presentation-Telc-2017.tex

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\title{Cohomology of unitary representations for Hermitian symmetric spaces}

\subtitle{ECI \et JČMF Workshop -- Telč 2017}

% \date{\today}

\date{}

\author{Vít Tuček}

\institute{Mathematical Institute of Charles University}

\titlegraphic{\hfill\includegraphics[height=1.7cm]{logo-en.pdf}}


\begin{document}

\metroset{titleformat frame=smallcaps}

\maketitle


\begin{frame}{Table of contents}

\setbeamertemplate{section in toc}[sections numbered]

\tableofcontents[hideallsubsections]

\end{frame}


%\begin{frame}{Animation}

% \begin{itemize}[<+- | alert@+>]

% \item \alert<4>{This is\only<4>{ really} important}

% \item Now this

% \item And now this

% \end{itemize}

%\end{frame}


\section{Hermitian symmetric spaces}


\begin{frame}{Definition -- Hermitian symmetric space}

\begin{itemize}%[<+- | alert@+>]
\item Compact symmetric space $H/K$ is called a Hermitian symmetric space if it admits $H$-invariant complex structure
\item $H/K \simeq G/P$ where $G$ is a complexification of $H$ and $P$ is a parabolic subgroup with \emph{abelian} radical
\item  Levi part of $P$ is complexification of the maximal compact subgroup $K$
\item compact roots $\Phi_c$ \et noncompact roots $\Phi_n$ \\
      \begin{center}
       $\lie{g} = \lie{p}_- \oplus \lie{k} \oplus \lie{p}_+, \quad \lie{p} = \lie{k} \oplus \lie{p}_+$
      \end{center}
%\item The noncompact dual $H^*$ is given on the level of Lie algebras by $\lie{h}^* = \lie{k} \oplus \imath \lie{m}$ where $\lie{h} = \lie{k} \oplus \lie{m}$
%\item Borel embedding $H^*/K \hookrightarrow G/P$
%\item Harish-Chandra embedding $H^*/K \hookrightarrow \lie{m}_+$ where $\lie{m}_+$ is the nilradical of $\lie{p}$
\end{itemize}

\end{frame}


\begin{frame}{Classification}

$
 I_{m,n} = \begin{pmatrix} \Id_m & 0 \\ 0 & -\Id_n \end{pmatrix} \quad J = \begin{pmatrix} 0 & \Id_n \\ -\Id_n & 0 \end{pmatrix} \quad X^\dag = \overline{X}^t
$

\vspace{0.2cm}
\begin{equation*}
\begin{aligned}
\alert{\mathrm{SU}(p,q)} &= \{ g \in \mathrm{GL}(p +q, \mathbb{C}) \,|\, g I_{p,q}g^\dag = I_{p,q} \}\\[0.5cm]
\mathrm{Sp}(n, \mathbb{C}) &= \{ g \in \mathrm{GL}(2n, \mathbb{C}) \,|\, g^t J g  = J \}\\
\alert{\mathrm{Sp}(n, \mathbb{R})} &= \mathrm{Sp}(n, \mathbb{C}) \cap \mathrm{SU}(n,n) \\[0.5cm]
\mathrm{O}(2n, \mathbb{C}) &= \{ g \in \mathrm{GL}(n, \mathbb{C}) \,|\, g^t JI_{n,n} g  = JI_{n,n} \}\\
\alert{\mathrm{SO}^*(2n)} &=  \mathrm{O}(2n, \mathbb{C}) \cap \mathrm{SU}(n,n) \\
\alert{\mathrm{SO}(2, n)} &= \{ g \in \mathrm{GL}(n, \mathbb{R})\,|\, g^t I_{2,n} g  = I_{2,n} \}
\end{aligned}
\end{equation*}

\alert{
\[
	\mathrm{E^{-25}_7 / U(1)E_6^\text{cpt} \quad E^{-14}_6 / U(1)Spin(10)} 
\]
}
\end{frame}

\begin{frame}{Classification}
\input{diagrams/dynkin_An_p.tikz}
\input{diagrams/dynkin_Cn_n.tikz}	
\input{diagrams/dynkin_Dn_n.tikz}	
\input{diagrams/dynkin_Dn_1.tikz}	
\input{diagrams/dynkin_Bn_1.tikz}
\end{frame}

\begin{frame}{Classification}
    \begin{tikzpicture}% E6 relative
        \node[nroot] (a1) [label=above:$\alpha_1$] {};
        \node[croot] (a3) [right=of a1] [label=above:$\alpha_3$] {};
        \node[croot] (a4) [right=of a3] [label=above:$\alpha_4$] {};
        \node[croot] (a5) [right=of a4] [label=above:$\alpha_5$] {};
        \node[croot] (a6) [right=of a5] [label=above:$\alpha_6$] {};
        \node[croot] (a2) [below=of a4] [label=right:$\alpha_2$] {};
        \draw (a1) to (a3) to (a4) to (a5) to (a6);
        \draw (a4) to (a2);
     \end{tikzpicture}
		
		     \begin{tikzpicture} % E7 relative
        \node[croot] (a1) [label=above:$\alpha_1$] {};
        \node[croot] (a3) [right=of a1] [label=above:$\alpha_3$] {};
        \node[croot] (a4) [right=of a3] [label=above:$\alpha_4$] {};
        \node[croot] (a5) [right=of a4] [label=above:$\alpha_5$] {};
        \node[croot] (a6) [right=of a5] [label=above:$\alpha_6$] {};
        \node[nroot] (a7) [right=of a6] [label=above:$\alpha_7$] {};
        \node[croot] (a2) [below=of a4] [label=right:$\alpha_2$] {};
        \draw (a1) to (a3) to (a4) to (a5) to (a6) to (a7);
        \draw (a4) to (a2);
     \end{tikzpicture}
\end{frame}


\begin{frame}{Invariant differential operators}
\begin{itemize}[<+- | alert@+>]
	\item For a homogeneous space $G/P$, $P$-representation $\mathbb{V}$ the associated homogeneous bundle is $\mathcal{V} = G \times_P \mathbb{V} \to G/P$
	\item \emph{invariant differential operators} between sections of two such bundles $\mathcal{V}$ and $\mathcal{W}$ must respect natural induced actions of $G$
	\begin{gather*}
\mathcal{D} \colon \Gamma^\infty(G/P, \mathcal{V}) \to \Gamma^\infty(G/P, \mathcal{W}) \\
\mathcal{D} \circ \widetilde{\rho_\mathbb{V}} =  \widetilde{\rho_\mathbb{W}} \circ \mathcal{D}.
\end{gather*}
\item (linear) differential operator $\mathcal{D}$ of order $k$ is given by a linear map from the $k$-th jet prolongation 
\[
D\colon \Gamma^\infty(G/P,\mathcal{J}^k \mathcal{V}) \to \Gamma^\infty(G/P, \mathcal{W})
\]
\item From invariance we equivalently have homomorphism $J^k \mathbb{V} \to \mathbb{W},$ where  $J^k \mathbb{V}$ denotes the algebraic jet prolongation of $\mathbb{V}.$
\item Passing to dual maps and taking the limit $k \to \infty$ we get 
\[
\Hom_{\mathfrak{p}}\left(\mathbb{W}^*, \, \mathfrak{U(g)\otimes_{U(p)}}\mathbb{V}^*\right) \simeq \Hom_{\mathfrak{g}}\left(\mathfrak{U(g)\otimes_{U(p)}}\mathbb{W}^*, \, \mathfrak{U(g)\otimes_{U(p)}}\mathbb{V}^*\right)
\]
\end{itemize}
 

