unitarizable_modules.tex

\chapter{Unitarizable highest weight modules}\label{ch:unitarizable}

One of the main open problem of representation theory is classification of unitary modules of Lie groups. Apart from the finite-dimensional representations (which are unitary for compact Lie groups) one of the classes of modules where a complete classification was obtained is the class of unitarizable modules of highest weight. It is precisely the intersection of BGG category $\mathcal{O}$ and Harish-Chandra $(\mathfrak{g}, K)$-modules. It was proved already by Harish-Chandra that these modules can occur only in the Hermitian symmetric case, see \cite{harish-chandra_representations_1955, harish-chandra_representations_1956-1} and \cite{harish-chandra_representations_1956} for details.

A sufficient and necessary condition for unitarizability of a highest weight module appears in \cite{garland_unitarizable_1981}. Independently, these modules were classified by Jakobsen in \cite{jakobsen_last_1981, jakobsen_hermitian_1983} with later development in \cite{jakobsen_intrinsic_1996}. Another, yet independent, approach was given by \cite{enright_classification_1983} with later simplification in \cite{joseph_annihilators_1992}. In \cite{adams_unitary_1987} it was shown that all unitarizable highest weight modules can be obtained via the derived functor construction from either a one-dimensional or a unipotent representation. This  parametrization given here fits in nicely with the Langlands classification and the coadjoint orbit picture.  

In this chapter we follow mainly the article \cite{enright_classification_1983} and reorganization of the unitarizable weight into integral cones that was presented in \cite{davidson_differential_1991}. 

\section{Classification}

The algebra $\lie{k}$ has a one-dimensional center which is complementary to the span of $\roots_c$. We pick a generator $\zeta$ of the center by requiring $\frac{2 \langle \zeta,\beta \rangle}{\langle \beta, \beta \rangle} = 1$, where $\beta$ is the unique maximal noncompact root\footnote{Equivalently, $\beta$ is the highest weight of $\lie{k}$-representation $\lie{p}_+$.} of $\roots^+$. Now any line $\lambda+z\zeta$ for $z\in\C$ can be uniquely written as $\lambda_0 + z\zeta$ with
\[
 \langle \lambda_0 + \rho,\beta \rangle = 0, \quad \rho =\sum_{\alpha\in\roots^+} \alpha.
\]

If $L(\lambda)$ is an irreducible unitarizable module for $\lambda=\lambda_0+z\zeta$, then $z$ must be real and the $K_\C$-finiteness implies that $\lambda$ must be $\roots^+_c$-dominant integral. Write the generalized Verma module $M(\lambda)$ as $S(\lie{p}_+)\otimes F(\lambda_0) \otimes \C_z\zeta$. If we fix a basis of $\lie{g}$ and $F(\lambda_0)$, we may view the modules $M(\lambda)$ as defined on the same vector space $S(\lie{p}_+)\otimes F(\lambda_0)$ where the action of $\lie{g}$ is given by polynomial expressions in $z$. Likewise, we may view the Shapovalov form on $M(\lambda)$ as a bilinear form on $S(\lie{p}_+) \otimes F(\lambda_0)$ with values in complex polynomials $\C[z]$.

A $\lie{g}$-module $M$ is called unitarizable if there exists (necessarily unique up to a multiple) positive definite contravariant form on $M$. The following theorem explains the structure of the set of $z\in\R$ such that the module $M(\lambda)$ is unitarizable.

\begin{figure}[H]
  \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:$C$] {};
  \end{tikzpicture}
  \end{center}\caption{Structure of unitarizable weights}\label{fig:struct} %TODO dodelat mark na 0
\end{figure}

\begin{theorem}[Theorem 2.4 of \cite{enright_classification_1983}]
 The set of real numbers $z$ with $L(\lambda_0 + z\zeta)$ a unitarizable $\lie{g}$-module is given by the set
\[
  \{z \in \mathbb{R}\, |\, z \leq A(\lambda_0)\} \cup \{ z=A(\lambda_0) + k C(\lambda_0)\,|\, z \leq B(\lambda_0) \et k\in \mathbb{N}_0\},
\]
where $A(\lambda_0)$, $B(\lambda_0)$ and $C(\lambda_0)$ are real numbers expressible in terms of certain root systems $Q(\lambda_0)$ and $R(\lambda_0)$ associated to $\lambda_0$. Moreover $C(\lambda_0)$ is independent of $\lambda_0$ and depends only on the type of $\lie{g}_0$. The values of $C$ are listed in Table \ref{tbl:C} and the structure of the set of unitarizable highest weights is depicted in Figure \ref{fig:struct}.
	
