# Irreducible unitary representations with non-zero relative Lie algebra
cohomology of the Lie group $SO_{0}(2,m)$

Ankita Pal, Pampa Paul  Department of Mathematics, Presidency University, 86/1
College Street, Kolkata 700073, India<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
###### Abstract.

By a theorem of D. Wigner, an irreducible unitary representation with non-zero
$({g},K)$-cohomology has trivial infinitesimal character, and hence up to
unitary equivalence, these are finite in number. We have determined the number
of equivalence classes of these representations and the Poincaré polynomial of
cohomologies of these representations for the Lie group $SO_{0}(2,m)$ for any
positive integer $m.$ We have also determined, among these, which are discrete
series representations and holomorphic discrete series representations.

###### 2020 Mathematics Subject Classification:

22E46, 17B10, 17B20, 17B22, 17B56.
Keywords and phrases: Lie group, Lie algebra, Dynkin diagram, $\theta$-stable
parabolic subalgebra, cohomological induction.

##  1\. Introduction

Let $G$ be a connected semisimple Lie group with finite centre, and $K$ be a
maximal compact subgroup of $G$ with Cartan involution $\theta.$ The
differential of $\theta$ at identity is denoted by the same notation $\theta.$
Let ${g}_{0}$ be the Lie algebra of $G,{k}_{0}$ be the subalgebra of ${g}_{0}$
corresponding to the Lie subgroup $K$ of $G,{h}_{0}$ be a $\theta$-stable
fundamental Cartan subalgebra of ${g}_{0},$ and
${g}={g}_{0}^{\mathbb{C}},{h}={h}_{0}^{\mathbb{C}}.$ Corresponding to a
$\theta$-stable parabolic subalgebra ${q}$ of ${g}_{0}$ containing ${h}_{0},$
and a linear function $\lambda$ on ${h}$ in a certain good range, there is a
cohomologically induced module $A_{q}(\lambda),$ which is an irreducible
unitary representation of $G$ with infinitesimal character $\chi_{\lambda}.$
These representations include all discrete series representations if
rank$(G)=$rank$(K).$ We are interested in those cohomologically induced module
$A_{q}(\lambda)$ for which the infinitesimal character $\chi_{\lambda}$ is
trivial, that is $\chi_{\lambda}$ is the infinitesimal character of the
trivial representation of $G$, and we denote it by $A_{q}.$

By a theorem of D. Wigner, an irreducible representation with non-zero
$({g},K)$-cohomology has trivial infinitesimal character. Hence there are only
finitely many irreducible unitary representations with non-zero
$({g},K)$-cohomology. In fact, the irreducible unitary representations with
non-zero relative Lie algebra cohomology are exactly the irreducible unitary
representations $A_{q}.$ See §2 for more details. So the representations
$A_{q}$ are important on its own. Apart from that Borel [1] has conjectured
that an irreducible unitary representation with non-zero $({g},K)$-cohomology
is an automorphic representation for a suitable uniform discrete subgroup of
$G.$ Millson and Raghunathan [10] have proved this conjecture for the group
$G=SO(n,1),$ by constructing geometric cycles and using Matsushima’s
isomophism [9]. So the representations $A_{q}$ are possible candidates of
automorphic representations of $G.$

Collingwood [3] has determined the representations $A_{q}$ and computed
cohomologies of these representations of $Sp(n,1),$ and the real rank one real
form of $F_{4}.$ Li and Schwermer [8] have determined the representations
$A_{q}$ and cohomologies of these representations for the connected non-
compact real Lie group of type $G_{2}.$ Mondal and Sankaran [11] have
determined certain representations $A_{q}$ of Hodge type $(p,p)$ when $G/K$ is
an irreducible Hermitian symmetric space. If $G$ is a complex simple Lie
group, the number of equivalence classes of the representations $A_{q},$ and
Poincaré polynomials of cohomologies of some of these representations have
been determined in [13]. In this article, we have determined the number of
equivalence classes of the representations $A_{q},$ and Poincaré polynomials
of cohomologies of these representations, when $G=SO_{0}(2,m)$ for any
positive integer $m.$ The main results are stated as follows:

######  Theorem 1.1.

(i) If $A$ is the number of equivalence classes of irreducible unitary
representations with non-zero $({g},K)$-cohomology of the Lie group
$SO_{0}(2,m)(m\in\mathbb{N})$, then

$A=\begin{cases}l(l+2)&\textrm{if }m=2l-1,\\\ l^{2}+4l-3&\textrm{if
}m=2l-2.\\\ \end{cases}$

(ii) An $A_{q}$ is unitarily equivalent to a discrete series representation of
$SO_{0}(2,m)$ with trivial infinitesimal character if and only if
$(-\Delta({u}\cap{p}_{-}))\cup\Delta({u}\cap{p}_{+})=\Delta_{n}^{+}.$ Also if
$m\neq 2,$ an $A_{q}$ is unitarily equivalent to a holomorphic discrete series
representation of $SO_{0}(2,m)$ with trivial infinitesimal character if and
only if $\Delta({u}\cap{p}_{+})=\phi\textrm{ or }\Delta_{n}^{+}.$ If
$D$(respectively, $D_{h}$) is the number of equivalence classes of discrete
series representations (respectively, holomorphic discrete series
representations) of $SO_{0}(2,m)$ with trivial infinitesimal character, then
$D=2l$ if $m=2l-1,$ or $2l-2.$ Also $D_{h}=2$ if $m\neq 2.$ If $m=2,$ then
$D_{h}=4.$

We have also determined Poincaré polynomials of cohomologies of these
representations in the Table 1, Table 2. The proof of Th.1.1 is given in §4.1.
We have used Remark 3.3(iii) of [11] to prove Th.1.1(i), and an alternative
approach of $\theta$-stable parabolic subalgebras to determine the Poincaré
polynomial of cohomologies of these representations.

##  2\. Irreducible unitary representations with non-zero $({g},K)$-cohomology

Let $G$ be a connected semisimple Lie group with finite centre, and $K$ be a
maximal compact subgroup of $G$ with Cartan involution $\theta.$ The
differential of $\theta$ at identity is denoted by the same notation $\theta.$
Let ${g}_{0}=$Lie$(G),{k}_{0}=$Lie$(K),$ and ${g}_{0}={k}_{0}\oplus{p}_{0}$ be
the Cartan decomposition corresponding to $\theta.$ Let
${g}={g}_{0}^{\mathbb{C}},{k}={k}_{0}^{\mathbb{C}}\subset{g},{p}={p}_{0}^{\mathbb{C}}\subset{g}.$

A $\theta$-stable parabolic subalgebra of ${g}_{0}$ is a parabolic subalgebra
${q}$ of ${g}$ such that (a) $\theta({q})={q}$, and (b) $\bar{{q}}\cap{q}={l}$
is a Levi subalgebra of ${q}$; where $\bar{\ }$ denotes the conjugation of
${g}$ with respect to ${g}_{0}$. By (b), ${l}$ is the complexification of a
real subalgebra ${l}_{0}$ of ${g}_{0}$. Also $\theta({l}_{0})={l}_{0}$ and and
${l}_{0}$ contains a maximal abelian subalgebra ${t}_{0}$ of ${k}_{0}$. Then
${h}_{0}={z}_{{g}_{0}}({t}_{0})$ is a $\theta$-stable Cartan subalgebra of
${g}_{0}.$ Let ${t}={t}_{0}^{\mathbb{C}}\subset{k},$ and
${h}={h}_{0}^{\mathbb{C}}\subset{g}.$ Note that ${t},{h}$ are Cartan
subalgebras of ${k},{g}$ respectively and ${h}\subset{q}.$ Let ${u}$ be the
nilradical of ${q}$ so that ${q}={l}\oplus{u}.$ Then ${u}$ is $\theta$-stable
and so ${u}=({u}\cap{k})\oplus({u}\cap{p}).$

If $V$ is a finite dimensional complex $A$-module, where $A$ is an abelian Lie
algebra; we denote by $\Delta(V)$( or by $\Delta(V,A)$), the set of all non-
zero weights of $V;$ by $V^{\alpha},$ the weight space of $V$ corresponding to
a weight $\alpha\in\Delta(V);$ and by $\delta(V)$ (or by $\delta(V,A)$), $1/2$
of the sum of elements in $\Delta(V)$ counted with their respective
multiplicities. Fix a maximal abelian subspace ${t}_{0}$ of ${k}_{0}$ and
${t}={t}_{0}^{\mathbb{C}}\subset{k}.$ Since ${k},{g}$ are ${t}$-modules (under
the adjoint action), we have

${k}={t}\oplus\sum_{\alpha\in\Delta({k},{t})}{k}^{\alpha},\textrm{and
}{g}={h}\oplus\sum_{\alpha\in\Delta({g},{t})}{g}^{\alpha}.$