\end{frame}


\begin{frame}{Invariant differential operators}
\begin{itemize}[<+- | alert@+>]
\item Classification in the homogeneous case $G/P$\\
 $\rightsquigarrow$ homomorphisms (resolutions) of parabolic Verma modules \\
 \begin{center}
 $M(\lambda) = \mathfrak{U(g)\otimes_{U(p)}}\mathbb{F}_\lambda$
 \end{center}
\item $H^i(\lie{p}_+, L) = H_i(\lie{p}_-, L) = \mathrm{Tor}_i^{\lie{p}_-}(\mathbb{C}, L)$ \et Vermas are $\lie{p}_-$-free
\item Natural extension to Cartan geometries modelled on $(G, P)$\\
 $\rightsquigarrow$ splitting operators for bundles associated to nilpotent Lie algebra cohomology
\end{itemize}
\end{frame}



\section{Unitarizable highest weight modules}
%

\begin{frame}{Classification}
\begin{itemize}
	\item Exist only for the Hermitian symmetric pairs. 
	\item Given by the simple quotient $L(\lambda)$ of $M(\lambda)$
	\item Hermitian product given by the \emph{Shapovalov form}
\end{itemize}
\end{frame}

\begin{frame}{Classification --- continued}
$\beta$ \ldots maximal non-compact root

any weight $\lambda \in \lie{h}^*$ can be written uniquely as $\lambda = \lambda_0 + z \zeta$ where $\langle \zeta, \beta^\vee \rangle = 1$ \et $\langle \lambda_0 + \rho,\beta \rangle = 0$

 %fixing the center of $\lie{k}$ pick $zeta$ such that  $\frac{2 \langle \zeta,\beta \rangle}{\langle \beta, \beta \rangle} = 1$ we can write any weight $\lambda$ uniquely as $\lambda = \lambda_0 + z \zeta$ with  $\langle \lambda_0 + \rho,\beta \rangle = 0$

\pause
set of $z \in \mathbb{C}$ for which the simple factor of Verma module $L(\lambda)$ is unitarizable:
	
\begin{center}
\begin{tikzpicture}
      \node[croot] at (2,0) [label=above:$A(\lambda_0)$] {};
      \node[croot] at (3,0) {};
      \node[croot] at (6,0) [label=above:$B(\lambda_0)$] {};
      \draw[thick] (0,0) to (2,0);
      \draw [dotted] (4,0) to (5,0);
      \draw[<->] (2,-0.5) -- (3,-0.5);
      \node at (2.5,-0.5) [label=below:$C$] {};
      \draw (1,-0.1) -- (1, 0.1);
      \node at (1, 0) [label=above:$0$] {};
  \end{tikzpicture}
\end{center}
	
	$A(\lambda_0)$, $B(\lambda_0)$ and $C(\lambda_0)$ are real numbers expressible in terms of certain root systems $Q(\lambda_0)$ and $R(\lambda_0)$ associated to $\lambda_0$
\pause	
\vspace{-0.5cm}
\begin{table}[h]
\[\begin{array}{c|ccccc}
\lie{g} & \mathrm{SU}(p,q) & \mathrm{Sp}(n,\mathbb{R}) & \mathrm{SO}^*(2n) & \mathrm{SO}(2,2n-2) & \mathrm{SO}(2,2n-1) \\\hline
C & 1 & \frac{1}{2} & 2 & n-2 & n-\frac{3}{2}
\end{array}\]
\end{table}
\end{frame}

\begin{frame}{Classification --- continued}
	    \begin{tikzpicture} % SU(p,q) ~ A_{n-1} relative
	\node[croot] (a1) [label=below:$\alpha_1$] {};
	\node[croot] (a2) [right= of a1] [label=below:$\alpha_2$] {};
	\node[croot] (a3) [right= of a2] [label=below:$\alpha_3$] {};
	\node (a4) [right= of a3] {};
	\node[nroot] (a5) [above right=of a4] [label=above:$-\beta$] {};
	\node (a6) [below right=of a5] {};
	\node[croot] (a7) [right=of a6] [label=below:$\alpha_{n-3}$] {};
	\node[croot] (a8) [right=of a7] [label=below:$\alpha_{n-2}$] {};
	\node[croot] (a9) [right=of a8] [label=below:$\alpha_{n-1}$] {};
	\draw (a1) to (a2) to (a3); \draw [dotted] (a3) to (a4);
	\draw [dotted](a6) to (a7);
	\draw (a7) to (a8) to (a9);
	\draw (a1) to (a5) to (a9);
     \end{tikzpicture}

$Q(\lambda_0)$ \ldots maximal connected subdiagram containing $-\beta$ such that its every compact simple root is orthogonal to $\lambda_0$.
\end{frame}

\begin{frame}{Classification --- continued}
\begin{theorem}[Theorem 2.10 of \cite{enright_classification_1983}]\label{thm:reduction_points}
 The first reduction point $A(\lambda_0)$ is given by\footnote{Or in other words, the number of reduction points equals the split rank of $Q(\lambda_0)$.}
 \[
   A(\lambda_0) = B(\lambda_0) - (\text{split rank of } Q(\lambda_0) -1) C.
 \]
\end{theorem}
\end{frame}

\begin{frame}{Classification --- continued}
\begin{theorem}[Theorem 2.8 of \cite{enright_classification_1983}]
 Let $\roots^+_{c,1} := \roots^+_c \cap Q(\lambda_0)$ and let $\roots^+_{c,2} := \roots^+_c \cap R(\lambda_0)$. Denote by $\rho_{c,i}$ half of the sum of roots in $\roots^+_{c,i}$.
 
 If $\lie{g}_0=\lie{so}(2,2n-1)$ and $Q(\lambda_0) \neq R(\lambda_0)$ then \[B(\lambda_0) = 1 + \frac{\langle \rho_{c,2},\beta\rangle}{\langle \beta, \beta \rangle}.\]
 In all other cases \[B(\lambda_0) = 1 + \frac{ \langle \rho_{c,1} + \rho_{c,2} , \beta \rangle}{\langle \beta, \beta \rangle}.\]
\end{theorem}
\end{frame}


\begin{frame}{Classification --- cone decomposition}

\emph{level of reduction} \ldots \\
$L(\lambda) = M(\lambda) / N(\lambda)$ where $N(\lambda) \subseteq M(\lambda)^k$ for $k\geq l$.

For $\lambda=(B - nC)\zeta + \lambda_0 \in \Lambda_r$ we have $l(\lambda) =n+1$

\pause

$a=(Q(\lambda_0), R(\lambda_0), l(\lambda_0))$

$C_a$  -- integral cone of $\lie{k}$-dominant integral elements in $\lie{h}^*_\R$ which are orthogonal to elements in $R$

The set of reduction points $\Lambda_r$ is the disjoint union of translated integral  cones $\lambda_a + C_a$.

\pause

\begin{lemma}
The cone $C_{Q, R, l}$ consists of positive integral multiples of weights of the form $\omega_i - (\omega_i, \beta^\vee) \zeta,$ where $\omega_i$ is a fundamental weight corresponding to simple root $\alpha_i$ that does not belong to $R$.
\end{lemma}
\end{frame}
%

\section{Nilpotent cohomology}

\begin{frame}{Enright's formula}

\begin{definition}\label{def:cohomology_roots}
Let $\Psi_\lambda$ be the set of roots orthogonal to $\lambda+\rho$.

\pause

 Denote by $\roots_{n,\lambda}^+$ the roots which satisfy the following conditions
 \begin{enumerate}
    \item $\alpha \in \roots_n^+$ and $(\lambda+\rho,\alpha^\vee)$ is a positive integer;
    \item $\alpha$ is orthogonal to $\Psi_\lambda$;
    \item $\alpha$ is short if there exist a long root in $\Psi_\lambda$.
 \end{enumerate}

\pause 

 Let $W_\lambda$ be the subgroup of $W$ which is generated by reflections $s_\alpha$ for $\alpha\in \roots_{n,\lambda}^+$.



