For the context for this post, see yesterday's.

Index

To compute the index we use lemma 2.19 and remark 2.20 of the Kuznetsov–Polishchuk paper. Combined they say the following.

Lemma Let $\beta$ be the simple root corresponding to the chosen maximal parabolic subgroup $P$, and $\xi$ the associated fundamental weight. Let $\overline{\beta}$ be the maximal root of the same length as $\beta$ such that the coefficient of $\beta$ in the expression of $\overline{\beta}$ is 1. Then the index of $G/P$ equals \begin{equation*} i_{G/P}=\frac{(\rho,\beta+\overline{\beta})}{(\xi,\beta)}. \end{equation*}

The Sage code for this is 

def index(D, k):
  R = RootSystem(D)
  L = R.root_lattice()

  beta = L.simple_root(k)

  length = beta.to_ambient().dot_product(beta.to_ambient())
  betabar = [alpha for alpha in L.roots()
      if alpha.coefficient(k) == 1
      and alpha.to_ambient().dot_product(alpha.to_ambient()) == length][-1]

  M = R.ambient_space()
  rho = M.rho()
  xi = M.fundamental_weight(k)
  beta = beta.to_ambient()
  betabar = betabar.to_ambient()

  return rho.dot_product(beta + betabar) / xi.dot_product(beta)

By accident I ran into these indices in the literature, they are also available in the proposition on page 124 of Akhiezer's Lie group actions in complex analysis.

$P_1$ $P_2$ $P_3$ $P_4$ $P_5$ $P_6$ $P_7$ $P_8$
$\mathrm{A}_1$ 2
$\mathrm{A}_2$ 3
$\mathrm{A}_3$ 4 4
$\mathrm{A}_4$ 5 5
$\mathrm{A}_5$ 6 6 6
$\mathrm{A}_6$ 7 7 7
$\mathrm{A}_7$ 8 8 8 8
...
$\mathrm{B}_2=\mathrm{C}_2$ 3 4
$\mathrm{B}_3=\mathrm{C}_3$ 5 4 6
$\mathrm{B}_4=\mathrm{C}_4$ 7 6 5 8
$\mathrm{B}_5=\mathrm{C}_5$ 9 8 7 6 10
$\mathrm{B}_6=\mathrm{C}_6$ 11 10 9 8 7 12
$\mathrm{B}_7=\mathrm{C}_7$ 13 12 11 10 9 8 14
...
$\mathrm{D}_4$ 6 5
$\mathrm{D}_5$ 8 7 6 8
$\mathrm{D}_6$ 10 9 8 7 10
$\mathrm{D}_7$ 12 11 10 9 8 12
...
$\mathrm{E}_6$ 12 11 9 7
$\mathrm{E}_7$ 17 14 11 8 10 13 18
$\mathrm{E}_8$ 23 17 13 9 11 14 19 29
 
$\mathrm{F}_4$ 8 5 7 11
 
$\mathrm{G}_2$ 5 3

Maybe in the future I have something useful to say about these derived categories, who knows.

It's not quite true that $\mathrm{B}_i=\mathrm{C}_i$ for the index, one has to reorder the indices a bit, but I didn't want to change the layout of the table compared to the previous ones. The table is given for $\mathrm{B}_i$. For $\mathrm{C}_i$ one has to put the entry for $P_i$ at the beginning, and shift all other entries to the right.