The discrete series representation corresponds to $z <0$ and limit of discrete series representation corresponds to $z=0$. For $z  < A(\lambda_0)$ we have $L(\lambda) = M(\lambda)$ (i.e. the generalized Verma modules are irreducible) and for all $z \geq A(\lambda_0)$ such that $L(\lambda)$ are unitarizable the generalized Verma modules are reducible.%, hence the elements of  $\{z=A(\lambda_0) + kC| \, z \leq B(\lambda_0) \et k\in \mathbb{N}_0 \}$ are called  reduction points or points of reducibility.

\begin{table}[h]
\[\begin{array}{c|ccccccc}
\lie{g}_0 & \mathrm{SU}(p,q) & \mathrm{Sp}(n,\mathbb{R}) & \mathrm{SO}^*(2n) & \mathrm{SO}(2,2n-2) & \mathrm{SO}(2,2n-1) & \mathrm{E}_6 & \mathrm{E}_7\\\hline
C & 1 & \frac{1}{2} & 2 & n-2 & n-\frac{3}{2} & 3 & 4
\end{array}\]\caption{Distance between points of reducibility}\label{tbl:C}
\end{table}
\end{theorem}

Let $\roots_c(\lambda_0) := \{ \alpha \in \roots_c | \langle \alpha,\lambda_0 \rangle = 0 \}$ and recall the definition of $\beta$ the maximal non-compact root. Take the root subsystem of $\roots$ generated by $\pm \beta$ and $\roots_c(\lambda_0)$ and decompose it into a disjoint union of simple root systems. Let $Q(\lambda_0)$ be the root system in this union which contains $\beta$.

If $\roots$ has two root lengths and if there are short compact roots $\alpha$ not orthogonal to $Q(\lambda_0)$ with $\frac{2 \langle \lambda_0,\alpha \rangle}{\langle \alpha, \alpha \rangle}  =1$, then let $\Psi$ be the root system generated by $\pm \beta, \roots_c(\lambda_0)$ and all such $\alpha$. Let $R(\lambda_0)$ be the simple component of $\Psi$ which contains $\beta$. If $\roots$ has only one root length or if no such $\alpha$ exists, then put $R(\lambda_0) = Q(\lambda_0)$.

Since these root systems are subsystems of $\roots$ and since each has compact and noncompact roots, each is a root system of a Hermitian symmetric pair.

There is a convenient way to construct $Q(\lambda_0)$ by means of Dynkin diagrams. Draw a Dynkin diagram of $\lie{g}$ and delete the unique node corresponding to simple noncompact root. Now adjoin to the resulting diagram $-\beta$ by the usual rules as when constructing extended Dynkin diagrams. The maximal connected subdiagram containing $-\beta$ such that its every compact simple root is orthogonal to $\lambda_0$ is the Dynkin diagram of $Q(\lambda_0)$. We illustrate on the case of $\lie{su}(p,q)$. Here the noncompact root $\beta$ is $\alpha_p$ and we get the following extended Dynkin diagram.
\begin{figure}[H]\label{fig:Q}
  \begin{center}
     \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}
  \end{center}\caption{The Dynkin diagram of $Q(\lambda_0)$} 
\end{figure}
If we write down $\lambda_0$ with respect to the basis of fundamental weights $\lambda_0 = \sum_i a_i \omega_i$ we see that $Q(\lambda_0)$ is a root system of $\lie{su}(p',q')$ where $p'$ and $q'$ are maximal such that the coefficients $a_1,a_2,\ldots,a_{p'}$ and $a_{n-q'+1},a_{n-q'+2}, \ldots, a_n$ are nonzero.

The root system $R(\lambda_0)$ is different from $Q(\lambda_0)$ only in two cases. The first one is $\lie{g}_0=\lie{sp}(n,\R)$ where $Q(\lambda_0) = \lie{sp}(n',\R)$ and $R(\lambda_0) = \lie{sp}(n'',\R)$ with $n' < n'' \leq n$. The second one is $\lie{g}_0=\lie{so}(2,2n-1)$ where $Q(\lambda_0) = \lie{su}(1,n-1)$ and $R(\lambda_0) = \lie{so}(2,2n-1)$ with $\lambda_0 = (\lambda_1,\frac{1}{2},\ldots,\frac{1}{2})$ in the $\epsilon$-basis.