\end{definition}

\end{frame}

\begin{frame}{Enright's formula -- continued}
 Let $\roots_\lambda$ be the subset of $\roots$ of elements $\beta$ with $s_\beta\in W_\lambda$ and let $\roots_{\lambda,c} = \roots_c \cap \roots_\lambda$, $\roots_{\lambda,c}^+ = \roots_{\lambda,c} \cap \roots^+$.

\pause 

$\roots_\lambda$ is a root subsystem and $(\roots_{\lambda,c}, \roots_{\lambda,n})$ is (\emph{reduced}) Hermitian symmetric pair

\pause
 
\begin{theorem}[Theorem 3.7 of \cite{davidson_differential_1991}]
 For unitarizable highest weight modules $L(\lambda)$ and for $i\in \mathbb{N}$ we have
\begin{equation*}
 H^i(\lie{p}_+, L(\lambda))\simeq \bigoplus_{w\in W^{c,i}_\lambda} F(\overline{w(\lambda+\rho)} - \rho)
\end{equation*}
where  $\overline{\lambda}$ is the unique $\roots_c^+$-dominant element in the $W_c$ orbit of $\lambda$ and  $W^{c,i}_\lambda = \{ w \in W_\lambda \, |\, w \rho \text{ is } \roots^+_{\lambda, c}\text{-dominant and } l_\lambda(w)=i \}$.
\end{theorem}

\end{frame}


\begin{frame}{Example -- nilpotent cohomology for $ \lie{so}(2,2n-2)$}

$\lie{g} = \lie{so}(2,2n-2)$ with $\lambda = (2-n)\omega_1$

$\lambda + \rho = (1,n-2,\ldots,1,0)$

 $\Psi_\lambda^+ = \{ \epsilon_1 - \epsilon_{n-1}\}$

The only noncompact root that is orthogonal to $\epsilon_1 -  \epsilon_{n-1}$ and whose scalar product with $\lambda + \rho$ is positive integral is $\alpha = \epsilon_1 + \epsilon_{n-1}$. 

Thus we get $\roots_{n,\lambda}^+ = \epsilon_1 + \epsilon_{n-1} = \roots_\lambda$. It follows that
\begin{align*}
 H^0(\lie{p}_+,L((2-n)\omega_1)) &= \mathbb{F}((2-n)\omega_1)\\
 H^1(\lie{p}_+,L((2-n)\omega_1)) &= \mathbb{F}(-n\omega_1)\\
 H^i(\lie{p}_+,L((2-n)\omega_1)) &= 0 \text{ for } i\geq 2.
\end{align*} 
\end{frame}

\begin{frame}{Example -- nilpotent cohomology for $ \lie{so}(2,2n-1)$}
Similarly for $\lie{g} = \lie{so}(2,2n-1)$ and $\lambda = (3/2 - n)\omega_1$ and we get that in that case
\begin{align*}
 H^0(\lie{p}_+,L((3/2-n)\omega_1)) &= \mathbb{F}((3/2-n)\omega_1)\\
 H^1(\lie{p}_+,L((3/2-n)\omega_1)) &= \mathbb{F}((-1/2-n)\omega_1)\\
 H^i(\lie{p}_+,L((3/2-n)\omega_1)) &= 0 \text{ for } i\geq 2.
\end{align*}
\end{frame}
%

\begin{frame}{$\mathrm{SO}(2,2n-2), n\geq 3$}
\begin{center}\begin{threeparttable}
\begin{tabular}{CCCCC}
  \text{Vertex } \lambda_a & \text{Weight } \mu_a &  Q(\lambda_a) = R(\lambda_a)& l(\lambda_a) \\ \hline
  -(2n-p-1)\omega_1+\omega_{p+1} & -(2n-p)\omega_1 + \omega_p  & \mathrm{SU}(1,p)\tnote{1} &  1 \\
  -(n+1)\omega_1 +\omega_{n-1} + \omega_n & -(n+2)\omega_1 + \omega_{n-2} & \mathrm{SU}(1,n-2) &  1 \\
  -(n-1)\omega_1 + \omega_n \tnote{2} & -n\omega_1+\omega_{n-1} &  \mathrm{SU}(1,n-1) & 1 \\
  -(n-2)\omega_1 & -n\omega_1 & \mathrm{SO}(2,2n-2) &  2\\
  0 & -2\omega_1 + \omega_2 & \mathrm{SO}(2,2n-2) &  1 \\
  -(n-1)\omega_1  + \omega_{n-1} \tnote{3} & -n\omega_1 + \omega_n &\mathrm{SU}(1,n-1) & 1
\end{tabular}\smallskip
\begin{tablenotes}
 \item [1] $1\leq p \leq n-3$ with Dynkin diagram of $R(\lambda_a)$:\\
 \begin{tikzpicture} % SO^*(2n) ~ D_n relative
	\node[nroot] (a1) [label=above:$-\beta$] {};
	\node[croot] (a2) [right= of a1] [label=above:$\alpha_2$] {};
	\node (a3) [right= of a2] {};
	\node (a4) [right= of a3] {};
	\node[croot] (a5) [right= of a4] [label=above:$\alpha_{p}$] {};
	\draw (a1) to (a2) to (a3);
	\draw [dotted] (a3) to (a4);
	\draw (a4) to (a5);
     \end{tikzpicture}
 \item [2] Dynkin diagram  of $R(\lambda_a)$:
 \begin{tikzpicture} % SO^*(2n) ~ D_n relative
	\node[nroot] (a1) [label=above:$-\beta$] {};
	\node[croot] (a2) [right= of a1] [label=above:$\alpha_2$] {};
	\node (a3) [right= of a2] {};
	\node (a4) [right= of a3] {};
	\node[croot] (a5) [right= of a4] [label=above:$\alpha_{n-1}$] {};
	\draw (a1) to (a2) to (a3);
	\draw [dotted] (a3) to (a4);
	\draw (a4) to (a5);
     \end{tikzpicture}
 \item [3] Dynkin diagram of $R(\lambda_a)$:
\begin{tikzpicture} % SO^*(2n) ~ D_n relative
	\node[nroot] (a1) [label=above:$-\beta$] {};
	\node[croot] (a2) [right= of a1] [label=above:$\alpha_2$] {};
	\node (a3) [right= of a2] {};
	\node (a4) [right= of a3] {};
	\node[croot] (a5) [right= of a4] [label=above:$\alpha_{n-2}$] {};
	\node[croot] (a6) [right= of a5] [label=above:$\alpha_{n}$] {};
	\draw (a1) to (a2) to (a3);
	\draw [dotted] (a3) to (a4);
	\draw (a4) to (a5) to (a6);
     \end{tikzpicture} 
\end{tablenotes}
\end{threeparttable}\end{center}
\end{frame}

\begin{frame}[allowframebreaks]{$\lambda =  (A - a_{n-1} - a_{n} - 2n + p + 1)\omega_1+(a_{p+1}+1)\omega_{p+1} + \sum_{i=p+2}^n a_i \omega_i$}

 
\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}  

\pagebreak

\begin{center}
 \[
 A = -2(a_{p+1} + \cdots + a_{n-2})
 \]
 
	\begin{tikzpicture}[>=latex,line join=bevel,scale=1, every node/.style={transform shape}]

  \node (node_0) at (0,10) [draw,draw=none] {$\left(A - a_{n-1} - a_{n} - 2n + p + 1,\,0,\,0,\ldots,\,0,\,0,\,a_{p+1}+1,\,a_{p+2},\ldots,\,a_{n}\right)$};

