/
cohomology_conformal_odd.tex
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cohomology_conformal_odd.tex
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\subsection[SO(2,2n-1)]{$\mathrm{SO}(2,2n-1) \sim B_n, n\geq 2$}\label{sec:conf_odd}
\subsubsection{Root system data}
\[ \alpha_i = \epsilon_i - \epsilon_{i+1}, i<n, \quad \alpha_n = \epsilon_n\]
\[ \omega_i = \epsilon_1 +\cdots+\epsilon_i, i<n, \quad \omega_n = \frac{1}{2}(\epsilon_1 +\cdots + \epsilon_n)\]
\begin{align*}
\roots &= \{ \pm \epsilon_i, \pm \epsilon_i \pm \epsilon_j | i\neq j, i,j=1\ldots n\}\\
\roots_c^+ &= \{ \epsilon_i\pm \epsilon_j | 2\leq i < j \leq n\} \cup \{ \epsilon_j|2\leq j \leq n \}\\
\roots_n^+ &= \{ \epsilon_1 \pm \epsilon_j | 2 \leq j \leq n \} \cup \{\epsilon_1\}
\end{align*}
\[\beta = \epsilon_1+\epsilon_2,\quad \rho = (n-\frac{1}{2},\ldots ,\frac{1}{2}),\quad \zeta = (1,0,\ldots,0)\]
\inserttikzfigure{diagrams/dynkin_Bn_1.tikz}{Marked Dynkin diagram for $\mathrm{SO}(2,2n-1)$}
\begin{center}\begin{threeparttable}
\begin{tabular}{CCCCC}
\text{Vertex } \lambda_a & \text{Weight } \mu_a & Q(\lambda_a) = R(\lambda_a)\tnote{1}& l(\lambda_a) \\ \hline
-(2n-p)\omega_1 +\omega_{p+1} & -(2n-p+1)\omega_1 + \omega_p & \mathrm{SU}(1,p)\tnote{2} & 1 \\
-(n+1)\omega_1 + 2\omega_n & -(n+2)\omega_1 + \omega_{n-1} & \mathrm{SU}(1,n-1) & 1 \\
0 & -2\omega_1 +\omega_2 &\mathrm{SO}(2,2n-1) & 1 \\
-(n-\frac{3}{2})\omega_1 & -(n+\frac{1}{2})\omega_1 & \mathrm{SO}(2,2n-1) & 2 \\
-(n-\frac{1}{2})\omega_1 + \omega_n & -(n+\frac{1}{2})\omega_1 +\omega_n & \mathrm{SU}(1,n-1) &1
\end{tabular}\smallskip
\begin{tablenotes}
\item [1] Except in the last row, where $R(\lambda_a)= \mathrm{SO}(2,2n-1)$.
\item [2] $1\leq p \leq n-2$
\end{tablenotes}
\caption{Vertices and root systems for $\mathrm{SO}(2,2n-1)$, $n\geq 2$}\label{tbl:so_odd}
\end{threeparttable}\end{center}
\subsubsection{Nilpotent cohomology in detail}
Scalar products of of $\rho$ with positive noncompact roots
\begin{equation}\label{eq:Bn_rho_scalar_posroots}
(\epsilon_1, \rho) = n - \frac{1}{2}, \quad (\epsilon_1 + \epsilon_j, \rho) = 2n-j, \quad (\epsilon_1 - \epsilon_j, \rho) = j - 1.
\end{equation}
\begin{enumerate}
\item $\lambda = (p-2n) \omega_1 + \omega_{p+1}$\\
The scalar products of positive noncompact roots with $\lambda+\rho$
\begin{align*}
(\epsilon_1, \lambda+\rho) &= p-n+\frac{1}{2} \\
(\epsilon_1+\epsilon_j,\lambda+\rho) &= \begin{cases}
p+2-j, & 1<j\leq p+1\\
p+1-j, & p+1 <j \leq n
\end{cases}\\
(\epsilon_1-\epsilon_j,\lambda+\rho) &= \begin{cases}
p-2n+j-1, & 1<j\leq p+1\\
p-2n+j, & p+1 <j \leq n
\end{cases}\\
\end{align*}
reveal that the set of singular roots is empty $\Psi^+_\lambda = \emptyset$ and that the set of generating roots is $\roots^+_{n,\lambda} = \{\epsilon_1 + \epsilon_j \,|\, 1<j\leq p+1 \}$. The generated root subsystem of type $A_p$ is
\[
\roots_\lambda = \{ \pm(\epsilon_1 + \epsilon_j \,|\, 1<j\leq p+1 \} \cup \{ \epsilon_i - \epsilon_j \,|\, 1 < i,j \leq p+1 \et i\neq j \}.