Note that $\Delta({k},{t})$ is actually the set of all non-zero roots of ${k}$
relative to the Cartan subalgebra ${t}.$ Choose a system of positive roots
$\Delta_{k}^{+}$ in $\Delta({k},{t}).$ If $x\in i{t}_{0}$ be such that
$\alpha(x)\geq 0$ for all $\alpha\in\Delta_{k}^{+},$ then
${q}_{x}={h}\oplus\sum_{\alpha\in\Delta({g},{t}),\alpha(x)\geq 0}{g}^{\alpha}$
is a $\theta$-stable parabolic subalgebra of ${g}_{0}$,
${l}_{x}={h}\oplus\sum_{\alpha\in\Delta({g},{t}),\alpha(x)=0}{g}^{\alpha}$ is
the Levi subalgebra of ${q}_{x},$ and
${u}_{x}=\sum_{\alpha\in\Delta({g},{t}),\alpha(x)>0}{g}^{\alpha}$ is the
nilradical of ${q}_{x}.$ If ${q}$ is a $\theta$-stable parabolic subalgebra of
${g}_{0}$, there exists $k\in K$ such that $Ad(k)({q})={q}_{x}.$

Now associated with a $\theta$-stable parabolic subalgebra ${q}$, we have an
irreducible unitary representation $\mathcal{R}^{S}_{q}(\mathbb{C})=A_{q}$ of
$G$ with trivial infinitesimal character, where $S=\textrm{dim}({u}\cap{k}).$
The associated $({g},K)$-module $A_{{q},K}$ contains an irreducible
$K$-submodule $V$ of highest weight (with respect to $\Delta^{+}_{k}$)
$2\delta({u}\cap{p},{t})=\sum_{\beta\in\Delta({u}\cap{p},{t})}\beta$ and it
occurs with multiplicity one in $A_{{q},K}$. Any other irreducible $K$-module
that occurs in $A_{{q},K}$ has highest weight of the form
$2\delta({u}\cap{p},{t})+\sum_{\gamma\in\Delta({u}\cap{p},{t})}n_{\gamma}\gamma,$
with $n_{\gamma}$ a non-negative integer [17, Th. 2.5]. The $({g},K)$-modules
$A_{{q},K}$ were first constructed, in general, by Parthasarathy [12]. Vogan
and Zuckerman [17] gave a construction of the $({g},K)$-modules $A_{{q},K}$
via cohomological induction and Vogan [15] proved that these are unitarizable.
Define an equivalence relation on the set of all $\theta$-stable parabolic
subalgebras of ${g}_{0},$ by ${q}$ is equivalent to ${q}^{\prime}$ if either
$Ad(k)({q})={q}^{\prime},$ for some $k\in K,$ or
${u}\cap{p}={u}^{\prime}\cap{p}.$ Also unitary equivalence is an equivalence
relation on the set of all irreducible unitary representations $A_{q}.$ Then
the set of all equivalence classes of $\theta$-stable parabolic subalgebras
are in one-one correspondence with the set of all equivalence classes of the
irreducible unitary representations $A_{q}$ [14, Prop. 4.5].

If ${q}$ is a $\theta$-stable parabolic subalgebra of ${g}$, then the Levi
subgroup $L=\\{g\in G:\textrm{Ad}(g)({q})={q}\\}$ is a connected reductive Lie
subgroup of $G$ with Lie algebra ${l}_{0}.$ As $\theta({l}_{0})={l}_{0},L\cap
K$ is a maximal compact subgroup of $L$. One has

$H^{r}({g},K;A_{{q},K})\cong H^{r-R({q})}({l},L\cap K;\mathbb{C}),$

where $R({q}):=\textrm{dim}({u}\cap{p})$. Let $Y_{q}$ denote the compact dual
of the Riemannian globally symmetric space $L/{L\cap K}$. Then
$H^{r}({l},L\cap K;\mathbb{C})\cong H^{r}(Y_{q};\mathbb{C})$. And hence

$H^{r}({g},K;A_{{q},K})\cong H^{r-R({q})}(Y_{q};\mathbb{C}).$

If $P_{q}(t)$ denotes the Poincaré polynomial of $H^{*}({g},K;A_{{q},K})$,
then by the above result, we have

$P_{q}(t)=t^{R({q})}P(Y_{q},t).$

Conversely, if $\pi$ is an irreducible unitary represention of $G$ with non-
zero $({g},K)$-cohomology, then $\pi$ is unitarily equivalent to $A_{q}$ for
some $\theta$-stable parabolic subalgebra ${q}$ of ${g}_{0}$ [17, Th. 4.1].
See also [16] for a beautiful description of the theory of $({g},K)$-modules
$A_{{q},K}.$

If rank$(G)=$ rank$(K)$ and ${q}$ is a $\theta$-stable Borel subalgebra that
is, ${q}$ is a Borel subalgebra of ${g}$ containing a Cartan subalgebra of
${k}$, then $A_{q}$ is a discrete series representation of $G$ with trivial
infinitesimal character. In this case, $R({q})=\frac{1}{2}\textrm{ dim}(G/K)$,
$L$ is a maximal torus in $K$ and hence

$H^{r}({g}^{\mathbb{C}},K;A_{{q},K})=\begin{cases}0&\textrm{if }r\neq
R({q}),\\\ \mathbb{C}&\textrm{if }r=R({q}).\end{cases}$

If we take ${q}={g}$, then $L=G$ and $A_{q}=\mathbb{C}$, the trivial
representation of $G$. If $G/K$ is Hermitian symmetric, choose a Borel-de
Siebenthal positive root system in $\Delta({g},{t})$ containing
$\Delta_{k}^{+},$ and a unique non-compact simple root $\nu;$ and define
${p}_{+}=\sum_{\beta\in\Delta({g},{t}),n_{\nu}(\beta)=1}{g}^{\beta},{p}_{-}=\sum_{\beta\in\Delta({g},{t}),n_{\nu}(\beta)=-1}{g}^{\beta};$
where $n_{\nu}(\beta)$ is the coefficient of $\nu$ in the decomposition of
$\beta$ into simple roots. Then
${p}={p}_{+}\oplus{p}_{-},{u}\cap{p}=({u}\cap{p}_{+})\oplus({u}\cap{p}_{-}).$
Define $R_{+}({q})=$dim$({u}\cap{p}_{+}),R_{-}({q})=$dim$({u}\cap{p}_{-}).$ So
$R({q})=R_{+}({q})+R_{-}({q}).$ One has a Hodge decomposition

$H^{r}({g},K;A_{{q},K})=\oplus_{p+q=r}H^{p,q}({g},K;A_{{q},K})=H^{p,q}({g},K;A_{{q},K})\cong
H^{p-R_{+}({q}),q-R_{-}({q})}(Y_{q};\mathbb{C});$

where $p+q=r,p-q=R_{+}({q})-R_{-}({q}).$ See [2, Ch. II, §4], [4], [17]. The
pair $(R_{+}({q}),R_{-}({q}))$ is referred to be the Hodge type of the
representation $A_{q}.$

##  3\. Discrete series representations

We follow the notations from the previous section. Assume that rank$(G)=$
rank$(K),$ so that $G$ admits discrete series representation. A non-singular
linear function $\lambda$ on $i{t}_{0}$ relative to $\Delta({g},{t})$ dominant
with respect to $\Delta_{k}^{+},$ defines uniquely a positive root system
$\Delta_{\lambda}^{+}$ of $\Delta({g},{t})$ containing $\Delta_{k}^{+}.$
Define
$\delta_{g}=\frac{1}{2}\sum_{\alpha\in\Delta_{\lambda}^{+}}\alpha,\delta_{k}=\frac{1}{2}\sum_{\alpha\in\Delta_{k}^{+}}\alpha.$
If $\lambda+\delta_{g}$ is analytically integral(that is, $\lambda+\delta_{g}$
is the differential of a Lie group homomorphism on the Cartan subgroup of $G$
corresponding to ${t}_{0}$), then there exists a discrete series
representation $\pi_{\lambda}$ with infinitesimal character
$\chi_{\lambda}$(it is the character of the Verma module of ${g}$ with highest
weight $\lambda-\delta_{g}$); the associated $({g},K)$-module
$\pi_{\lambda,K}$ contains an irreducible $K$-submodule with highest weight
$\Lambda=\lambda+\delta_{g}-2\delta_{k}$ and it occurs with multiplicity one
in $\pi_{\lambda,K}.$ Any other irreducible $K$-module that occurs in
$\pi_{\lambda,K}$ has highest weight of the form
$\Lambda+\sum_{\alpha\in\Delta_{\lambda}^{+}}n_{\alpha}\alpha,$ with
$n_{\alpha}$ a non-negative integer. Upto unitary equivalence these are all
discrete series representations of $G.$ This $\lambda$ is called the Harish-
Chandra parameter, $\Lambda$ is called the Blattner parameter of the discrete
series representation $\pi_{\lambda}.$ The positive root system
$\Delta_{\lambda}^{+}$ is called the Harish-Chandra root order corresponding
to $\lambda.$ If $G/K$ is Hermitian symmetric, then $\pi_{\lambda}$ is a
holomorphic discrete series representation if and only if the Harish-Chandra
root order corresponding to $\lambda$ is a Borel-de Siebenthal positive root
system. See [6], [7].