  \node (node_1) at (0,9) [draw,draw=none] {$\left(A - a_{n-1} - a_{n} - 2n + p,\,0,\,0,\ldots,\,0,\,1,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};

  \node (node_2) at (0,8) [draw,draw=none] {$\left(A - a_{n-1} - a_{n} - 2n + p-1,\,0,\,0,\ldots,\,1,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};

  \node (node_3) at (0,7) [draw,draw=none] {$\left(A - a_{n-1} - a_{n} - 2n + 1,\,0,\,1,\ldots,\,0,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};

  \node (node_4) at (0,6) [draw,draw=none] {$\left(A - a_{n-1} - a_{n} - 2n + 2,\,1,\,0,\ldots,\,0,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};
  
  \node (node_5) at (0,5) [draw,draw=none] {$\left(A - a_{n-1} - a_{n} - 2n +2,\,0,\,0,\ldots,\,0,\,0,\,a_{p+1},\,a_{p+2},\ldots,\,a_{n}\right)$};

  \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}
\end{center}
  
\end{frame}

\begin{frame}{$\lambda = -(a+b+n+1)\omega_1 + (a+1)\omega_{n-1} + (b+1)\omega_n$}
\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]{$\epsilon_{1} + \epsilon_{n-1}$};
\draw[fill=black] (2.5 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{n-2} - \epsilon_{n-1}$};
\draw[fill=black] (5 cm, 0 cm) circle (.1cm) node[below=4pt]{$\epsilon_{n-3} - \epsilon_{n-4}$};
\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}  

\begin{center}

\begin{tikzpicture}[>=latex,line join=bevel,scale=1, every node/.style={transform shape}]

  \node (node_0) at (0,10) [draw,draw=none] {$\left(-(a+b+n+1), 0, 0, \ldots, 0, 0, a+1, b+1 \right)$};

  \node (node_1) at (0,9) [draw,draw=none] {$\left(-(a+b+n+2), 0, 0, \ldots, 0, 1, a, b \right)$};

  \node (node_2) at (0,8) [draw,draw=none] {$\left(-(a+b+n+3), 0, 0, \ldots, 1, 0, a, b \right)$};

  \node (node_3) at (0,7) [draw,draw=none] {$\left(-(a+b+2n-3), 0, 1, \ldots, 0, 0, a, b \right)$};

  \node (node_4) at (0,6) [draw,draw=none] {$\left(-(a+b+2n-2), 1, 0, \ldots, 0, 0, a, b \right)$};
  
  \node (node_5) at (0,5) [draw,draw=none] {$\left(-(a+b+2n-2), 0, 0, \ldots, 0, 0, a, b \right)$};

  \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}
\end{center}
	
\end{frame}

\begin{frame}{$\lambda = -(t+n-1)\omega_1 + (t+1)\omega_n$}
For $t=0$ we obtain set of singular roots $\Psi^+_\lambda = \{ \epsilon_1-\epsilon_n\}$ and the set of generating roots $\roots^+_{n,\lambda} = \{ \epsilon_1 + \epsilon_n \}$. This gives the subsystem of type $A_1$
  \[
   \roots_\lambda = \{ \epsilon_1 + \epsilon_n, -\epsilon_1 - \epsilon_n\}
  \]
  and the only nontrivial cohomology is in degree $1$ with weight $-n\omega_1 + \omega_{n-1}$. 
\end{frame}

\begin{frame}{$\lambda = -(t+n-1)\omega_1 + (t+1)\omega_n$}
  
  For $t\geq 1$ we get no singular roots $\Psi^+_\lambda = \emptyset$ and the generated subsystem is of type $A_{n-1}.$
	
	\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}  

\end{frame}

\begin{frame}{$\lambda = -(t+n-1)\omega_1 + (t+1)\omega_n$}
\begin{center}

\begin{tikzpicture}[>=latex,line join=bevel,scale=1, every node/.style={transform shape}]

  \node (node_0) at (0,10) [draw,draw=none] {$\left(-t-n+1, 0, 0, \ldots, 0, 0, t+1\right)$};
  \node (node_1) at (0,9) [draw,draw=none] {$\left(-t-n, 0, 0, \ldots, 0, 1, t \right)$};
  \node (node_2) at (0,8) [draw,draw=none] {$\left(-t-n-1, 0, 0, \ldots, 1, 0, t \right)$};
  \node (node_3) at (0,7) [draw,draw=none] {$\left(-t-2n+4, 0, 1, \ldots, 0, 0, t \right)$};
  \node (node_4) at (0,6) [draw,draw=none] {$\left(-t-2n+3, 1, 0, \ldots, 0, 0, t \right)$};
  \node (node_5) at (0,5) [draw,draw=none] {$\left(-t-2n+3, 0, 0, \ldots, 0, 0, t \right)$};

  \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}
\end{center}
\end{frame}


\begin{frame}{$\lambda = -(t+n-1)\omega_1  + (t+1)\omega_{n-1}$}
For $t=0$ we have similarly to the previous case a singular weight with only nontrivial cohomology of degree one with weight $-n\omega_1 + \omega_n$

Fro $t\geq 1$
	\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}  
\end{frame}


\begin{frame}{$\lambda = -(t+n-1)\omega_1  + (t+1)\omega_{n-1}$}
\begin{center}
\begin{tikzpicture}[>=latex,line join=bevel,scale=1, every node/.style={transform shape}]

  \node (node_0) at (0,10) [draw,draw=none] {$\left(-t-n+1, 0, 0, \ldots, 0, t+1, 0\right)$};
  \node (node_1) at (0,9) [draw,draw=none] {$\left(-t-n, 0, 0, \ldots,  0, t, 1 \right)$};
  \node (node_2) at (0,8) [draw,draw=none] {$\left(-t-n-1, 0, 0, \ldots, 1,  t, 0 \right)$};
  \node (node_3) at (0,7) [draw,draw=none] {$\left(-t-2n+4, 0, 1, \ldots, 0,  t, 0 \right)$};
  \node (node_4) at (0,6) [draw,draw=none] {$\left(-t-2n+3, 1, 0, \ldots, 0,  t, 0 \right)$};
  \node (node_5) at (0,5) [draw,draw=none] {$\left(-t-2n+3, 0, 0, \ldots, 0,  t, 0 \right)$};