\]
\begin{figure}[H]
\centering
\begin{tikzpicture}
\draw (0 cm,0) -- (6 cm,0);
\draw (8 cm,0) -- (10 cm,0);
\draw[fill=white] (0 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{1} + \epsilon_{p+1}$};
\draw[fill=black] (2 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{p} - \epsilon_{p+1}$};
\draw[fill=black] (4 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{p-1} - \epsilon_{p}$};
\node (node_a) at (6 cm, 0 cm) {};
\node (node_b) at (8 cm, 0 cm) {};
\draw[fill=black] (10 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{2} - \epsilon_{3}$};
\draw [dotted] (node_a) to (node_b);
\end{tikzpicture}
\caption{The reduced Hermitian symmetric pair for $\lambda = (p-2n) \omega_1 + \omega_{p+1}$}
\end{figure}
The integral cone is in this case
\[
C = \{ a_1\omega_1 + a_{p+1}\omega_{p+1} + \cdots + a_n \omega_n \,|\, a_1 + 2( a_{p+1} + \cdots + a_{n-1}) + a_n = 0 \}
\]
and one can easily check that $\Psi^+_\lambda = \Psi^+_{\lambda+\mu}$ for all $\mu \in C$ and thus the translation principle from the section \ref{sec:translation} applies.
\begin{figure}[H]
\centering
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
\node (node_0) at (0,10) [draw,draw=none] {$\left(A - a_{n} - 2n + p,\,0,\,0,\ldots,\,0,\,0,\,a_{p+1}+1,\,a_{p+2},\ldots,\,a_{n}\right)$};
\node (node_1) at (0,8) [draw,draw=none] {$\left(A - a_{n} - 2n + p-1,\,0,\,0,\ldots,\,0,\,1,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};
\node (node_2) at (0,6) [draw,draw=none] {$\left(A - a_{n} - 2n + p-2,\,0,\,0,\ldots,\,1,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};
\node (node_3) at (0,4) [draw,draw=none] {$\left(A - a_{n} - 2n+2,\,0,\,1,\ldots,\,0,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};
\node (node_4) at (0,2) [draw,draw=none] {$\left(A - a_{n} - 2n + 1,\,1,\,0,\ldots,\,0,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};
\node (node_5) at (0,0) [draw,draw=none] {$\left(A - a_{n} - 2n + 1,\,0,\,0,\ldots,\,0,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};
\draw [black,->] (node_0) edge (node_1);
\draw [black,->] (node_1) edge (node_2);
\draw [dotted] (node_2) to (node_3);
\draw [black,->] (node_3) edge (node_4);
\draw [black,->] (node_4) edge (node_5);
%
\end{tikzpicture}
\caption{Nilpotent cohomology / BGG resolution, $A = -2(a_{p+1} + \cdots + a_{n-1})$}
\end{figure}
\item $\lambda = -(n+1)\omega_1 + 2\omega_n $\\
Scalar products of positive noncompact roots with $\lambda+\rho$
\begin{align*}
(\epsilon_1, \lambda+\rho) &= -\frac{1}{2} \\
(\epsilon_1+\epsilon_j,\lambda+\rho) &= n+1-j \\
(\epsilon_1-\epsilon_j,\lambda+\rho) &= -n-2+j
\end{align*}
show that the set of singular roots is again empty $\Psi^+_\lambda = \emptyset$ and the set of generating roots is $\roots^+_{n,\lambda} = \{\epsilon_1 + \epsilon_j \,|\, 1<j\leq n \}$. The generated root subsystem of type $A_{n-1}$ is
\[
\roots_\lambda = \{ \pm(\epsilon_1 + \epsilon_j \,|\, 1<j\leq p+1 \} \cup \{ \epsilon_i - \epsilon_j \,|\, 1 < i,j \leq p+1 \et i\neq j \}.