##  4\. Irreducible unitary representations with non-zero $({g},K)$-cohomology
of the Lie group $SO_{0}(2,m)$

Let $I_{2,m}=\left(\begin{array}[]{ccc}-I_{2}&0\\\ 0&I_{m}\\\
\end{array}\right)$, where $I_{m}$ denotes the identity matrix of order $m$.
Let $G$ be the group $SO_{0}(2,m)$, the connected component of the group
$\\{g\in SL(m+2,\mathbb{R}):g^{t}I_{2,m}g=I_{2,m}\\}$. Then $G$ is a Lie group
with Lie algebra ${g}_{0}={so}(2,m)$ =
$\\{\left(\begin{array}[]{ccc}X_{1}&X_{2}\\\ X_{2}^{t}&X_{3}\\\
\end{array}\right):\textrm{all }X_{i}\textrm{ real},X_{1},X_{3}\textrm{ skew
symmetric of order }2\textrm{ and }m\textrm{ respectively},X_{2}\textrm{
arbitrary}\\}$. The map $\theta:G\longrightarrow G$ given by
$\theta(g)=I_{2,m}gI_{2,m}$ for all $g\in G$ is a Cartan involution with
maximal compact subgroup $K=\\{\left(\begin{array}[]{ccc}A&0\\\ 0&B\\\
\end{array}\right):A\in SO(2),B\in SO(m)\\}\cong SO(2)\times SO(m)$. The
differential of $\theta$ at the identity element of $G$ is the map $X\mapsto
I_{2,m}XI_{2,m}$ for all $X\in{g}_{0}$, and is denoted by the same notation
$\theta:{g}_{0}\longrightarrow{g}_{0}$. Then
$\theta:{g}_{0}\longrightarrow{g}_{0}$ is a Cartan involution and
${g}_{0}={k}_{0}\oplus{p}_{0}$ is the Cartan decomposition corresponding to
$+1$ and $-1$-eigenspaces of $\theta$. Note that
${k}_{0}=\\{\left(\begin{array}[]{ccc}A&0\\\ 0&B\\\
\end{array}\right):A\in{so}(2),B\in{so}(m)\\}\cong{so}(2)\oplus{so}(m)$, and
it is the Lie subalgebra of ${g}_{0}$ corresponding to the connected Lie
subgroup $K$ of $G$. Note that $G/K$ is an irreducible Hermitian symmetric
space of non-compact type.

The complexification of ${g}_{0}$ is ${g}={so}(m+2,\mathbb{C})$, and

${g}=\begin{cases}{b}_{l}&\textrm{if }m=2l-1,\\\ {\delta}_{l}&\textrm{if
}m=2l-2.\end{cases}$

Let ${k}={k}_{0}^{\mathbb{C}}\subset{g},{p}={p}_{0}^{\mathbb{C}}\subset{g},$
and ${t}^{\prime}_{0}$ be a maximal abelian subspace of ${so}(m)$. Then
${t}_{0}={so}(2)\oplus{t}^{\prime}_{0}$ be a maximal abelian subspace of
${k}_{0}$, and ${h}={t}_{0}^{\mathbb{C}}$ is a Cartan subalgebra of ${k}$ as
well as of ${g}$. Let $\Delta=\Delta({g},{h})$ be the set of all non-zero
roots of ${g}$ with respect to the Cartan subalgebra ${h},$ similarly
$\Delta_{k}=\Delta({k},{h})$ be the set of all non-zero roots of ${k}$ with
respect to ${h},$ and $\Delta_{n}=\Delta\setminus\Delta_{k}=$ the set of all
non-compact roots of ${g}$ with respect to ${h}$. Then
${k}={h}+\sum_{\alpha\in\Delta_{k}}{g}^{\alpha},{p}=\sum_{\alpha\in\Delta_{n}}{g}^{\alpha},$
where ${g}^{\alpha}$ is the root subspace of ${g}$ of the root
$\alpha\in\Delta$. Let $B$ denote the Killing form of ${g}$. For any linear
function $\lambda$ on ${h},$ there exists unique $H_{\lambda}\in{h}$ such that

$\lambda(H)=B(H,H_{\lambda})\textrm{ for all }H\in{h}.$

Put $\langle\lambda,\mu\rangle=B(H_{\lambda},H_{\mu})$ for any linear
functions $\lambda,\mu$ on ${h}$,
$H_{\alpha}^{*}=2H_{\alpha}/\alpha(H_{\alpha})$ for all $\alpha\in\Delta,$ and
${h}_{\mathbb{R}}=\sum_{\alpha\in\Delta}\mathbb{R}H_{\alpha}$. Then
${h}_{\mathbb{R}}=i{t}_{0}.$

For $m\neq 2,$ let $\Delta^{+}$ be a Borel-de Siebenthal positive root system
of $\Delta$ with a unique non-compact simple root $\phi_{1}$, that is

$n_{\phi_{1}}(\alpha)=\begin{cases}0&\textrm{if }\alpha\in\Delta_{k},\\\ \pm
1&\textrm{if }\alpha\in\Delta_{n}.\end{cases}$

If $m=2,$ then $\Delta^{+}=\\{\phi_{1},\phi_{2}\\},$ where both of $\phi_{1},$
and $\phi_{2}$ are non-compact and simple. Let
$\Delta_{k}^{+}=\Delta^{+}\cap\Delta_{k},\Delta_{n}^{+}=\Delta^{+}\cap\Delta_{n},$
and $\Delta_{n}^{-}=-\Delta_{n}^{+}$. Write
${p}_{+}=\sum_{\alpha\in\Delta_{n}^{+}}{g}^{\alpha},$ and
${p}_{-}=\sum_{\alpha\in\Delta_{n}^{-}}{g}^{\alpha}$. Then
${p}={p}_{+}\oplus{p}_{-}$ is the irreducible decomposition of ${p}$ under the
adjoint representation of ${k},$ if $m\neq 2$. For $m\neq 2,$ let
$\Phi_{k}=\\{\phi_{2},\phi_{3},\ldots,\phi_{l}\\}$ be the set of all simple
roots in in $\Delta_{k}^{+}$. Then
$\Phi=\\{\phi_{1},\phi_{2},\ldots,\phi_{l}\\}$ is the set of all simple roots
in $\Delta$. In the diagrams of this article, the non-compact roots are
represented by black vertices.

Figure 1. Dynkin diagram of
${g}={so}(m+2,\mathbb{C})$$\phi_{1}$$\phi_{2}$$\phi_{3}$$\phi_{l-1}$$\phi_{l}$${b}_{l}:$(
if
$m=2l-1$)$\phi_{1}$$\phi_{2}$$\phi_{3}$$\phi_{l-3}$$\phi_{l-2}$$\phi_{l-1}$$\phi_{l}$${\delta}_{l}:$(
if $m=2l-2\neq 2$)$\phi_{1}$$\phi_{2}$${\delta}_{2}:$( if $m=2$)

Since $A_{\textrm{Ad}(k)({q})}$ is unitarily equivalent to $A_{q}$ for all
$k\in K,$ to determine all unitarily inequivalent $A_{q},$ it is sufficient to
determine all $\theta$-stable parabolic subalgebras ${q}$ of ${g}_{0}$ which
contain ${h}\oplus\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha}$. Let ${q}$ be a
$\theta$-stable parabolic subalgebra of ${g}_{0}$ containing
${h}\oplus\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha}$. Then there exists
$x\in{h}_{\mathbb{R}}$ such that ${q}={q}_{x}={h}\oplus\sum_{\alpha(x)\geq
0,\alpha\in\Delta}{g}^{\alpha}={l}_{x}\oplus{u}_{x},$ where
${l}_{x}={h}\oplus\sum_{\alpha(x)=0,\alpha\in\Delta}{g}^{\alpha}$ is the Levi
subalgebra of ${q}_{x},$ and
${u}_{x}=\sum_{\alpha(x)>0,\alpha\in\Delta}{g}^{\alpha}$ is the nilradical of
${q}_{x}$. Note that $\alpha(x)\geq 0$ for all $\alpha\in\Delta_{k}^{+}.$
Write $\Delta({u}_{x}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta(x)>0\\},$
and $\Delta({u}_{x}\cap{p}_{-})=\\{\beta\in\Delta_{n}^{-}:\beta(x)>0\\}$. For
$x,y\in{h}_{\mathbb{R}},A_{{q}_{x}}$ is unitarily equivalent to $A_{{q}_{x}}$
iff
$\Delta({u}_{x}\cap{p}_{+})\cup\Delta({u}_{x}\cap{p}_{-})=\Delta({u}_{y}\cap{p}_{+})\cup\Delta({u}_{y}\cap{p}_{-})$.
So we will determine all possible candidates of
$\Delta({u}_{x}\cap{p}_{+})\cup\Delta({u}_{x}\cap{p}_{-}),$ where
$x\in{h}_{\mathbb{R}}$ with $\alpha(x)\geq 0$ for all
$\alpha\in\Delta_{k}^{+}$. For $x\in{h}_{\mathbb{R}}$ with $\alpha(x)\geq 0$
for all $\alpha\in\Delta_{k}^{+},$ we may write $x=H_{\lambda}$ for some
linear function $\lambda$ on ${h}_{\mathbb{R}}$ with
$\langle\lambda,\alpha\rangle\geq 0$ for all $\alpha\in\Delta_{k}^{+}$. We
write ${q}_{\lambda}={q}_{x},{l}_{\lambda}={l}_{x},$ and
${u}_{\lambda}={u}_{x}$. Thus
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\langle\lambda,\beta\rangle>0\\},$
and
$\Delta({u}_{\lambda}\cap{p}_{-})=\\{\beta\in\Delta_{n}^{-}:\langle\lambda,\beta\rangle>0\\}$.
Clearly $\Delta({u}_{\lambda}\cap{p}_{+})$ and
$-\Delta({u}_{\lambda}\cap{p}_{-})$ are disjoint subsets of $\Delta_{n}^{+}$.