  \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}
\end{center}
\end{frame}

\begin{frame}{Example -- scalar products with positive roots}
$\mathrm{SO}^*(16)$: $\lambda = \left(a_{5} + 1\right)\omega_{5} + a_6\omega_6 + a_7\omega_7 - \left(2 \, a_{5} + 2 \, a_{6} + a_{7} + 8\right)\omega_{8}$
\begin{tikzpicture}[>=latex,line join=bevel,scale=0.65, every node/.style={transform shape}]
%%
\node (node_2) at (308.5bp,61.5bp) [draw,draw=none] {$ \epsilon_{3} + \epsilon_{5}$};
  \node (node_4) at (173.5bp,8.5bp) [draw,draw=none] {$\epsilon_{2} + \epsilon_{7}$};
  \node (node_3) at (269.5bp,8.5bp) [draw,draw=none] {$\epsilon_{3} + \epsilon_{6}$};
  \node (node_9) at (200.5bp,220.5bp) [draw,draw=none] {$\epsilon_{1} + \epsilon_{4}$};
  \node (node_8) at (149.5bp,114.5bp) [draw,draw=none] {$ \epsilon_{1} + \epsilon_{6}$};
  \node (node_7) at (221.5bp,61.5bp) [draw,draw=none] {$\epsilon_{2} + \epsilon_{6}$};
  \node (node_6) at (236.5bp,114.5bp) [draw,draw=none] {$\epsilon_{2} + \epsilon_{5}$};
  \node (node_5) at (308.5bp,114.5bp) [draw,draw=none] {$ \epsilon_{3} + \epsilon_{4}$};
  \node (node_14) at (236.5bp,326.5bp) [draw,draw=none] {$\epsilon_{1} + \epsilon_{2}$};
  \node (node_13) at (348.5bp,8.5bp) [draw,draw=none] {$\epsilon_{4} + \epsilon_{5})$};
  \node (node_12) at (108.5bp,61.5bp) [draw,draw=none] {$ \epsilon_{1} + \epsilon_{7}$};
  \node (node_11) at (200.5bp,167.5bp) [draw,draw=none] {$\epsilon_{1} + \epsilon_{5}$};
  \node (node_10) at (55.5bp,8.5bp) [draw,draw=none] {$ \epsilon_{1} + \epsilon_{8}$};
  \node (node_1) at (272.5bp,167.5bp) [draw,draw=none] {$\epsilon_{2} + \epsilon_{4}$};
  \node (node_0) at (272.5bp,220.5bp) [draw,draw=none] {$\epsilon_{2} + \epsilon_{3}$};
  \node (node_15) at (236.5bp,273.5bp) [draw,draw=none] {$\epsilon_{1} + \epsilon_{3}$};
  \draw [black,->] (node_12) ..controls (120.42bp,77.332bp) and (129.52bp,88.646bp)  .. (node_8);
  \draw [black,->] (node_8) ..controls (164.56bp,130.56bp) and (176.35bp,142.35bp)  .. (node_11);
  \draw [black,->] (node_10) ..controls (71.149bp,24.558bp) and (83.403bp,36.35bp)  .. (node_12);
  \draw [black,->] (node_6) ..controls (246.92bp,130.26bp) and (254.79bp,141.41bp)  .. (node_1);
  \draw [black,->] (node_1) ..controls (250.71bp,183.93bp) and (232.91bp,196.54bp)  .. (node_9);
  \draw [black,->] (node_3) ..controls (280.84bp,24.332bp) and (289.49bp,35.646bp)  .. (node_2);
  \draw [black,->] (node_4) ..controls (187.6bp,24.483bp) and (198.55bp,36.114bp)  .. (node_7);
  \draw [black,->] (node_9) ..controls (210.92bp,236.26bp) and (218.79bp,247.41bp)  .. (node_15);
  \draw [black,->] (node_0) ..controls (262.08bp,236.26bp) and (254.21bp,247.41bp)  .. (node_15);
  \draw [black,->] (node_11) ..controls (200.5bp,182.81bp) and (200.5bp,193.03bp)  .. (node_9);
  \draw [black,->] (node_7) ..controls (225.73bp,76.88bp) and (228.78bp,87.262bp)  .. (node_6);
  \draw [black,->] (node_13) ..controls (336.87bp,24.332bp) and (327.99bp,35.646bp)  .. (node_2);
  \draw [black,->] (node_15) ..controls (236.5bp,288.81bp) and (236.5bp,299.03bp)  .. (node_14);
  \draw [black,->] (node_5) ..controls (298.08bp,130.26bp) and (290.21bp,141.41bp)  .. (node_1);
  \draw [black,->] (node_1) ..controls (272.5bp,182.81bp) and (272.5bp,193.03bp)  .. (node_0);
  \draw [black,->] (node_7) ..controls (199.71bp,77.935bp) and (181.91bp,90.539bp)  .. (node_8);
  \draw [black,->] (node_2) ..controls (286.71bp,77.935bp) and (268.91bp,90.539bp)  .. (node_6);
  \draw [black,->] (node_6) ..controls (226.08bp,130.26bp) and (218.21bp,141.41bp)  .. (node_11);
  \draw [black,->] (node_3) ..controls (255.4bp,24.483bp) and (244.45bp,36.114bp)  .. (node_7);
  \draw [black,->] (node_2) ..controls (308.5bp,76.805bp) and (308.5bp,87.034bp)  .. (node_5);
  \draw [black,->] (node_4) ..controls (154.02bp,24.784bp) and (138.37bp,37.061bp)  .. (node_12);
%
\end{tikzpicture}
\end{frame}


\begin{frame}{Example -- scalar products with positive roots}
$\mathrm{SO}^*(16)$: $\lambda = \left(a_{5} + 1\right)\omega_{5} + a_6\omega_6 + a_7\omega_7 - \left(2 \, a_{5} + 2 \, a_{6} + a_{7} + 8\right)\omega_{8}$
\begin{tikzpicture}[>=latex,line join=bevel,scale=0.65, every node/.style={transform shape}]
%%
\node (node_2) at (308.5bp,61.5bp) [draw,draw=none] {$(2, \epsilon_{3} + \epsilon_{5})$};
  \node (node_4) at (173.5bp,8.5bp) [draw,draw=none] {$(-a_{5} - a_{6}, \epsilon_{2} + \epsilon_{7})$};
  \node (node_3) at (269.5bp,8.5bp) [draw,draw=none] {$(-a_{5}, \epsilon_{3} + \epsilon_{6})$};
  \node (node_9) at (200.5bp,220.5bp) [draw,draw=none] {$(5, \epsilon_{1} + \epsilon_{4})$};
  \node (node_8) at (149.5bp,114.5bp) [draw,draw=none] {$(-a_{5} + 2, \epsilon_{1} + \epsilon_{6})$};
  \node (node_7) at (221.5bp,61.5bp) [draw,draw=none] {$(-a_{5} + 1, \epsilon_{2} + \epsilon_{6})$};
  \node (node_6) at (236.5bp,114.5bp) [draw,draw=none] {$(3, \epsilon_{2} + \epsilon_{5})$};
  \node (node_5) at (308.5bp,114.5bp) [draw,draw=none] {$(3, \epsilon_{3} + \epsilon_{4})$};
  \node (node_14) at (236.5bp,326.5bp) [draw,draw=none] {$(7, \epsilon_{1} + \epsilon_{2})$};
  \node (node_13) at (348.5bp,8.5bp) [draw,draw=none] {$(1, \epsilon_{4} + \epsilon_{5})$};
  \node (node_12) at (108.5bp,61.5bp) [draw,draw=none] {$(-a_{5} - a_{6} + 1, \epsilon_{1} + \epsilon_{7})$};
  \node (node_11) at (200.5bp,167.5bp) [draw,draw=none] {$(4, \epsilon_{1} + \epsilon_{5})$};
  \node (node_10) at (55.5bp,8.5bp) [draw,draw=none] {$(-a_{5} - a_{6} - a_{7}, \epsilon_{1} + \epsilon_{8})$};
  \node (node_1) at (272.5bp,167.5bp) [draw,draw=none] {$(4, \epsilon_{2} + \epsilon_{4})$};
  \node (node_0) at (272.5bp,220.5bp) [draw,draw=none] {$(5, \epsilon_{2} + \epsilon_{3})$};
  \node (node_15) at (236.5bp,273.5bp) [draw,draw=none] {$(6, \epsilon_{1} + \epsilon_{3})$};
  \draw [black,->] (node_12) ..controls (120.42bp,77.332bp) and (129.52bp,88.646bp)  .. (node_8);
  \draw [black,->] (node_8) ..controls (164.56bp,130.56bp) and (176.35bp,142.35bp)  .. (node_11);
  \draw [black,->] (node_10) ..controls (71.149bp,24.558bp) and (83.403bp,36.35bp)  .. (node_12);
  \draw [black,->] (node_6) ..controls (246.92bp,130.26bp) and (254.79bp,141.41bp)  .. (node_1);
  \draw [black,->] (node_1) ..controls (250.71bp,183.93bp) and (232.91bp,196.54bp)  .. (node_9);
  \draw [black,->] (node_3) ..controls (280.84bp,24.332bp) and (289.49bp,35.646bp)  .. (node_2);
  \draw [black,->] (node_4) ..controls (187.6bp,24.483bp) and (198.55bp,36.114bp)  .. (node_7);
  \draw [black,->] (node_9) ..controls (210.92bp,236.26bp) and (218.79bp,247.41bp)  .. (node_15);
  \draw [black,->] (node_0) ..controls (262.08bp,236.26bp) and (254.21bp,247.41bp)  .. (node_15);
  \draw [black,->] (node_11) ..controls (200.5bp,182.81bp) and (200.5bp,193.03bp)  .. (node_9);
  \draw [black,->] (node_7) ..controls (225.73bp,76.88bp) and (228.78bp,87.262bp)  .. (node_6);
  \draw [black,->] (node_13) ..controls (336.87bp,24.332bp) and (327.99bp,35.646bp)  .. (node_2);
  \draw [black,->] (node_15) ..controls (236.5bp,288.81bp) and (236.5bp,299.03bp)  .. (node_14);
  \draw [black,->] (node_5) ..controls (298.08bp,130.26bp) and (290.21bp,141.41bp)  .. (node_1);
  \draw [black,->] (node_1) ..controls (272.5bp,182.81bp) and (272.5bp,193.03bp)  .. (node_0);
  \draw [black,->] (node_7) ..controls (199.71bp,77.935bp) and (181.91bp,90.539bp)  .. (node_8);
  \draw [black,->] (node_2) ..controls (286.71bp,77.935bp) and (268.91bp,90.539bp)  .. (node_6);
  \draw [black,->] (node_6) ..controls (226.08bp,130.26bp) and (218.21bp,141.41bp)  .. (node_11);
  \draw [black,->] (node_3) ..controls (255.4bp,24.483bp) and (244.45bp,36.114bp)  .. (node_7);
  \draw [black,->] (node_2) ..controls (308.5bp,76.805bp) and (308.5bp,87.034bp)  .. (node_5);
  \draw [black,->] (node_4) ..controls (154.02bp,24.784bp) and (138.37bp,37.061bp)  .. (node_12);
%
\end{tikzpicture}
\end{frame}



