with a polarization such that $N(P)=1$.

I misunderstood what was going on in Hesselink's tables. It looks as though the first bad examples in each series are the Richardson orbits for the stabilizer of a line in $Sp(6)$, the stabilizer of a 4-space in $SO(9)$, and the stabilizer of a 5-space in $SO(12)$ (that one I'm less confident I got right). If I understand correctly, inducing these up should give bad examples of higher rank in these series.

The revelant information is in the article of Hesselink: Polarizations in the classical groups.

Fix a parabolic $P$ and a Richardson element $u$. Let $N(u,P)$ be the the number of conjugates of $P$ that contain $u$ (the number of polarizations of $u$). In Hesselink's notation, this is $N_1(P)$.

EDIT: I realized I had misread Hesslink; $N(P)$ is not the same for all polarizations.

Theorem (Fu): A nilpotent orbit closure $\bar{O}$ has a symplectic resolution if and only if $O$ is normal and Richardson with a polarization such that $N(P)=1$.

I misunderstood what was going on in Hesselink's tables. You should look for entries that only have one conjugacy class of polarizations which have $N_1=2$ (this is stronger than what you need, but all of Hesselink's examples have this form). It looks as though the first bad examples in each series are the Richardson orbits for the stabilizer of a line in $Sp(6)$ (denoted $C_2$), the stabilizer of a 4-space in $SO(9)$ (denoted $A_3$), and the stabilizer of a 5-space in $SO(12)$ (denoted $A_4$; that one I'm less confident I got right). If I understand correctly, inducing these up should give bad examples of higher rank in these series.