Now we begin our proofs with this elementary lemma.

######  Lemma 4.1.

[11, Remark 3.3(iii)] Let $\lambda$ be a linear function on ${h}_{\mathbb{R}}$
such that $\langle\lambda,\alpha\rangle\geq 0$ for all
$\alpha\in\Delta_{k}^{+}$.
(i) Let $\beta,\gamma\in\Delta_{n}$ be such that $\gamma>\beta,$ and both
belong to $\Delta_{n}^{+}$ or $\Delta_{n}^{-}$. Then
$\langle\lambda,\beta\rangle>0\implies\langle\lambda,\gamma\rangle>0$.
(ii) Let $\phi\in\Delta_{k}^{+}$ be simple and $\beta\in\Delta_{n}$. If
$\beta-\phi\in\Delta,\langle\lambda,\beta-\phi\rangle=0,$ and
$\langle\lambda,\beta\rangle>0,$ then $\langle\lambda,\phi\rangle>0$.
(iii) Let $\phi\in\Delta_{k}^{+}$ be simple and $\beta\in\Delta_{n}$. If
$\beta+\phi\in\Delta,\langle\lambda,\beta+\phi\rangle=0,$ and
$\langle\lambda,\beta\rangle=0,$ then $\langle\lambda,\phi\rangle=0$. If
$\beta-\phi\in\Delta,\langle\lambda,\beta-\phi\rangle=0,$ and
$\langle\lambda,\beta\rangle=0,$ then $\langle\lambda,\phi\rangle=0$.

###### Proof.

(i) Let $\beta,\gamma\in\Delta_{n}$ be such that $\gamma>\beta$ and both
belong to $\Delta_{n}^{+}$ or $\Delta_{n}^{-}$. Then $\gamma=\beta+\sum_{2\leq
i\leq l}n_{i}\phi_{i},$ where $n_{i}\in\mathbb{N}\cup\\{0\\}$ for all $2\leq
i\leq l$. Since $\langle\lambda,\beta\rangle>0,$ and
$\langle\lambda,\phi_{i}\rangle\geq 0$ for all $2\leq i\leq l,$ we have
$\langle\lambda,\gamma\rangle>0$.

(ii)
$\langle\lambda,\beta-\phi\rangle=0\implies\langle\lambda,\phi\rangle=\langle\lambda,\beta\rangle>0$.

(iii)
$\langle\lambda,\beta+\phi\rangle=0\implies\langle\lambda,\phi\rangle=-\langle\lambda,\beta\rangle=0,$
and
$\langle\lambda,\beta-\phi\rangle=0\implies\langle\lambda,\phi\rangle=\langle\lambda,\beta\rangle=0$.
∎

Lemma 4.1(i) says that $\Delta({u}_{\lambda}\cap{p}_{+})$ is either empty or a
set of the form $\cup_{1\leq i\leq
r}\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{i}\\},$ and
$\Delta({u}_{\lambda}\cap{p}_{-})$ is either empty or a set of the form
$\cup_{1\leq j\leq
s}\\{-\beta\in\Delta_{n}^{-}:-\beta\geq-\eta_{j}\\}=\cup_{1\leq j\leq
s}(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\eta_{j}\\})$, where
$\\{\xi_{1},\xi_{2},\ldots,\xi_{r}\\},\\{\eta_{1},\eta_{2},\ldots,\eta_{s}\\}$
are sets of pairwise non-comparable roots in $\Delta_{n}^{+}$.

If ${g}={b}_{l}(l\geq 2)$, then
$\Delta_{n}^{+}=\\{\phi_{1},\phi_{1}+\phi_{2},\ldots,\phi_{1}+\phi_{2}+\cdots+\phi_{l},\phi_{1}+\phi_{2}+\cdots+2\phi_{l},\phi_{1}+\phi_{2}+\cdots+2\phi_{l-1}+2\phi_{l},\ldots,\phi_{1}+2\phi_{2}+\cdots+2\phi_{l}\\}$.
If ${g}={b}_{1}$, then $\Delta_{n}^{+}=\\{\phi_{1}\\}$. If
${g}={\delta}_{l}(l\geq 4)$, then
$\Delta_{n}^{+}=\\{\phi_{1},\phi_{1}+\phi_{2},\ldots,\phi_{1}+\phi_{2}+\cdots+\phi_{l-2},\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}+\phi_{l-1},\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}+\phi_{l},\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}+\phi_{l-1}+\phi_{l},\phi_{1}+\phi_{2}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l},\ldots,\phi_{1}+2\phi_{2}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}$.
If ${g}={\delta}_{2}$, then $\Delta_{n}^{+}=\\{\phi_{1},\phi_{2}\\}$. If
${g}={\delta}_{3}$, then
$\Delta_{n}^{+}=\\{\phi_{1},\phi_{1}+\phi_{2},\phi_{1}+\phi_{3},\phi_{1}+\phi_{2}+\phi_{3}\\}$.

Figure 2. Diagram of
$\Delta_{n}^{+}$$\phi_{1}$$\phi_{1}+\phi_{2}$$\phi_{1}+\phi_{2}+\cdots+\phi_{l}$$\phi_{1}+\phi_{2}+\cdots+2\phi_{l}$$\phi_{1}+\phi_{2}+\cdots+2\phi_{l-1}+2\phi_{l}$$\phi_{1}+2\phi_{2}+\cdots+2\phi_{l-1}+2\phi_{l}$$\Delta_{n}^{+}:$(for
${g}={b}_{l}$,$l\geq
2$)$\phi_{2}$$\phi_{3}$$\phi_{l}$$\phi_{l}$$\phi_{l-1}$$\phi_{l-2}$$\phi_{2}$$\phi_{1}$$\phi_{1}+\phi_{2}$$\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}$$\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}+\phi_{l-1}$$(=\xi_{1})$$\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}+\phi_{l}$$(=\xi_{2})$$\phi_{1}+\phi_{2}+\cdots+\phi_{l}$$\phi_{1}+\phi_{2}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}$$\phi_{1}+2\phi_{2}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}$$\Delta_{n}^{+}:$(for
${g}={\delta}_{l}$,$l\geq
3$)$\phi_{2}$$\phi_{3}$$\phi_{l-2}$$\phi_{l-1}$$\phi_{l}$$\phi_{l}$$\phi_{l-1}$$\phi_{l-2}$$\phi_{l-3}$$\phi_{2}$

$\phi_{1}$$\phi_{2}$$\Delta_{n}^{+}:$(for ${g}={\delta}_{2}$)

In the Figure 2, the vertices represent roots in $\Delta_{n}^{+}$. Two roots
$\beta,\gamma\in\Delta_{n}^{+}$ are joined by a line with an arrow in the
direction of $\gamma$ if $\gamma=\beta+\phi$ for some simple root
$\phi\in\Delta_{k}^{+}$. In this case, the simple root $\phi$ is given on one
side of the line.

###  4.1. Proof of Th.1.1

Let $\omega_{1},\omega_{2},\ldots,\omega_{l}$ be the fundamental weights of
${g}$ corresponding to the simple roots $\phi_{1},\phi_{2},\ldots,\phi_{l}$
respectively.