\begin{frame}{Example -- scalar products with positive roots}
\begin{tikzpicture}[>=latex,line join=bevel,scale=0.7, every node/.style={transform shape}]
%%
\node (node_2) at (308.5bp,61.5bp) [draw,draw=none] {$(2, \epsilon_{3} + \epsilon_{5})$};
  \node (node_4) at (173.5bp,8.5bp) [draw,draw=none] {$(\alert{\bm{-a_{5} - a_{6}}}, \epsilon_{2} + \epsilon_{7})$};
  \node (node_3) at (269.5bp,8.5bp) [draw,draw=none] {$(\alert{\bm{-a_{5}}}, \epsilon_{3} + \epsilon_{6})$};
  \node (node_9) at (200.5bp,220.5bp) [draw,draw=none] {$(5, \epsilon_{1} + \epsilon_{4})$};
  \node (node_8) at (149.5bp,114.5bp) [draw,draw=none] {$(\alert{\bm{-a_{5} + 2}}, \epsilon_{1} + \epsilon_{6})$};
  \node (node_7) at (221.5bp,61.5bp) [draw,draw=none] {$(\alert{\bm{-a_{5} + 1}}, \epsilon_{2} + \epsilon_{6})$};
  \node (node_6) at (236.5bp,114.5bp) [draw,draw=none] {$(3, \epsilon_{2} + \epsilon_{5})$};
  \node (node_5) at (308.5bp,114.5bp) [draw,draw=none] {$(3, \epsilon_{3} + \epsilon_{4})$};
  \node (node_14) at (236.5bp,326.5bp) [draw,draw=none] {$(7, \epsilon_{1} + \epsilon_{2})$};
  \node (node_13) at (348.5bp,8.5bp) [draw,draw=none] {$(1, \epsilon_{4} + \epsilon_{5})$};
  \node (node_12) at (108.5bp,61.5bp) [draw,draw=none] {$(\alert{\bm{-a_{5} - a_{6} + 1}}, \epsilon_{1} + \epsilon_{7})$};
  \node (node_11) at (200.5bp,167.5bp) [draw,draw=none] {$(4, \epsilon_{1} + \epsilon_{5})$};
  \node (node_10) at (55.5bp,8.5bp) [draw,draw=none] {$(\alert{\bm{-a_{5} - a_{6} - a_{7}}}, \epsilon_{1} + \epsilon_{8})$};
  \node (node_1) at (272.5bp,167.5bp) [draw,draw=none] {$(4, \epsilon_{2} + \epsilon_{4})$};
  \node (node_0) at (272.5bp,220.5bp) [draw,draw=none] {$(5, \epsilon_{2} + \epsilon_{3})$};
  \node (node_15) at (236.5bp,273.5bp) [draw,draw=none] {$(6, \epsilon_{1} + \epsilon_{3})$};
  \draw [black,->] (node_12) ..controls (120.42bp,77.332bp) and (129.52bp,88.646bp)  .. (node_8);
  \draw [black,->] (node_8) ..controls (164.56bp,130.56bp) and (176.35bp,142.35bp)  .. (node_11);
  \draw [black,->] (node_10) ..controls (71.149bp,24.558bp) and (83.403bp,36.35bp)  .. (node_12);
  \draw [black,->] (node_6) ..controls (246.92bp,130.26bp) and (254.79bp,141.41bp)  .. (node_1);
  \draw [black,->] (node_1) ..controls (250.71bp,183.93bp) and (232.91bp,196.54bp)  .. (node_9);
  \draw [black,->] (node_3) ..controls (280.84bp,24.332bp) and (289.49bp,35.646bp)  .. (node_2);
  \draw [black,->] (node_4) ..controls (187.6bp,24.483bp) and (198.55bp,36.114bp)  .. (node_7);
  \draw [black,->] (node_9) ..controls (210.92bp,236.26bp) and (218.79bp,247.41bp)  .. (node_15);
  \draw [black,->] (node_0) ..controls (262.08bp,236.26bp) and (254.21bp,247.41bp)  .. (node_15);
  \draw [black,->] (node_11) ..controls (200.5bp,182.81bp) and (200.5bp,193.03bp)  .. (node_9);
  \draw [black,->] (node_7) ..controls (225.73bp,76.88bp) and (228.78bp,87.262bp)  .. (node_6);
  \draw [black,->] (node_13) ..controls (336.87bp,24.332bp) and (327.99bp,35.646bp)  .. (node_2);
  \draw [black,->] (node_15) ..controls (236.5bp,288.81bp) and (236.5bp,299.03bp)  .. (node_14);
  \draw [black,->] (node_5) ..controls (298.08bp,130.26bp) and (290.21bp,141.41bp)  .. (node_1);
  \draw [black,->] (node_1) ..controls (272.5bp,182.81bp) and (272.5bp,193.03bp)  .. (node_0);
  \draw [black,->] (node_7) ..controls (199.71bp,77.935bp) and (181.91bp,90.539bp)  .. (node_8);
  \draw [black,->] (node_2) ..controls (286.71bp,77.935bp) and (268.91bp,90.539bp)  .. (node_6);
  \draw [black,->] (node_6) ..controls (226.08bp,130.26bp) and (218.21bp,141.41bp)  .. (node_11);
  \draw [black,->] (node_3) ..controls (255.4bp,24.483bp) and (244.45bp,36.114bp)  .. (node_7);
  \draw [black,->] (node_2) ..controls (308.5bp,76.805bp) and (308.5bp,87.034bp)  .. (node_5);
  \draw [black,->] (node_4) ..controls (154.02bp,24.784bp) and (138.37bp,37.061bp)  .. (node_12);
%
\end{tikzpicture}
\end{frame}


{\setbeamercolor{palette primary}{fg=black, bg=white}

\begin{frame}[standout]

Thank you for attention!

