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cohomology_sp.tex
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cohomology_sp.tex
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\subsection[Sp(n,R)]{$\mathrm{Sp}(n,\R) \sim C_n, n \geq 2$}
\subsubsection{Root system data}
\[ \alpha_i = \epsilon_i - \epsilon_{i+1}, i< n,a_n = 2\epsilon_n, \quad \omega_i=\epsilon_1+\cdots+\epsilon_i\]
\begin{align*}
\roots &= \{\pm \epsilon_i \pm \epsilon_j | i,j = 1,\ldots, n \}\setminus\{0\}\\
\roots_c^+ &= \{ \epsilon_i-\epsilon_j | 1\leq i < j \leq n\}\\
\roots_n^+ &= \{ \epsilon_i + \epsilon_j | 1 \leq i \leq j \leq n \}
\end{align*}
\[\beta = 2\epsilon_1,\quad \rho = (n,\ldots ,1),\quad \zeta = (1,1,\ldots,1)\]
\inserttikzfigure{diagrams/dynkin_Cn_n.tikz}{Marked Dynkin diagram of $\mathrm{Sp}(n,\R)$}
The reduction points of unitarizable highest weight modules are the following integral translated cones $\lambda_a + C_a$:
Let $a=(Q,R,l)$, $R=\mathrm{Sp}(r,\R)$ and $1\leq l \leq r \leq n$. Then
\[
C_a = \{ a_r\omega_r + \cdots + a_n\omega_n \,|\, a_n=-(a_r+\cdots + a_{n-1}) \}
\]
and
\begin{gather*}
\lambda_a = \omega_q + \omega_r - (2+n-\frac{1}{2}(r+q-l+1))\omega_n\\
\mu_a= \omega_{q-l} + \omega_{r-l} - (2+n-\frac{1}{2}(r+q-l+1))\omega_n\\
1\leq q\leq r\leq n,\quad 1\leq l \leq q\\
Q(\lambda_a) = \mathrm{Sp}(q,\R),\quad R(\lambda_a)= \mathrm{Sp}(r,\R)
\end{gather*}
\subsubsection{Nilpotent cohomology in detail}
Scalar products of $\rho$ with noncompact roots
\begin{align*}
(\epsilon_i - \epsilon_j, \rho ) & = n-i+1 - (n-j+1) = j - i \\
(\epsilon_i + \epsilon_j, \rho ) & = n-i+1 + (n-j+1) = 2n+2-i-j.
\end{align*}
The $i$th coordinate of $\lambda$ with respect to the $\epsilon$-basis is
\[
(\epsilon_i, \lambda) = \begin{cases}
2+ \frac{1}{2}(r+q-l+1) - n -2, & 1\leq i \leq q \\
1 + \frac{1}{2}(r+q-l+1) - n -2, & q < i \leq r \\
\frac{1}{2}(r+q-l+1) - n -2, & r < i.
\end{cases}
\]
Computation of $(\epsilon_i - \epsilon_j, \lambda + \rho)$ for $j > i$ leads to the following table
\begin{center}
\begin{tabular}{C|CCC}
& 1 < j \leq q & q < j \leq r & r < j \leq n \\[2pt]\hline
1\leq i \leq q & j - i & 1+j-i & 2+j-i \\
q < i \leq r & & j-i & 1+j-i \\
r < i \leq n & & & j-i
\end{tabular}
\end{center}
and scalar products of the remaining positive roots $\epsilon_i + \epsilon_j$, $j\geq i$ are
\begin{center}
\begin{tabular}{C|CCC}
& 1 < j \leq q & q < j \leq r & r < j \leq n \\[2pt]\hline
1\leq i \leq q & 3+m-i-j & 2 + m -i-j & 1 + m -i-j \\
q < i \leq r & & 1 + m-i-j & m -i-j \\
r < i \leq n & & & m-1-i-j
\end{tabular}
\end{center}
where
\[
m = r+q-l.
\]
We see again that only noncompact roots can be orthogonal to $\lambda+\rho$ in accordance with the lemma \ref{lem:singular_are_noncompact}. If a positive noncompact root $\epsilon_i + \epsilon_j$ belongs to $\Psi^+_\lambda$, then
\begin{enumerate}
\item $1\leq i \leq q$
\begin{enumerate}
\item $1 \leq j \leq q$: \[ 3+r-l \leq i \leq \min \left\{ q, \frac{3+m}{2} \right \}\]
\item $q < j \leq r$: \[ 2+q-l \leq i \leq \min \{ 1+r-l, q \} \]
\item $r < j \leq n$: \[ \max \{ 1, 1+m-n \} \leq i \leq q-l \]
\end{enumerate}
\item $q < i \leq r$
\begin{enumerate}
\item $q < j \leq r$: \[ 1+q \leq i \leq \frac{1+m}{2} \]
\item $r < j \leq n$: \[ m-n \leq i < q-l \]
\end{enumerate}
\item $r < i \leq n$, \quad $r < j \leq n$: \[ \max \{ r+1, m-n-1 \} \leq i \leq \frac{m-1}{2}. \]
\end{enumerate}
The set of indices 2a is empty, because $q-l < q$; similarly the third set is empty since $r+1 > \frac{m-1}{2}$.
A singular long root exists if and only if
\[
m \text{ is odd and } (3+r \leq q+l \text{ or } q+l < 1+r)
\]
or alternatively a singular root doesn't exist if and only if
\[
m \text{ is even or } m \text{ is odd and either } q+l = 2+r \text{ or } q+l = 1+r.
\]
Two positive noncompact roots are orthogonal if and only if the intersection of their indices is empty, i.e.
\[
(\epsilon_i + \epsilon_j,\epsilon_k + \epsilon_l) = 0 \Longleftrightarrow \{i,j\} \cap \{k,l\} = \emptyset.
\]
\begin{figure}[H]
\centering
\resizebox{\textwidth}{!}{%
\input{diagrams/nroots_C5_5.tikz}
\input{diagrams/bruhat_C5_5.tikz} %
}
\caption{Poset of noncompact roots and the BGG graph for $\mathrm{Sp}(5,\R)$}
\end{figure}