(i) ${g}={b}_{l}(l>1):$ Lemma 4.1(i) and the diagram of $\Delta_{n}^{+}$ in
Figure 2 show that $\Delta({u}_{\lambda}\cap{p}_{+})$ is either empty or a set
of the form $\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},$ and
$\Delta({u}_{\lambda}\cap{p}_{-})$ is either empty or a set of the form
$\\{-\beta\in\Delta_{n}^{-}:-\beta\geq-\eta\\}=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\eta\\}$,
and $\Delta({u}_{\lambda}\cap{p}_{+}),-\Delta({u}_{\lambda}\cap{p}_{-})$ are
disjoint subsets of $\Delta_{n}^{+}$, where $\xi,\eta\in\Delta_{n}^{+}.$
Let $\Delta({u}_{\lambda}\cap{p}_{-})$ be empty. Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},$
where $\xi>\phi_{1}+\phi_{2}+\cdots+\phi_{l}$ is not possible. For then
$\xi=\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l},$ where $2\leq
i\leq l$. So $\langle\lambda,\phi_{i}\rangle>0,$ by Lemma 4.1(ii). Again
$\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{i}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{i-1}\rangle=0.$
Thus $\langle\lambda,\phi_{i}\rangle=0,$ a contradiction. If
$\xi\leq\phi_{1}+\phi_{2}+\cdots+\phi_{l},$ then
$\xi=\phi_{1}+\cdots+\phi_{i},$ for some $1\leq i\leq l,$ and
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},\Delta({u}_{\lambda}\cap{p}_{-})=\phi,$
where $\lambda=\omega_{i}.$ Also
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=\phi,$
for $\lambda=0.$ Thus the number of equivalence classes of irreducible unitary
representations with non-zero $({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=\phi,$ is $l+1$.
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},$
where $1\leq i\leq l-1.$ Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},$
where
$\xi>\phi_{1}+\cdots+\phi_{l},\xi\neq\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l}$
is not possible. Because
$\xi>\phi_{1}+\cdots+\phi_{l},\xi\neq\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l}$
implies $\xi=\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l}$ for some
$2\leq j\leq l,j\neq i+1.$ Then
$\langle\lambda,\phi_{i+1}\rangle>0,\langle\lambda,\phi_{j}\rangle>0,$ by
Lemma 4.1(ii). If $2\leq j\leq i,$ then
$\langle\lambda,\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l}\rangle=0,\langle\lambda,\phi_{1}+\cdots+\phi_{i+1}+2\phi_{i+2}+\cdots+2\phi_{l}\rangle=0$.
Thus $\langle\lambda,\phi_{i+1}\rangle=0$, a contradiction. If $i+2\leq j\leq
l,$ then
$\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{j}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{j-1}\rangle=0$.
Thus $\langle\lambda,\phi_{j}\rangle=0$, a contradiction. If
$\xi\leq\phi_{1}+\phi_{2}+\cdots+\phi_{l},$ then
$\xi=\phi_{1}+\cdots+\phi_{j},$ for some $i+1\leq j\leq l,$ and
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},$
where
$\lambda=\frac{\omega_{i+1}}{\langle\omega_{i+1},\phi_{i+1}\rangle}+\frac{\omega_{j}}{\langle\omega_{j},\phi_{j}\rangle}-\frac{\omega_{1}}{\langle\omega_{1},\phi_{1}\rangle}.$
If $\xi=\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l},$ then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},$
for
$\lambda=\frac{\omega_{i+1}}{\langle\omega_{i+1},\phi_{i+1}\rangle}-\frac{\omega_{1}}{\langle\omega_{1},\phi_{1}\rangle}.$
Also $\Delta({u}_{\lambda}\cap{p}_{+})=\phi$ is not possible, for
$\langle\lambda,\phi_{i+1}\rangle>0,$ by Lemma 4.1(ii); and
$\langle\lambda,\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l}\rangle=0,\langle\lambda,\phi_{1}+\cdots+\phi_{i+1}+2\phi_{i+2}+\cdots+2\phi_{l}\rangle=0$.
Thus $\langle\lambda,\phi_{i+1}\rangle=0$, a contradiction. Hence the number
of equivalence classes of irreducible unitary representations with non-zero
$({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},$
is $l-i+1$ for all $1\leq i\leq l-1.$
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l}\\}.$
Since
$\xi>\phi_{1}+\cdots+\phi_{l},\xi=\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l}$
for some $2\leq j\leq l.$ Now
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l}\\},$
for
$\lambda=\frac{\omega_{l}}{\langle\omega_{l},\phi_{l}\rangle}+\frac{\omega_{j}}{\langle\omega_{j},\phi_{j}\rangle}-\frac{3\omega_{1}}{\langle\omega_{1},\phi_{1}\rangle}.$
Also
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l}\\},$
for
$\lambda=\frac{\omega_{l}}{\langle\omega_{l},\phi_{l}\rangle}-\frac{2\omega_{1}}{\langle\omega_{1},\phi_{1}\rangle}.$
Thus the number of equivalence classes of irreducible unitary representations
with non-zero $({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l}\\},$
is $l.$
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l}\\},2\leq
i\leq l.$ Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l}\\},$
for
$\lambda=\frac{\omega_{i-1}}{\langle\omega_{i-1},\phi_{i-1}\rangle}-\frac{2\omega_{1}}{\langle\omega_{1},\phi_{1}\rangle}$
if $3\leq i\leq l,$ and $\lambda=-\omega_{1}$ if $i=2.$ If
$\xi=\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l}$ for some $2\leq
j<i,$ then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l}\\},$
for
$\lambda=\frac{\omega_{i-1}}{\langle\omega_{i-1},\phi_{i-1}\rangle}+\frac{\omega_{j}}{\langle\omega_{j},\phi_{j}\rangle}-\frac{3\omega_{1}}{\langle\omega_{1},\phi_{1}\rangle}.$
Hence the number of equivalence classes of irreducible unitary representations
with non-zero $({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l}\\},$
is $i-1$ for all $2\leq i\leq l.$
Thus the number of equivalence classes of irreducible unitary representations
with non-zero $({g},K)$-cohomology of the Lie group $SO_{0}(2,2l-1)(l>1)$ is
$A=(l+1)+l+\cdots+2+l+1+\cdots+(l-1)=3l+(l-1)l/2+(l-1)l/2=l(l+2).$

${g}={b}_{1}:$ In this case $\Delta_{n}^{+}=\\{\phi_{1}\\}.$ If $\lambda=0,$
then $\Delta({u}_{\lambda}\cap{p}_{-}),\Delta({u}_{\lambda}\cap{p}_{+})$ are
empty. $\lambda=\omega_{1}$ implies
$\Delta({u}_{\lambda}\cap{p}_{-})=\phi,\Delta({u}_{\lambda}\cap{p}_{+})=\\{\phi_{1}\\};$
and $\lambda=-\omega_{1}$ implies
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=\\{-\phi_{1}\\}.$
Thus $A=3=l(l+2).$