The following two theorems finish the general classification of unitarizable highest weight modules.

\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}


\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}

Now let's see what we can tell about the maximal module $J(\lambda)$ of the generalized Verma module $M(\lambda)$. %The following was proved in \cite{davidson_differential_1991}.

\begin{theorem}[Theorem 3.1 of \cite{davidson_differential_1991}]
 Suppose $L(\lambda) = M(\lambda)/J(\lambda)$ is unitarizable and $J(\lambda)\neq 0$. Then
 \begin{enumerate}
  \item $H^1(\lie{p}_-,L(\lambda))$ is an irreducible $\lie{k}$-module
  \item $J(\lambda)$ is generated over $S(\lie{p}_+)$ by an irreducible $\lie{k}$-submodule $J(\lambda)^0$ isomorphic to $H^1(\lie{p}_-,L(\lambda))$.
 \end{enumerate}
\end{theorem}


%Let $J(\lambda)$ denote the maximal submodule of the generalized Verma module $M(\lambda)$. For all unitarizable highest weight modules we have  $J(\lambda)$  generated by $\lie{U(g)}$ from an irreducible $\lie{k}$-module.
In particular, the generator of $J(\lambda)$ must be contained in some component $S^k(\lie{p}_+)\otimes F(\lambda)$. This $k$ is called \emph{level of reduction} of $L(\lambda)$ and is denoted by $l(\lambda)$.

\begin{definition}
 The set of reduction points $\Lambda_r$ is the union of all reduction points. Explicitly
 \[
 \Lambda_r := \{ \lambda = z\zeta + \lambda_0 \in \lie{h}^* | z = A(\lambda_0) +kC, k\in\N_0, z \leq B(\lambda_0) \}.
 \]

 For $\lambda \in \Lambda_r$ let $a(\lambda) := (Q(\lambda_0),R(\lambda_0),l(\lambda))$ and let $\mathcal{A}$ denote the set of all such triples as $\lambda$ ranges over $\Lambda_r$. For $a\in\mathcal{A}$, let $\Lambda_a$ denote the set of all $\lambda\in\Lambda_r$ with $a(\lambda)=a$.
\end{definition}

Now we can look more closely at the structure of the set of reduction points. 

\begin{corollary}[of \ref{thm:reduction_points}]
 Let $a=(Q,R,l)\in\mathcal{A}$ and let $\lambda\in\Lambda_a$. If we write $\lambda= z\zeta + \lambda_0$, then $z=B(\lambda)-(l-1)C$ and on the other hand for $\lambda=(B - nC)\zeta + \lambda_0 \in \Lambda_r$ we have $l(\lambda) =n+1$.
\end{corollary}

Let $\lie{h}^*_\R$ denote the real span of the roots. A \emph{cone} with vertex zero (in $\lie{h}^*_\R$) is the intersection of a (nonempty) collection of closed half spaces. Each cone $C$ is thus determined by a finite set $\{h_i \in \lie{h}|i=1,\ldots,k\}$ with $C= \{ \lambda \in \lie{h}^*_\R | \lambda(h_i) \geq 0, i=1,\ldots k\}$. An \emph{integral cone} will be the intersection of a cone with the set of all $\lie{k}$-integral points of $\lie{h}^*.$ For an integral element $\nu\in\lie{h}^*$, a translated cone with vertex $\nu$ is a set of the form $\nu + C$ with $C$ some integral cone.

\begin{definition}
 For $a=(Q,R,l)\in\mathcal{A}$, let $C_a$ be the integral cone of $\lie{k}$-dominant integral elements in $\lie{h}^*_\R$ which are orthogonal to elements in $R$.
\end{definition}

%Since the root system $R$ always contains $\beta$, the representations $L(\lambda)$ are in fact finite-dimensional for $\lambda\in C_a$.