\end{frame}

}


\appendix


\begin{frame}[allowframebreaks]{References}


% \bibliography{../thesis}

% \bibliographystyle{abbrv}

\printbibliography


\end{frame}

\begin{frame}{$\mathrm{SU}(p,q), p+q = n \geq 2$}
\begin{gather*}
  \lambda_a=\omega_{p'} + \omega_{n-q'} - (n+l+1-p'-q')\omega_p   \\
  1\leq p' \leq p,\quad 1\leq q' \leq q,\quad 1\leq l \leq \min(p',q')\\
  Q(\lambda_a)=R(\lambda_a)=\mathrm{SU}(p',q') \\
  (  \mu_a = \omega_{p'-l}+\omega_{n-q'+l}-(n+l+1-p'-q')\omega_p)
\end{gather*}

\begin{multline*}
 C_a = \{a_{p'}\omega_{p'} + \cdots + a_p\omega_p + \cdots + a_{n-q'}\omega_{n-q'} \,|\, \\
\quad  a_p=-a_{p'}-\cdots -a_{p-1}-a_{p+1} - \cdots - a_{n-q'} \}.
\end{multline*}

\end{frame}

\begin{frame}{$\mathrm{Sp}(n,\R), n \geq 2$}
\begin{gather*}
  \lambda_a = \omega_q + \omega_r - (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) \\
   (\mu_a= \omega_{q-l} + \omega_{r-l} - (2+n-\frac{1}{2}(r+q-l+1))\omega_n)
\end{gather*}
\[
 C_a = \{ a_r\omega_r + \cdots + a_n\omega_n \,|\, a_n=-(a_r+\cdots + a_{n-1}) \}
\]
\end{frame}

\begin{frame}{$\mathrm{SO}^*(2n), n\geq 4$}

\begin{center}
\resizebox{\textwidth}{!}{
\begin{threeparttable}
\begin{tabular}{CCCCC}
  \text{Vertex } \lambda_a & \text{Weight } \mu_a &  Q(\lambda_a) = R(\lambda_a)& l(\lambda_a) \\ \hline
  \omega_2 - (2n-2)\omega_n & -(2n-2)\omega_n & \mathrm{SU}(1,1) &  1 \\
  \omega_p -2(n-p+l) \omega_n & \omega_{p-2l}-2(n-p+l)\omega_n & \mathrm{SO}^*(2p)\tnote{1} & 1\leq l \leq \floor{\frac{p}{2}} \\
  \omega_{n-1} - (1+2l)\omega_n & \omega_{n-1-2l} - 2(1+l)\omega_n &  \mathrm{SO}^*(2n-2) & 1\leq l \leq \floor{\frac{n-1}{2}} \\
  -(2l-2)\omega_n & \omega_{n-2l} - 2l\omega_n &  \mathrm{SO}^*(2n)  & 1\leq l \leq \floor{\frac{n}{2}} \\
  \omega_1 +\omega_{q+1} - (2n-q)\omega_n & \omega_q -(2n-q)\omega_n & \mathrm{SU}(1,q)\tnote{2} & 1\\
  \omega_1 +\omega_{n-1} - (n+1)\omega_n & \omega_{n-2} - (n+2)\omega_n &  \mathrm{SU}(1,n-2)  & 1 \\
  \omega_1 -(n-1)\omega_n & \omega_{n-1} - n\omega_n &  \mathrm{SU}(1,n-1)  & 1
\end{tabular}
\smallskip
\begin{tablenotes}
 \item [1] $3 \leq p \leq n-2$
 \item [2] $2 \leq q \leq n-3$
\end{tablenotes}
\end{threeparttable}
}
\end{center}
\end{frame}

\begin{frame}{$\mathrm{SO}^*(2n), n\geq 4$}
Let $a=(Q,R,l)$, $Q=R$. Then for $R=\mathrm{SO}^*(2p)$, $3\leq p\leq n$
\[
 C_a = \{a_p\omega_p+\cdots + a_n\omega_n \,|\, a_n = -2a_p - \cdots -2a_{n-2} - a_{n-1}\}
\]
and for  $R=\mathrm{SU}(1,q)$, $1\leq q \leq n-1$
\[
 C_a = \{a_1\omega_1 + a_{q+1}\omega_{q+1} + \cdots + a_n\omega_n \,|\, a_n = -(a_1 + 2a_{q+1} + \cdots + 2a_{n-2} + a_{n-1})\}.
\]

\end{frame}

\begin{frame}{$E_6$}
\begin{center}\begin{threeparttable}\label{tbl:e6}
\begin{tabular}{CCCCC}
  \text{Vertex } \lambda_a & \text{Weight } \mu_a & Q(\lambda_a) = R(\lambda_a) & l(\lambda_a) \\ \hline
  -12 \omega_1 + \omega_2 & -12 \omega_1 & \mathrm{SU}(1,1) & 1\\
  -12 \omega_1 + \omega_4 & -12 \omega_1 + \omega_2 & \mathrm{SU}(1,2)& 1\\
  -12 \omega_1 + \omega_3 + \omega_5 & -12 \omega_1 + \omega_4 & \mathrm{SU}(1,3) & 1\\
  -9 \omega_1 + \omega_5 \tnote{1} & -10 \omega_1 + \omega_3 & \mathrm{SU}(1,4) & 1\\
  -10 \omega_1 + \omega_3 + \omega_6 \tnote{2} & -10 \omega_1 + \omega_5 & \mathrm{SU}(1,4) & 1\\
  -8 \omega_1 + \omega_3 & -8 \omega_1+ \omega_6 & \mathrm{SU}(1,5) & 1 \\
  -5 \omega_1 +  \omega_6 & -6 \omega_1+ \omega_2 & \mathrm{SO}(2,8) & 1 \\
  -8 \omega_1 + \omega_6 & -9 \omega_1 & \mathrm{SO}(2,8) & 2 \\
  0 & -2 \omega_1+ \omega_3 & EIII & 1 \\
  -3 \omega_1 & -5 \omega_1 + \omega_6 & EIII & 2
\end{tabular}
\smallskip
\begin{tablenotes}
 \item [1] Dynkin diagram
    \begin{tikzpicture} % E6 relative
        \node[croot] (a1) [label=above:$-\beta$] {};
        \node[croot] (a2) [right=of a1] [label=above:$\alpha_2$] {};
        \node[croot] (a3) [right=of a2] [label=above:$\alpha_4$] {};
        \node[croot] (a4) [right=of a4] [label=above:$\alpha_3$] {};
	\draw (a1) to (a2) to (a3) to (a4);
     \end{tikzpicture}
 \item [2] Dynkin diagram
 \begin{tikzpicture} % E6 relative
        \node[croot] (a1) [label=above:$-\beta$] {};
        \node[croot] (a2) [right=of a1] [label=above:$\alpha_2$] {};
        \node[croot] (a3) [right=of a2] [label=above:$\alpha_4$] {};
        \node[croot] (a4) [right=of a3] [label=above:$\alpha_5$] {};
	\draw (a1) to (a2) to (a3) to (a4);
     \end{tikzpicture}
\end{tablenotes}
\end{threeparttable}\end{center}
\end{frame}