${g}={\delta}_{l}(l\geq 3):$ Lemma 4.1(i) and the diagram of $\Delta_{n}^{+}$
in Figure 2 show that $\Delta({u}_{\lambda}\cap{p}_{+})$ is either empty or a
set of the form $\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},$ or
$\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{1}\textrm{ or }\xi_{2}\\},$ and
$\Delta({u}_{\lambda}\cap{p}_{-})$ is either empty or a set of the form
$\\{-\beta\in\Delta_{n}^{-}:-\beta\geq-\eta\\}=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\eta\\},$
or $-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\textrm{ or }\xi_{2}\\},$ where
$\xi,\eta\in\Delta_{n}^{+};\xi_{1}=\phi_{1}+\cdots+\phi_{l-2}+\phi_{l-1},\xi_{2}=\phi_{1}+\cdots+\phi_{l-2}+\phi_{l}.$
Let $\Delta({u}_{\lambda}\cap{p}_{-})$ be empty. Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},$
where $\xi\geq\phi_{1}+\phi_{2}+\cdots+\phi_{l}$ is not possible. For then
$\xi=\phi_{1}+\cdots+\phi_{l},$ or
$\xi=\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l},$
where $2\leq i\leq l-2$. If $\xi=\phi_{1}+\cdots+\phi_{l},$ then
$\langle\lambda,\phi_{l-1}\rangle>0,\langle\lambda,\phi_{l}\rangle>0,$ by
Lemma 4.1(ii). Again
$\langle\lambda,\phi_{1}+\cdots+\phi_{l-1}\rangle=0,\langle\lambda,\phi_{1}+\cdots+\phi_{l-2}+\phi_{l}\rangle=0,\langle\lambda,\phi_{1}+\cdots+\phi_{l-2}\rangle=0$.
Thus $\langle\lambda,\phi_{l-1}\rangle=0,\langle\lambda,\phi_{l}\rangle=0,$ a
contradiction. If
$\xi=\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}(2\leq
i\leq l-2),$ then $\langle\lambda,\phi_{i}\rangle>0,$ by Lemma 4.1(ii). Again
$\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{i}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{i-1}\rangle=0$.
Thus $\langle\lambda,\phi_{i}\rangle=0$, a contradiction. If
$\xi<\phi_{1}+\phi_{2}+\cdots+\phi_{l},$ then $\xi=\phi_{1}+\cdots+\phi_{i},$
for some $1\leq i\leq l-1,$ or $\xi=\xi_{2}$ and
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},\Delta({u}_{\lambda}\cap{p}_{-})=\phi,$
for $\lambda=\omega_{i},\omega_{l}$ respectively. Also
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{1}\textrm{
or }\xi_{2}\\},\Delta({u}_{\lambda}\cap{p}_{-})=\phi,$ for
$\lambda=\omega_{l-1}+\omega_{l},$ and
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=\phi,$
for $\lambda=0.$ Thus the number of equivalence classes of irreducible unitary
representations with non-zero $({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=\phi,$ is $l+2$.
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},$
where $1\leq i\leq l-3,l\geq 4.$ Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},$
where
$\xi\geq\phi_{1}+\cdots+\phi_{l},\xi\neq\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}$
is not possible. For if $\xi=\phi_{1}+\cdots+\phi_{l},$ then
$\langle\lambda,\phi_{l-1}\rangle>0,\langle\lambda,\phi_{l}\rangle>0$ by Lemma
4.1(ii). Again
$\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{l-1}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}+\phi_{l}\rangle=0.$
Thus $\langle\lambda,\phi_{l-1}\rangle=0,\langle\lambda,\phi_{l}\rangle=0,$ a
contradiction. Now
$\xi>\phi_{1}+\cdots+\phi_{l},\xi\neq\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}$
implies
$\xi=\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}$
for some $2\leq j\leq l-2,j\neq i+1.$ Then
$\langle\lambda,\phi_{i+1}\rangle>0,\langle\lambda,\phi_{j}\rangle>0,$ by
Lemma 4.1(ii). If $2\leq j\leq i,$ then
$\langle\lambda,\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l}\rangle=0,\langle\lambda,\phi_{1}+\cdots+\phi_{i+1}+2\phi_{i+2}+\cdots+2\phi_{l}\rangle=0$.
Thus $\langle\lambda,\phi_{i+1}\rangle=0$, a contradiction. If $i+2\leq j\leq
l-2,$ then
$\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{j}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{j-1}\rangle=0$.
Thus $\langle\lambda,\phi_{j}\rangle=0$, a contradiction. If
$\xi<\phi_{1}+\phi_{2}+\cdots+\phi_{l},$ then $\xi=\phi_{1}+\cdots+\phi_{j},$
for some $i+1\leq j\leq l-1,$ or $\xi=\xi_{2}.$ Now
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{j}\\}(i+1\leq
j\leq
l-1),\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\}$
for
$\lambda=\omega_{i+1}+\omega_{j}-\omega_{1};\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{2}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\}$
for $\lambda=\omega_{i+1}+\omega_{l}-\omega_{1};$ and
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l},\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\}$
for $\lambda=\omega_{i+1}-\omega_{1}.$ Also
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{1}\textrm{
or
}\xi_{2}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\}$
for $\lambda=\omega_{i+1}+\omega_{l-1}+\omega_{l}-\omega_{1}.$ Again
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi$ is not possible, for
$\langle\lambda,\phi_{i+1}\rangle>0,$ by Lemma 4.1(ii); and
$\langle\lambda,\phi_{1}+\cdots+\phi_{i}+2\phi_{i+1}+\cdots+2\phi_{l}\rangle=0,\langle\lambda,\phi_{1}+\cdots+\phi_{i+1}+2\phi_{i+2}+\cdots+2\phi_{l}\rangle=0$.
Thus $\langle\lambda,\phi_{i+1}\rangle=0$, a contradiction. Hence the number
of equivalence classes of irreducible unitary representations with non-zero
$({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},$
is $l-i+2$ for all $1\leq i\leq l-3.$
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}\\}.$
Then $\langle\lambda,\phi_{l-1}\rangle>0,\langle\lambda,\phi_{l}\rangle>0$ by
Lemma 4.1(ii). Hence $\Delta({u}_{\lambda}\cap{p}_{+})=\phi,$ or
$\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi\\},$ where
$\xi>\phi_{1}+\cdots+\phi_{l}$ is not possible. For then
$\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{l}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{l-1}\rangle=0,\langle\lambda,\phi_{1}+\phi_{2}+\cdots+\phi_{l-2}+\phi_{l}\rangle=0.$
Thus $\langle\lambda,\phi_{l}\rangle=0,\langle\lambda,\phi_{l-1}\rangle=0,$ a
contradiction. Now
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{1}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}\\}$
for
$\lambda=2\omega_{l-1}+\omega_{l}-\omega_{1};\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{2}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}\\}$
for $\lambda=\omega_{l-1}+2\omega_{l}-\omega_{1};$ and
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}\\}$
for $\lambda=\omega_{l-1}+\omega_{l}-\omega_{1}.$ Also
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{1}\textrm{
or
}\xi_{2}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}\\}$
for $\lambda=2\omega_{l-1}+2\omega_{l}-\omega_{1}.$ Hence the number of
equivalence classes of irreducible unitary representations with non-zero
$({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}\\},$
is $4.$
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\},$
where $a=l-1,l;$ and $b=l\textrm{ if }a=l-1,\textrm{and }b=l-1\textrm{ if
}a=l.$ Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l-2}+\phi_{b}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\}$
for
$\lambda=2\omega_{b}-\omega_{1};\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\}$
for
$\lambda=\omega_{a}+2\omega_{b}-2\omega_{1};\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}(2\leq
j\leq
l-2),\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\}$
for $\lambda=\omega_{j}+\omega_{b}-2\omega_{1}(a=l-1,l).$ Also
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\}$
for $\lambda=\omega_{b}-\omega_{1}.$ Thus the number of equivalence classes of
irreducible unitary representations with non-zero $({g},K)$-cohomology for
which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\}(a=l-1,l),$
is $l.$
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\textrm{
or }\xi_{2}\\}.$ Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\textrm{
or }\xi_{2}\\}$ for $\lambda=2\omega_{l-1}+2\omega_{l}-3\omega_{1};$
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}(2\leq
j\leq
l-2),\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\textrm{
or }\xi_{2}\\}$ for $\lambda=\omega_{j}+\omega_{l-1}+\omega_{l}-3\omega_{1};$
and
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\textrm{
or }\xi_{2}\\}$ for $\lambda=\omega_{l-1}+\omega_{l}-2\omega_{1}.$ Thus the
number of equivalence classes of irreducible unitary representations with non-
zero $({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\textrm{
or }\xi_{2}\\}$ is $l-1.$
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}.$
Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}(2\leq
j\leq
l-2),\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}$
for
$\lambda=\omega_{j}+\omega_{l-2}-3\omega_{1};\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}$
for $\lambda=\omega_{l-2}-2\omega_{1}.$ Thus the number of equivalence classes
of irreducible unitary representations with non-zero $({g},K)$-cohomology for
which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},$
is $l-2.$
Let
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}(2\leq
i\leq l-2),l\geq 4.$ Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}$
for $\lambda=\omega_{i-1}-2\omega_{1}$ if $3\leq i\leq l-2,$ and
$\lambda=-\omega_{1}$ if $i=2.$ Also
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{j-1}+2\phi_{j}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}(2\leq
j\leq
i-1),\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}$
for $\lambda=\omega_{i-1}+\omega_{j}-3\omega_{1}.$ Thus the number of
equivalence classes of irreducible unitary representations with non-zero
$({g},K)$-cohomology for which
$\Delta({u}_{\lambda}\cap{p}_{-})=-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}(2\leq
i\leq l-2),$ is $i-1.$
Hence $A=(l+2)+(l+1)+\cdots+5+4+l+l+(l-1)+(l-2)+(l-3)+\cdots+1=l^{2}+4l-3.$

${g}={\delta}_{2}:$ In this case $\Delta_{n}^{+}=\\{\phi_{1},\phi_{2}\\}.$ If
$\lambda=0,$ then
$\Delta({u}_{\lambda}\cap{p}_{-}),\Delta({u}_{\lambda}\cap{p}_{+})$ are empty.
$\lambda=\omega_{i}(1\leq i\leq 2)$ implies
$\Delta({u}_{\lambda}\cap{p}_{-})=\phi,\Delta({u}_{\lambda}\cap{p}_{+})=\\{\phi_{i}\\};\lambda=\omega_{1}+\omega_{2}$
implies
$\Delta({u}_{\lambda}\cap{p}_{-})=\phi,\Delta({u}_{\lambda}\cap{p}_{+})=\\{\phi_{1},\phi_{2}\\};\lambda=-\omega_{i}(1\leq
i\leq 2)$ implies
$\Delta({u}_{\lambda}\cap{p}_{-})=\\{-\phi_{i}\\},\Delta({u}_{\lambda}\cap{p}_{+})=\phi;\lambda=\omega_{1}-\omega_{2}$
implies
$\Delta({u}_{\lambda}\cap{p}_{-})=\\{-\phi_{2}\\},\Delta({u}_{\lambda}\cap{p}_{+})=\\{\phi_{1}\\};\lambda=\omega_{2}-\omega_{1}$
implies
$\Delta({u}_{\lambda}\cap{p}_{-})=\\{-\phi_{1}\\},\Delta({u}_{\lambda}\cap{p}_{+})=\\{\phi_{2}\\};$
and $\lambda=-\omega_{1}-\omega_{2}$ implies
$\Delta({u}_{\lambda}\cap{p}_{+})=\phi,\Delta({u}_{\lambda}\cap{p}_{-})=\\{-\phi_{1},-\phi_{2}\\}.$
Thus $A=9=l^{2}+4l-3.$