For concrete calculations the following lemma can be useful. 

\begin{lemma}[Section 4.3 of \cite{enright_resolutions_2004}]
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}
\begin{proof}
 Any  $\lie{k}$-dominant integral weight can be written in the form $\mu = \sum_i a_i \omega_i + b \zeta,$ where $a_i$ are non-negative integers. Such a weight is perpendicular to $R$ if and only if it is perpendicular to all compact simple roots contained in $R$ and to the noncompact root $-\beta$, i.e. to the simple roots of the root system $R$. The crucial observation here is that for each Hermitian symmetric space the weight $\zeta$ is in fact the fundamental weight corresponding to the simple noncompact root. Hence we have 
\[
0 = \left(\sum a_i \omega_i + b \zeta, \alpha_i^\vee \right) = a_i 
\] 
for all compact simple roots of  $R$ and 
\[
0 = \left(\sum a_i \omega_i + b \zeta, \beta^\vee \right)
\]
for the noncompact root. Recalling the definition of $\zeta$ and solving for $b$ we see that the cone consists of vectors of the form $\sum a_i (\omega_i - (\omega_i, \alpha_i^\vee)\zeta)$ where the sum is only indices whose simple roots are not in $R$.
\end{proof}

\begin{proposition}[Proposition 6.6 of \cite{davidson_differential_1991}]
 The set of reduction points $\Lambda_r$ is the disjoint union of the sets $\Lambda_a, a\in\mathcal{A}$.

 Each set $\Lambda_a$ is a translated integral cone with vertex $\lambda_a + C_a$. We list the vertices $\lambda_a$ in sections \ref{sec:su} through \ref{sec:exceptional}.
\end{proposition}
\begin{proof}
 The first statement is trivial and the second one follows by case by case computations.
\end{proof}

The next proposition gives an alternative way to compute the highest weight of the maximal submodule.

\begin{proposition}[Proposition 6.8 of \cite{davidson_differential_1991}]
 Suppose $\lambda\in\Lambda_a$ with $a=(Q,R,l)$. Let $u$ and $v$ denote respectively the unique elements of maximal length in the Weyl groups for the positive root systems $Q\cap \roots^+_c$ and $R\cap \roots^+_c$ and let $\mu$ denote the highest weight of the maximal submodule $J(\lambda)$. Then
 \[
  \mu = \lambda + \frac{1}{2}(u \mu_l + v \mu_l)
 \]
 in all cases except when $\lie{g}_0=\lie{so}(2,2n-1)$ and $Q\neq R$. In this case
 \[
  \mu = \lambda + \frac{1}{2}(\mu_l + v\mu_l).
 \]

 Moreover, in all cases $F(\mu)$ occurs with multiplicity one in $M(\lambda+\rho)$ and in all cases $F(\mu)$ is a PRV component of a tensor product in $S(\lie{p}_+)\otimes F(\lambda)$.
\end{proposition}

The next theorem  deals with effect of a sort of `translation functor' on unitarizable highest weight modules.
\begin{theorem}[Factorization theorem 6.15 of \cite{davidson_differential_1991}]
 Fix $a\in\mathcal{A}$ and let $\lambda = \lambda_a + \lambda'\in\Lambda_a$. Let $J(\lambda)^0$ and $J(\lambda_a)^0$ be the $\lie{k}$-modules that generate $J(\lambda)$ and $J(\lambda_a)$. Extend the $\lie{k}$-equivariant projection \[P:F(\lambda_a)\otimes F(\lambda') \to F(\lambda)\] to a mapping $\widetilde{P}:M(\lambda_a) \otimes F(\lambda') \to M(\lambda)$ by \[\widetilde{P}(F\otimes v) (T) := P(F(T)\otimes v),\]
  where we have used that $M(\lambda_a) = S(\lie{p}_+)\otimes F(\lambda_a)$.

Let $\mu$ and $\mu_a$ denote the highest weights of  $J(\lambda)^0$ and $J(\lambda_a)^0$. Then
 \begin{enumerate}
  \item $\mu  = \mu_a+\lambda'$
  \item $P(J(\lambda_a)^0\otimes F(\lambda')) = J(\lambda)^0$ and 
  \item $P(J(\lambda_a)\otimes F(\lambda')) = J(\lambda)$.
 \end{enumerate}
\end{theorem}

\section{Nilpotent cohomology of unitarizable highest weight modules}

The convention employed in this section is that we omit the terms whose indices are outside natural boundaries. 

\begin{definition}\label{def:cohomology_roots}
Let $\Psi_\lambda$ be the set if roots in $\roots$ which are orthogonal to $\lambda+\rho$ and let $\Psi_\lambda^+ = \Psi_\lambda \cap \roots^+$. 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}
 
 Let $W_\lambda$ be the subgroup of $W$ which is generated by reflections $s_\alpha$ for $\alpha\in \roots_{n,\lambda}^+$.
 