\]
The integral cone is in this case \[C = \{t(-\omega_1 + \omega_n) \,|\, t\in\mathbb{N}_0 \}\] and $\Psi^+_\lambda = \Psi^+_{\lambda+\mu}$ for all $\mu \in C$.
\begin{figure}[H]
\centering
\begin{tikzpicture}
\draw (0 cm,0) -- (6 cm,0);
\draw (8 cm,0) -- (10 cm,0);
\draw[fill=white] (0 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{1} + \epsilon_{n}$};
\draw[fill=black] (2.5 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{n-1} - \epsilon_{n}$};
\draw[fill=black] (5 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{n-2} - \epsilon_{n-1}$};
\node (node_a) at (6 cm, 0 cm) {};
\node (node_b) at (8 cm, 0 cm) {};
\draw[fill=black] (10 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{2} - \epsilon_{3}$};
\draw [dotted] (node_a) to (node_b);
\end{tikzpicture}
\caption{The reduced Hermitian symmetric pair for $\lambda = -(n+1)\omega_1 + 2\omega_n$}
\end{figure}
\begin{figure}[H]
\centering
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
\node (node_0) at (0,10) [draw,draw=none] {$\left(-t-n-1, 0, 0, \ldots, 0, 0, t+2\right)$};
\node (node_1) at (0,8) [draw,draw=none] {$\left(-t-n-2, 0, 0, \ldots, 0, 1, t \right)$};
\node (node_2) at (0,6) [draw,draw=none] {$\left(-t-n-3, 0, 0, \ldots, 1, 0, t \right)$};
\node (node_3) at (0,4) [draw,draw=none] {$\left(-t-2n+2, 0, 1, \ldots, 0, 0, t \right)$};
\node (node_4) at (0,2) [draw,draw=none] {$\left(-t-2n+1, 1, 0, \ldots, 0, 0, t \right)$};
\node (node_5) at (0,0) [draw,draw=none] {$\left(-t-2n+1, 0, 0, \ldots, 0, 0, t \right)$};
\draw [black,->] (node_0) edge (node_1);
\draw [black,->] (node_1) edge (node_2);
\draw [dotted] (node_2) to (node_3);
\draw [black,->] (node_3) edge (node_4);
\draw [black,->] (node_4) edge (node_5);
%
\end{tikzpicture}
\caption{Nilpotent cohomology / BGG resolution for $\lambda = -(t+n+1)\omega_1 + (t+2)\omega_n$}
\end{figure}
\item $\lambda = 0 $\\
In this case the scalar products of positive roots with $\lambda+\rho$ are of course given by \eqref{eq:Bn_rho_scalar_posroots} and there are no singular roots $\Psi^+_\lambda = \emptyset$. We have $\roots^+_{n,\lambda} = \roots^+_n$, the root subsystem is
\[
\roots_\lambda = \roots
\]
and the Kostant's formula applies.
\item $\lambda = (\frac{3}{2} - n)\omega_1 $\\
The scalar products of positive noncompact roots with $\lambda+\rho$ are
\begin{align*}
(\epsilon_1, \lambda+\rho) &= 1 \\
(\epsilon_1+\epsilon_j,\lambda+\rho) &= n+\frac{3}{2}-j \\
(\epsilon_1-\epsilon_j,\lambda+\rho) &= -n + \frac{1}{2} + j.
\end{align*}
The set of singular roots is empty $\Psi^+_\lambda = \emptyset$ and the integrality conditions of the definition \ref{def:cohomology_roots} imply that $\roots^+_{n,\lambda} = \{ \epsilon_1 \}$. It follows that
\[
\roots_\lambda = \{ \pm \epsilon_1 \}
\]
and that all the nontrivial cohomologies are contained in the table \ref{tbl:so_odd}.
\item $\lambda = -(n-\frac{1}{2})\omega_1 + \omega_n $\\
Sscalar products of positive noncompact roots with $\lambda+\rho$
\begin{align*}
(\epsilon_1, \lambda+\rho) &= \frac{1}{2} \\
(\epsilon_1+\epsilon_j,\lambda+\rho) &= n+\frac{3}{2}-j \\
(\epsilon_1-\epsilon_j,\lambda+\rho) &= -n-\frac{1}{2}+j
\end{align*}
again show that there are no singular roots $\Psi^+_\lambda = \emptyset$ and since $\epsilon_1^\vee = 2\epsilon_1$ we have $\roots^+_{n,\lambda} = \{ \epsilon_1 \}$. This yields the subsystem
\[
\roots_\lambda = \{ \pm \epsilon_1 \}
\]
which is of type $A_1$ and the nontrivial cohomologies are given by table \ref{tbl:so_odd}.