(ii) An irreducible unitary representation $\pi$ of $SO_{0}(2,m)$ with trivial
infinitesimal character is a discrete series representation if and only if
$\pi$ is unitarily equivalent to $A_{b},$ where ${b}$ is a Borel subalgebra of
${g}$ containing ${h}+\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha}.$ If ${b}$ is
a Borel subalgebra of ${g}$ containing
${h}+\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha},$ then
${b}={b}_{\lambda}={h}\oplus{u}_{\lambda}$ for some linear function $\lambda$
on ${h}_{\mathbb{R}}$ with $\langle\lambda,\alpha\rangle\neq 0$ for all
$\alpha\in\Delta,$ and $\langle\lambda,\alpha\rangle>0$ for all
$\alpha\in\Delta_{k}^{+}.$ Since $\langle\lambda,\beta\rangle\neq 0$ for all
$\beta\in\Delta_{n}^{+},$ for the irreducible unitary representation $A_{b}$
we have
$(-\Delta({u}_{\lambda}\cap{p}_{-}))\cup\Delta({u}_{\lambda}\cap{p}_{+})=\Delta_{n}^{+}.$

Conversely suppose that $\lambda$ be a linear function on ${h}_{\mathbb{R}}$
such that $\langle\lambda,\alpha\rangle\geq 0$ for all
$\alpha\in\Delta_{k}^{+},$ and
$(-\Delta({u}_{\lambda}\cap{p}_{-}))\cup\Delta({u}_{\lambda}\cap{p}_{+})=\Delta_{n}^{+}.$
Since $\langle\lambda,\alpha\rangle\geq 0$ for all $\alpha\in\Delta_{k}^{+},$
and $\langle\lambda,\beta\rangle\neq 0$ for all
$\beta\in\Delta_{n},\lambda=\sum_{1\leq i\leq
l}\frac{c_{i}\omega_{i}}{\langle\omega_{i},\phi_{i}\rangle},$ where $c_{1}$ is
a non-zero real number and $c_{i}$ is a non-negative real number for all
$2\leq i\leq l.$ If $c_{1}>0,$ let $d_{1}=c_{1};$ and $d_{i}=c_{i}\textrm{ if
}c_{i}\neq 0,\textrm{and }d_{i}=1\textrm{ if }c_{i}=0;$ for all $2\leq i\leq
l.$ If $c_{1}<0,$ let $\\{i:2\leq i\leq
l,c_{i}=0\\}=\\{i_{1},i_{2},\ldots,i_{k}\\},$ and
$m_{j}=\textrm{max}\\{n_{\phi_{i_{j}}}(\beta):\beta\in\Delta_{n}^{+},\langle\lambda,\beta\rangle<0\\}$
for all $1\leq j\leq k.$ Assume that ${g}\neq\delta_{2},$ and if
${g}=\delta_{l}(l\geq 3),$
$-\Delta({u}_{\lambda}\cap{p}_{-})\neq\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{i}\\},$
where $i=1,2.$ Then if
$\beta\in-\Delta({u}_{\lambda}\cap{p}_{-}),\beta^{\prime}\in\Delta({u}_{\lambda}\cap{p}_{+});\beta<\beta^{\prime},$
and so $n_{\phi_{i_{j}}}(\beta)\leq n_{\phi_{i_{j}}}(\beta^{\prime})$ for all
$1\leq j\leq k.$ In this case, if $c_{1}<0,$ let $d_{1}=c_{1}-\sum_{1\leq
j\leq k}m_{j};$ and $d_{i}=c_{i}\textrm{ if }c_{i}\neq 0,\textrm{and
}d_{i}=1\textrm{ if }c_{i}=0;$ for all $2\leq i\leq l.$ Let
$\lambda^{\prime}=\sum_{1\leq i\leq
l}\frac{d_{i}\omega_{i}}{\langle\omega_{i},\phi_{i}\rangle},$ for any
$c_{1}\in\mathbb{R}\setminus\\{0\\}.$ Then
$\langle\lambda^{\prime},\alpha\rangle>0$ for all $\alpha\in\Delta_{k}^{+};$
and if $\beta\in\Delta_{n},\
\langle\lambda,\beta\rangle<0\implies\langle\lambda^{\prime},\beta\rangle<0,$
and
$\langle\lambda,\beta\rangle>0\implies\langle\lambda^{\prime},\beta\rangle>0.$
Thus ${q}_{\lambda^{\prime}}$ is a Borel subalgebra of ${g}$ containing
${h}+\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha},$ and
$\Delta({u}_{\lambda^{\prime}}\cap{p}_{-})=\Delta({u}_{\lambda}\cap{p}_{-}),\Delta({u}_{\lambda^{\prime}}\cap{p}_{+})=\Delta({u}_{\lambda}\cap{p}_{+}).$
Thus $A_{{q}_{\lambda}}$ is is unitarily equivalent to
$A_{{q}_{\lambda^{\prime}}},$ which is a discrete series representation with
trivial infinitesimal character.
Let ${g}=\delta_{l}(l\geq 3),$ and
$-\Delta({u}_{\lambda}\cap{p}_{-})=\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{i}\\},$
where $i=1,2;$ that is
$-\Delta({u}_{\lambda}\cap{p}_{-})=\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\}(a=l-1,l).$
Then
$\Delta({u}_{\lambda}\cap{p}_{+})=\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l-2}+\phi_{b}\\},$
where $b=l\textrm{ if }a=l-1,\textrm{and }b=l-1\textrm{ if }a=l.$ Clearly
$\langle\lambda,\phi_{b}\rangle>0,$ that is $c_{b}>0.$ If $c_{a}=0,$ let
$d_{1}=c_{1}-\sum_{1\leq j\leq k}m_{j};d_{a}=c_{a}+1;d_{b}=c_{b}+1;$ and for
all $2\leq i\leq l-2,\ d_{i}=c_{i}\textrm{ if }c_{i}\neq 0,\textrm{and
}d_{i}=1\textrm{ if }c_{i}=0.$ If $c_{a}\neq 0,$ let $d_{1}=c_{1}-\sum_{1\leq
j\leq k}m_{j};$ and $d_{i}=c_{i}\textrm{ if }c_{i}\neq 0,\textrm{and
}d_{i}=1\textrm{ if }c_{i}=0;$ for all $2\leq i\leq l.$
$\lambda^{\prime}=\sum_{1\leq i\leq
l}\frac{d_{i}\omega_{i}}{\langle\omega_{i},\phi_{i}\rangle}.$ Since
$\beta\in-\Delta({u}_{\lambda}\cap{p}_{-}),\beta^{\prime}\in\Delta({u}_{\lambda}\cap{p}_{+})\implies
n_{\phi_{i_{j}}}(\beta)\leq n_{\phi_{i_{j}}}(\beta^{\prime})$ for all $1\leq
j\leq k,i_{j}\neq l-1,l;$ we have $\langle\lambda^{\prime},\alpha\rangle>0$
for all $\alpha\in\Delta_{k}^{+};$ and if $\beta\in\Delta_{n},\
\langle\lambda,\beta\rangle<0\implies\langle\lambda^{\prime},\beta\rangle<0,$
and
$\langle\lambda,\beta\rangle>0\implies\langle\lambda^{\prime},\beta\rangle>0.$
As above $A_{{q}_{\lambda}}$ is is unitarily equivalent to
$A_{{q}_{\lambda^{\prime}}},$ which is a discrete series representation with
trivial infinitesimal character.
Let ${g}={\delta}_{2}.$ Since
$(-\Delta({u}_{\lambda}\cap{p}_{-}))\cup\Delta({u}_{\lambda}\cap{p}_{+})=\Delta_{n}^{+},$
the candidates of
$(-\Delta({u}_{\lambda}\cap{p}_{-}),\Delta({u}_{\lambda}\cap{p}_{+}))$ are
$(\phi,\\{\phi_{1},\phi_{2}\\}),(\\{\phi_{1}\\},\\{\phi_{2}\\}),(\\{\phi_{2}\\},\\{\phi_{1}\\}),$
and $(\\{\phi_{1},\phi_{2}\\},\phi).$ The corresponding $\lambda^{\prime}$ are
$\omega_{1}+\omega_{2},-\omega_{1}+\omega_{2},\omega_{1}-\omega_{2},-\omega_{1}-\omega_{2}$
resectively. Then $A_{{q}_{\lambda}}$ is is unitarily equivalent to
$A_{{q}_{\lambda^{\prime}}},$ which is a discrete series representation with
trivial infinitesimal character.