 Let $\roots_\lambda$ be the subset of $\roots$ of elements $\beta$ with $s_\beta\in W_\lambda$ and let $\roots_{\lambda,c} = \roots_c \cap \roots_\lambda$, $\roots_{\lambda,c}^+ = \roots_{\lambda,c} \cap \roots^+$.
 
 Finally, define  $W^{c,i}_\lambda = \{ w \in W_\lambda \, |\, w \rho \text{ is } \roots^+_{\lambda, c}\text{-dominant and } l_\lambda(w)=i \}$.
\end{definition}

\begin{theorem}[3.7 \cite{davidson_differential_1991}]\label{thm:cohomology}
 Let $L$ be unitarizable with highest weight $\lambda $. Then for $i\in \mathbb{N}$ we have
\begin{equation}\label{eq:cohomology}
 H^i(\lie{p}_+,L)\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 $\lambda$.
\end{theorem}

It turns out that $\roots_\lambda$ is actually a root system of a simple Lie algebra and moreover it's intersection with noncompact roots is nonempty and gives a decomposition of $\roots_\lambda$ into compact and noncompact roots. In other words, we obtain a smaller Hermitian symmetric pair defined by $\lambda$.

\begin{definition}
 The Hermitian symmetric pair $(\lie{g}_\lambda, \lie{k}_\lambda)$ attached to $(\roots_{\lambda,c}, \roots_\lambda \cap \roots_n)$ is called \emph{reduced Hermitian symmetric pair} associated to $\lambda$.
\end{definition}

A priori, it is not clear that $W_\lambda$ is a Weyl group. The following theorem deals with this issue. 

\begin{theorem}[Theorem 3.3 of \cite{dyer_reflection_1990}, \cite{deodhar_note_1989}]
	Let $(W, R)$ be a Weyl group generated by a set of simple reflections $R$ and let $T = \bigcup_{w \in W} wRw^{-1}$ be the set of all reflections. 
	If $G$ is any subgroup of a Weyl group $W$ that is generated by reflections, then it is a Coxeter group. Let $N(w) = \{ t \in T \, | \, l(tw) < l(w) \}$ where $l$ denotes the length function of $(W, R).$ The set $\{ t \in T \,|\, N(t) \cap G = \{t\} \}$ is a set of Coxeter generators for $G$.
\end{theorem}

We can use this theorem to find the simple roots of the reduced Hermitian symmetric pair $(\lie{g}_\lambda, \lie{k}_\lambda).$ The proof of the formula \eqref{eq:cohomology} is based on the fact that for unitarizable highest weights the Enright Shelton equivalences \ref{sec:es_equivalence} translate the problem to calculation of nilpotent Lie algebra cohomology of the reduced Hermitian pair with values in a finite dimensional representation. Hence one can just calculate the BGG diagram of minimal representatives of the pair $(\lie{g}_\lambda, \lie{k}_\lambda)$ using the classical algorithm and then apply the embedding of $(\lie{g}_\lambda, \lie{k}_\lambda)$ into $(\lie{g}, \lie{k}).$

\begin{remark}
 The $\lie{k}$-weight of the first cohomology is given by $\lambda_0-\rho$ where $\lambda_0$ is the unique $\roots_c^+$ dominant element in the $W_c$ orbit of $s_{\gamma_0}\lambda$ for the unique noncompact simple root $\gamma_0 \in \roots_\lambda^+$.
\end{remark}