\end{enumerate}
%%%%%%%%%%%% ALL ROOTS
%
% Before we compute the cohomology we first do some preliminary calculations and write down the scalar products of positive roots $\roots^+ = \{ \epsilon_i \pm \epsilon_j \,|\, 1\leq i < j \leq n \} \cup \{ \epsilon_i \,|\, 1\leq i \leq n\}$ with $\rho$
% \begin{equation}\label{eq:Bn_rho_scalar_posroots}
% \begin{split}
% (\epsilon_i, \rho) & = n + \frac{1}{2} - i \\
% (\epsilon_i + \epsilon_j, \rho) & = n + \frac{1}{2} - i + n + \frac{1}{2} - j = 2n+1-i-j \\
% (\epsilon_i - \epsilon_j, \rho) & = n + \frac{1}{2} - i - (n + \frac{1}{2} - j) = j - i.
% \end{split}
% \end{equation}
%
% \begin{enumerate}
% \item $\lambda = (p-2n) \omega_1 + \omega_{p+1}$\\
% The scalar products of positive roots with $\lambda+\rho$
% \begin{gather*}
% (\epsilon_i, \lambda+\rho) = \begin{cases}
% p-n+\frac{1}{2}, & i=1\\
% \frac{3}{2} + n -i, & 1 < i \leq p+1 \\
% \frac{1}{2} + n - i, & p+1 < i
% \end{cases}\\
% (\epsilon_i+\epsilon_j,\lambda+\rho) = \begin{cases}
% p+2-j, & i=1, 1<j\leq p+1\\
% 2n+3-i-j,& 1<i<j\leq p+1 \\
% 2n+2 -i -j, & 1<i\leq p+1 <j \leq n \\
% 2n+1-i-j, & p+1 < i < j \leq n
% \end{cases}\\
% (\epsilon_i-\epsilon_j,\lambda+\rho) = \begin{cases}
% p+2n+j-i, & i=1, 1<j\leq p+1\\
% j-i,& 1<i<j\leq p+1 \\
% 1+j-i, & 1<i\leq p+1 <j \leq n \\
% j-i, & p+1 < i < j \leq n.
% \end{cases}\\
% \end{gather*}
% \item $\lambda = -(n+1)\omega_1 + 2\omega_n $\\
% The scalar products of positive roots with $\lambda+\rho$
% \begin{gather*}
% (\epsilon_i, \lambda+\rho) = \begin{cases}
% -\frac{1}{2}, & i = 1 \\
% n+\frac{3}{2} - i, & 1<i\leq n
% \end{cases}\\
% (\epsilon_i+\epsilon_j,\lambda+\rho) = \begin{cases}
% n+1-j, & i=1, 1<j\leq n\\
% 2n+3-i-j, & 1 < i < j \leq n
% \end{cases}\\
% (\epsilon_i-\epsilon_j,\lambda+\rho) = \begin{cases}
% -n-2+j, & i=1, 1<j\leq n \\
% j-i, & 1 < i < j \leq n
% \end{cases}
% \end{gather*}
% \item $\lambda = 0 $\\
% In this case the scalar products of positive roots with $\lambda+\rho$ are of course given by \eqref{eq:Bn_rho_scalar_posroots}.
% \item $\lambda = (\frac{3}{2} - n)\omega_1 $\\
% The scalar products of positive roots with $\lambda+\rho$
% \begin{gather*}
% (\epsilon_i, \lambda+\rho) = \\
% (\epsilon_i+\epsilon_j,\lambda+\rho) = \\
% (\epsilon_i-\epsilon_j,\lambda+\rho) =
% \end{gather*}
% \item $\lambda = -(n-\frac{1}{2})\omega_1 + \omega_n $\\
% The scalar products of positive roots with $\lambda+\rho$
% \begin{gather*}
% (\epsilon_i, \lambda+\rho) = \\
% (\epsilon_i+\epsilon_j,\lambda+\rho) = \\
% (\epsilon_i-\epsilon_j,\lambda+\rho) =
% \end{gather*}
% \end{enumerate}
\begin{figure}[H]
\centering
\input{diagrams/nroots_B5_1.tikz}
\input{diagrams/bruhat_B5_1.tikz}
\caption{Poset of noncompact roots for $\mathrm{SO}(2,9)$}
\end{figure}