\begin{frame}{$E_7$}

\begin{center}\begin{threeparttable}
\begin{tabular}{CCCCC}
  \text{Vertex } \lambda_a & \text{Weight } \mu_a & Q(\lambda_a) = R(\lambda_a) & l(\lambda_a) \\ \hline
  \omega_1 - 18 \omega_7 & -18 \omega_7 & \mathrm{SU}(1,1) & 1 \\
  \omega_3 - 18 \omega_7 & \omega_1 -18 \omega_7 & \mathrm{SU}(1,2) & 1 \\
  \omega_4 - 18 \omega_7 & \omega_3 -18 \omega_7 & \mathrm{SU}(1,3) & 1 \\
  \omega_2 + \omega_5 - 18 \omega_7 & \omega_4 - 18 \omega_7 & \mathrm{SU}(1,4) & 1 \\
  \omega_5 -15 \omega_7 \tnote{1} &  \omega_2 - 15\omega_7 & \mathrm{SU}(1,5) & 1 \\
  \omega_2 + \omega_6 - 16 \omega_7 \tnote{2} &  \omega_5 - 16 \omega_7 & \mathrm{SU}(1,5) & 1 \\
  \omega_2 - 13 \omega_7 &  \omega_6 - 14 \omega_7 & \mathrm{SU}(1,6) & 1 \\
  \omega_6 - 10 \omega_7 &  \omega_1 - 10 \omega_7 & \mathrm{SO}(2,10) & 1 \\
  \omega_6 - 14 \omega_7 & -14 \omega_7 & \mathrm{SO}(2,10) & 2 \\
  0 &  \omega_6 - 2 \omega_7 & EVII & 1 \\
  -4 \omega_7 &  \omega_1 - 6 \omega_7 & EVII & 2 \\
  -8 \omega_7 & -10 \omega_7 & EVII & 3
\end{tabular}
\smallskip
\begin{tablenotes}
 \item [1] Dynkin diagram
    \begin{tikzpicture} % E6 relative
        \node[croot] (a1) [label=above:$-\beta$] {};
        \node[croot] (a2) [right=of a1] [label=above:$\alpha_1$] {};
        \node[croot] (a3) [right=of a2] [label=above:$\alpha_3$] {};
        \node[croot] (a4) [right=of a3] [label=above:$\alpha_4$] {};
        \node[croot] (a5) [right=of a4] [label=above:$\alpha_2$] {};
	\draw (a1) to (a2) to (a3) to (a4) to (a5);
     \end{tikzpicture}
 \item [2] Dynkin diagram
 \begin{tikzpicture} % E6 relative
        \node[croot] (a1) [label=above:$-\beta$] {};
        \node[croot] (a2) [right=of a1] [label=above:$\alpha_1$] {};
        \node[croot] (a3) [right=of a2] [label=above:$\alpha_3$] {};
        \node[croot] (a4) [right=of a3] [label=above:$\alpha_4$] {};
        \node[croot] (a5) [right=of a4] [label=above:$\alpha_5$] {};
	\draw (a1) to (a2) to (a3) to (a4) to (a5);
     \end{tikzpicture}
\end{tablenotes}
\end{threeparttable}\end{center}
\end{frame}


\begin{frame}{Calderbank--Diemer construction}
\begin{itemize}[<+- | alert@+>]
	\item Certain Neumann series has to converge $\rightsquigarrow$ formal globalization
	\item formal globalization has a well defined $P$-action\\ \hspace{1cm} $\rightsquigarrow$ highest / lowest weight
	\item operator in the Neumann series constructed using Kostant's Hodge decomposition $\rightsquigarrow$ unitarizable modules
\end{itemize}
\end{frame}

\begin{frame}{Enright--Shelton equivalence 1}

\begin{theorem}[\cite{enright_categories_1987}, \cite{enright_highest_1989}] 
\hspace{1em} \\[-1em]
Suppose that either $\roots$ has one root length or that it has roots of two lengths and all the roots in $J$ are short. Then there exists an equivalence of categories
	\[
	\mathcal{E}: \mathcal{O}^\lie{p}_\mu \to \mathcal{O}^\lie{p'}_{\mathrm{reg}},
	\]
	where $\lie{p}'$ is a parabolic subalgebra of Hermitian type of a complex simple Lie algebra $\lie{g}'$ of rank \alert{smaller} or equal to the rank of $\lie{g}.$ Moreover, there is an isomorphism of posets $W^{\Sigma, J} \to W'^{\Sigma'}$ such that the functor $\mathcal{E}$ sends parabolic Verma modules $M_\Sigma(w)$ to parabolic Verma modules $M_{\Sigma'} (w')$ and similarly for simple modules.

\end{theorem}

\end{frame}

\begin{frame}{Enright--Shelton equivalence  2}
\begin{theorem}[\cite{enright_categories_1987}, \cite{enright_highest_1989}] 
 Suppose that $\roots$ has two root lengths and $J$ contains a long root. Then there exists an equivalence of categories
	\[
	\mathcal{E}: \mathcal{O}^\lie{p}_\mu \to \mathcal{O}^\lie{p'}_{\mathrm{reg}} \oplus  \mathcal{O}^\lie{p'}_{\mathrm{reg}},
	\]
	where $\lie{p}'$ is a parabolic subalgebra of Hermitian type of a complex simple Lie algebra $\lie{g}'$ of rank \alert{smaller} or equal to the rank of $\lie{g}.$ More precisely the poset $W^{\Sigma, J}$ is a disjoint union $W^{\Sigma, J}_\mathrm{odd} \cup W^{\Sigma, J}_{\mathrm{even}}$ of two poset and the category $\mathcal{O} ^\lie{p}_\mu$ has a decomposition into a direct sum $\mathcal{O} ^\lie{p}_\mu = \mathcal{O} ^\lie{p}_{\mu, \mathrm{even}} \oplus \mathcal{O} ^\lie{p}_{\mu, \mathrm{odd}}$ such that all extensions between modules in different summands is zero. There exists isomorphisms of posets $ W^{\Sigma, J}_{\mathrm{odd}} \to W'^{\Sigma'}$ and $ W^{\Sigma, J}_{\mathrm{even}} \to W'^{\Sigma'}$ and corresponding equivalences of categories $\mathcal{E}_{\mathrm{odd}}: \mathcal{O} ^\lie{p}_{\mu, \mathrm{odd}} \to  \mathcal{O}^\lie{p'}_{\mathrm{reg}}$ and $\mathcal{E}_{\mathrm{even}}: \mathcal{O} ^\lie{p}_{\mu, \mathrm{even}} \to  \mathcal{O}^\lie{p'}_{\mathrm{reg}}$ such that $M_\Sigma(w) \to M_{\Sigma'} (w')$ and similarly for their simple quotients.

\end{theorem}

\end{frame}

\begin{frame}{Enright--Shelton equivalences --- example}

Consider $(\lie{sl}_6, \lie{sl}_3 \oplus \lie{sl}_3)$ with $\mu = (0,1,2\, |\, 3,3,4)$. 

Permutations of $\mu + \rho$ which are highest weights  of simple modules plus $\rho$ have their first three and last three entries strictly decreasing.  There are only six possible cases, namely
\[
(4, 3, 2\, |\, 3, 1, 0) \quad (4, 3, 1\, |\,3, 2, 0) \quad (4, 3, 0\, |\,3, 2, 1)
\]
and their counterparts with first three and last three entries swapped. Now these three weights are equivalent to 
\[
(4,  2\, |\,  1, 0) \quad (4,  1\, |\, 2, 0) \quad (4,  0\, |\, 2, 1).
\]
%If we impose that the weight for $\lie{sl}(6)$ has to have entries increasing by $1$ this mapping on weights has inverse since we know that we have to put the number $3$ into two places and there's only one way to do it in each case so that one obtains $\lie{k}$-dominant weight. This directly generalizes to all other $\lie{sl}$ cases as any weight can be brought by translation functors to a weight that starts with $0$, has entries which are nondecreasing and which increase only by $1$.

\end{frame}

\end{document}