\begin{lemma}\label{lem:singular_are_noncompact}
 Let $\lambda$ be a highest weight of a unitarizable highest weight module. If a positive root is orthogonal to $\lambda + \rho$ then it must be noncompact.
 \[
  \alpha \in \roots^+: \alpha \perp \lambda + \rho \Longrightarrow \alpha \in \roots^+_n
 \]
\end{lemma}
\begin{proof}
 Every positive roots can be written as a positive linear combination of simple roots, i.e. $\alpha = \sum_i c_i \alpha_i$ where $c_i \geq 0$. The fundamental weights form a basis of $\lie{h}^*$ and thus $\lambda = \sum_i k_i \omega_i$. Now we just use the defining property of fundamental weights $\frac{2(\alpha_i,\omega_j)}{(\alpha_i,\alpha_i)} = \delta_{ij}$ to compute
 \begin{align*}
  (\alpha,\lambda+\rho) & = \sum_{i,j} \left(  c_i k_j (\alpha_i,\omega_j) + c_i (\alpha_i,\omega_j) \right ) \\
			& = \sum_{i,j} \left( c_i k_j \frac{(\alpha_i,\alpha_i)}{2} \delta_{ij} + c_i \frac{(\alpha_i,\alpha_i)}{2} \delta_{ij} \right ) \\
			& = \sum_i c_i \frac{(\alpha_i,\alpha_i)}{2} (k_i + 1).
 \end{align*}
 If $\lambda$ is a highest weight of a unitarizable module, then all but one of the coefficients $k_i$ are non-negative and the only possibly negative coefficient corresponds to the fundamental weight dual to the coroot of the unique noncompact simple root - let's denote it's index by $i_0$. If the scalar product $(\alpha, \lambda+\rho)$ is zero, then $c_{i_0}$ must be nonzero -- all the remaining terms in the sum are non-negative. But $c_{i_0} \neq 0$ is equivalent to $\alpha$ being noncompact.
\end{proof}


\begin{example} 
 Let us take $\lie{g} = \lie{so}(2,2n-2)$ with $\lambda = (2-n)\omega_1$. Then in the epsilon basis we have $\lambda + \rho = (1,n-2,\ldots,1,0)$ and $\Psi_\lambda^+ = \{ \epsilon_1 - \epsilon_{n-1}\}$. 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)) &= F((2-n)\omega_1)\\
 H^1(\lie{p}_-,L((2-n)\omega_1)) &= F(-n\omega_1)\\
 H^i(\lie{p}_-,L((2-n)\omega_1)) &= 0 \text{ for } i\geq 2.
\end{align*} 

Similarly there is only one root generating $W_\lambda$ for $\lie{g} = \lie{so}(2,2n-1)$ and $\lambda = (\frac{3}{2} - n)\omega_1$ and we get that in that case
\begin{align*}
 H^0(\lie{p}_-,L((\frac{3}{2}-n)\omega_1)) &= F((\frac{3}{2}-n)\omega_1)\\
 H^1(\lie{p}_-,L((\frac{3}{2}-n)\omega_1)) &= F((-\frac{1}{2}-n)\omega_1)\\
 H^i(\lie{p}_-,L((\frac{3}{2}-n)\omega_1)) &= 0 \text{ for } i\geq 2.
\end{align*}

Moreover, by inspecting the tables \ref{tbl:so_even} and \ref{tbl:so_odd}, we see that in both of these cases the cone $C_a$ is empty.
\end{example}

\begin{remark}
 The formula \eqref{eq:cohomology} is actually stated a little bit differently in \cite{enright_analogues_1988}. Namely, the finite dimensional modules appearing in the cohomology are $F(\overline{w}\cdot lambda)$ where $\overline{w}$ is the minimal length representant of $w$. The same formula appears in a recent article \cite{enright_diagrams_2014}. However, this formula is wrong as the following example shows.

Consider $\lie{su}(1,2)$ and weight $\lambda = -(a+2)\omega_1 + (a+1)\omega_2$. The reduced Hermitian pair is of type $A_1$ and it's given by $\{\epsilon_1 - \epsilon_3 \}$. The associated reflection is $s_{\epsilon_1 - \epsilon_3} = s_1 s_2 s_1$ and its minimal coset representative is $s_1 s_2$. It's affine action on $\lambda$ gives $-2\omega_1 - (a+2)\omega_2$ which is not $\roots_c^+$-dominant. On the other hand the (normal) action of $s_1 s_2 s_1$ on $\lambda + \rho$ gives $ -(a+2)\omega_1 + (a+1)\omega_2$ which is $\roots_c^+$-dominant and hence the first cohomology is $F(-(a+3)\omega_1 + a\omega_2).$
\end{remark}

Cohomologies of all unitarizable modules for the two expceptional types are explicitely computed in \cite{enright_resolutions_2004-1}. The papers \cite{enright_hilbert_2004}, \cite{enright_resolutions_2004} treat also certain weights for the classical types $\lie{su}(p,q)$, $\lie{sp}(n,\R)$ and $\lie{so}^*(2n).$  The orthogonal cases $\lie{so}(2,2n-2)$, $\lie{so}(2,2n-1)$ are rather easy and are calculated completely in sections \ref{sec:conf_even} and \ref{sec:conf_odd}. For the other classical types we calculate some of the data that go into the formula \eqref{eq:cohomology} and calculate the cohomologies completely in small ranks in appendix \ref{app:cohomology}. We borrow idea from \cite{enright_resolutions_2004-1} and calculate possible $\Psi_\lambda^+ $ by labeling the poset of noncompact roots by scalar products of the associated coroots with $\lambda + \rho + \mu$ for $\mu \in C$. We show only nodes / roots where the scalar product is not always negative. 

In general the combinatorics behind the calculations is rather involved. In most cases the reduced Hermitian pair doesn't stay the same on the cone which is due to the fact that these cones can intersect facets in nontrivial ways. 

\begin{example}
To illustrate the situation, here is an example for $\mathrm{SO}^*(16)$ and cone of unitarizable weights $\left(a_{5} + 1\right)\omega_{5} + a_6\omega_6 + a_7\omega_7 - \left(2 \, a_{5} + 2 \, a_{6} + a_{7} + 8\right)\omega_{8}.$

\begin{figure}[ht]
  \centering
      \begin{tikzpicture}[>=latex,line join=bevel,]
%%
\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}
  \caption{Nonnegative scalar products with noncompact roots}
\end{figure}

We can see that the set of singular weights $\Psi_\lambda^+$ is generically empty, but for $a_5 = a_6 = a_7$ it contains the roots $\epsilon_1 - \epsilon_8$ and $\epsilon_2 + \epsilon_7$ and for small values of $a_5$ even more. It is however clear that the cone can be written as a union of smaller cones such that $\Psi_\lambda^+$ remains the same on each of them. 

\end{example}

\clearpage

\input{cohomology_sl}

\clearpage

\input{cohomology_sp} 

\clearpage

\input{cohomology_so}

\clearpage

\input{cohomology_conformal_even}

\clearpage

\input{cohomology_conformal_odd}

\clearpage
\subsection[Exceptional cases]{Exceptional cases}

\subsubsection{$\mathrm{E_6}$}

\begin{center}
    \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}
\end{center}
%
%\[\alpha_2 = \epsilon_1+\epsilon_2,\quad \alpha_i = \epsilon_{i-1}-\epsilon_{i-2},\quad (3\leq i \leq 6)\]
%\[\alpha_1 = \frac{1}{2}(\epsilon_1-\epsilon_2-\epsilon_3- \cdots -\epsilon_6 -\epsilon_7 + \epsilon_8)\]
%
%\[\roots_c^+ = \{ \pm\epsilon_i + \epsilon_j | 1\leq i < j \leq  5\}\]
%\[\roots_n^+ = \left\{  \frac{1}{2}\sum_{i=1}^5 (-1)^{\nu(i)}\epsilon_i - \epsilon_5 -\epsilon_7 + \epsilon_8\, |\, \sum_{i=1}^5 \nu(i)\text{ is even} \right \} \]
%\[\beta = \frac{1}{2}(\epsilon_1+\epsilon_2+\epsilon_3+\epsilon_4+\epsilon_5-\epsilon_6-\epsilon_7+\epsilon_8) =\alpha_1+2\alpha_2+2\alpha_3+3\alpha_4+2\alpha_5+\alpha_6\]
%\[\quad \rho = (0,1,2,3,4,-4,-4,4),\quad \zeta = (0,0,0,0,0,\frac{-2}{3},\frac{-2}{3},\frac{2}{3})\]
 %
%\begin{figure}[H]
  %\centering
  %\input{diagrams/nroots_E6_1.tikz} 
  %\caption{Poset of noncompact roots for $\mathrm{E}_6$}
%\end{figure}  

% \begin{table}[h]
\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}\caption{Vertices and root systems for $\mathrm{E}_6$}
\end{threeparttable}\end{center}
% \end{table}

For full cohomologies of unitarizable modules see \cite{enright_resolutions_2004-1}.

\subsubsection{$\mathrm{E_7}$}\label{sec:exceptional}

\begin{center}
     \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{center}

 %
%\begin{figure}[H]
  %\centering
  %\input{diagrams/nroots_E7_7.tikz} 
  %\caption{Poset of noncompact roots for $\mathrm{E}_7$}
%\end{figure}  

\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}\caption{Vertices and root systems for $\mathrm{E}_7$}
\end{threeparttable}\end{center}
% \end{table}

For full cohomologies of unitarizable modules see \cite{enright_resolutions_2004-1}.