The Blattner parameter of the discrete series representation
$A_{{b}_{\lambda}},$ where ${b}_{\lambda}={h}\oplus{u}_{\lambda}$ for some
linear function $\lambda$ on ${h}_{\mathbb{R}}$ with
$\langle\lambda,\alpha\rangle\neq 0$ for all $\alpha\in\Delta,$ and
$\langle\lambda,\alpha\rangle>0$ for all $\alpha\in\Delta_{k}^{+},$ is
$\sum_{\beta\in\Delta({u}_{\lambda}\cap{p}_{-})\cup\Delta({u}_{\lambda}\cap{p}_{+})}\beta.$
If $m\neq 2,{g}$ is simple and in this case the discrete series representation
$A_{{b}_{\lambda}}$ is a holomorphic discrete series representation if and
only if the Blattner parameter is $\sum_{\beta\in\Delta_{n}^{+}}\beta,$ or
$\sum_{\beta\in\Delta_{n}^{-}}\beta;$ that is
$\Delta({u}_{\lambda}\cap{p}_{+})$ is either $\Delta_{n}^{+}$ or empty. For
since ${g}$ is simple, the only Borel-de Siebenthal positive root system
containing $\Delta_{k}^{+}$ are $\Delta_{k}^{+}\cup\Delta_{n}^{+},$ and
$\Delta_{k}^{+}\cup\Delta_{n}^{-}.$ Hence the number of equivalence classes of
holomorphic discrete series representations of $SO_{0}(2,m)(m\neq 2)$ with
trivial infinitesimal character is $2.$ If ${g}={\delta}_{2},$ any positive
root system is Borel-de Siebenthal positive root system, and so any discrete
series representation with trivial infinitesimal character is holomorphic.

Let ${g}={b}_{l}(l\geq 2).$ The candidates of
$(\Delta({u}_{\lambda}\cap{p}_{-}),\Delta({u}_{\lambda}\cap{p}_{+}))$ for
which
$(-\Delta({u}_{\lambda}\cap{p}_{-}))\cup\Delta({u}_{\lambda}\cap{p}_{+})=\Delta_{n}^{+},$
are
$(\phi,\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{i+1}\\})(1\leq
i\leq
l-1),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l-1}+2\phi_{l}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{i-2}+2\phi_{i-1}+\cdots+2\phi_{l}\\})(3\leq
i\leq
l),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+2\phi_{2}+\cdots+2\phi_{l}\\},\phi).$
Thus the number of equivalence classes of discrete series representations of
$SO_{0}(2,m)$ with trivial infinitesimal character is $2l\textrm{ if
}m=2l-1,l\geq 2.$
If ${g}={b}_{1},$ the candidates of
$(\Delta({u}_{\lambda}\cap{p}_{-}),\Delta({u}_{\lambda}\cap{p}_{+}))$ for
which
$(-\Delta({u}_{\lambda}\cap{p}_{-}))\cup\Delta({u}_{\lambda}\cap{p}_{+})=\Delta_{n}^{+},$
are $(\phi,\\{\phi_{1}\\}),(\\{-\phi_{1}\\},\phi)$. Thus the number is $2.$
Let ${g}={\delta}_{l}(l\geq 3).$ The candidates of
$(\Delta({u}_{\lambda}\cap{p}_{-}),\Delta({u}_{\lambda}\cap{p}_{+}))$ for
which
$(-\Delta({u}_{\lambda}\cap{p}_{-}))\cup\Delta({u}_{\lambda}\cap{p}_{+})=\Delta_{n}^{+},$
are
$(\phi,\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{i+1}\\})(1\leq
i\leq l-3,l\geq
4),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l-2}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{1}\textrm{
or
}\xi_{2}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{2}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{2}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{1}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\xi_{1}\textrm{
or
}\xi_{2}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{l}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{l-3}+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\}),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+\cdots+\phi_{i-1}+2\phi_{i}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+\cdots+\phi_{i-2}+2\phi_{i-1}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\})(3\leq
i\leq l-2,l\geq
4),(-\\{\beta\in\Delta_{n}^{+}:\beta\leq\phi_{1}+2\phi_{2}+\cdots+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},\phi).$
Thus the number of equivalence classes of discrete series representations of
$SO_{0}(2,m)$ with trivial infinitesimal character is $2l\textrm{ if
}m=2l-2,l\geq 3.$
If ${g}={\delta}_{2},$ then obviously the number is $4.$

######  Remark 4.2.

Note that the set of all Hodge types $(R_{+}({q}),R_{-}({q}))$ of irreducible
unitary representations $A_{q}$ is given by

$\ \begin{cases}\\{(i,j):i,j\in\mathbb{N}\cup\\{0\\},l\leq
i+j\leq|\Delta_{n}^{+}|\\}\cup\\{(i,i):0\leq
i\leq[\frac{|\Delta_{n}^{+}|}{2}]\\}&\textrm{if }{g}={b}_{l},\\\
\\{(i,j):i,j\in\mathbb{N}\cup\\{0\\},l-1\leq
i+j\leq|\Delta_{n}^{+}|\\}\cup\\{(i,i):0\leq
i\leq[\frac{|\Delta_{n}^{+}|}{2}]\\}&\textrm{if }{g}={\delta}_{l};\\\
\end{cases}$

where $|\Delta_{n}^{+}|$ is the number of roots in $\Delta_{n}^{+}.$ The
discrete series representations with trivial infinitesimal character
correspond to the set
$\\{(i,j):i,j\in\mathbb{N}\cup\\{0\\},i+j=|\Delta_{n}^{+}|\\}.$

##  5\. Poincaré polynomials of cohomologies of the irreducible unitary
representations with non-zero $({g},K)$-cohomology of the Lie group
$SO_{0}(2,m)$

To determine the Poincaré polynomial of $H^{*}({g},K;A_{{q},K}),$ we need to
determine the spaces $Y_{q},$ and to do so we need to analyze the
$\theta$-stable parabolic subalgebras containing
${h}\oplus\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha}$ more closely. Note that
a parabolic subalgebra of ${g}$ is $\theta$-stable if and only if it contains
a maximal abelian subspace of ${k}_{0}.$ Thus a parabolic ${q}$ subalgebra of
${g}$ which contains ${h}\oplus\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha},$ is
$\theta$-stable. So there exists a positive root system $\Delta_{q}^{+}$ of
$\Delta$ containing $\Delta_{k}^{+}$ and a subset $\Gamma$ of $\Phi_{q},$ the
set of all simple roots in $\Delta_{q}^{+},$ such that ${q}={l}\oplus{u},$
where ${l}={h}\oplus\sum_{n_{\phi}(\alpha)=0\textrm{ for all
}\phi\in\Gamma}{g}^{\alpha}$ is the Levi subalgebra of ${q},$ and
${u}=\sum_{n_{\phi}(\alpha)>0\textrm{ for some }\phi\in\Gamma}{g}^{\alpha}$ is
the nilradical of ${q};$ where
$\alpha=\sum_{\phi\in\Phi_{q}}n_{\phi}(\alpha)\phi\in\Delta.$ Note that the
Levi subalgebra ${l}$ is the direct sum of an $|\Gamma|$-dimensional centre
and a semisimple Lie algebra whose Dynkin diagram is the subdiagram of the
dynkin diagram of ${g}$ consisting of the vertices $\Phi_{q}\setminus\Gamma.$

If ${q}$ is a parabolic subalgebra which contains
${h}\oplus\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha},$ there are many positive
root systems of $\Delta$ containing
$\Delta_{k}^{+}\cup\Delta({u}\cap{p}_{-})\cup\Delta({u}\cap{p}_{+}).$ For
example,
$\Delta_{k}^{+}\cup\Delta({u}\cap{p}_{-})\cup(\Delta_{n}^{+}\setminus(-\Delta({u}\cap{p}_{-})))$
is a positive root system of $\Delta,$ as we have seen in the proof of Th.
1.1(ii) that there exists a non-singular linear function $\lambda^{\prime}$ on
${h}_{\mathbb{R}}$ such that $\lambda^{\prime}$ is dominant with respect to
$\Delta_{k}^{+}\cup\Delta({u}\cap{p}_{-})\cup(\Delta_{n}^{+}\setminus(-\Delta({u}\cap{p}_{-}))).$
We define
$\Delta_{q}^{+}=\Delta_{k}^{+}\cup\Delta({u}\cap{p}_{-})\cup(\Delta_{n}^{+}\setminus(-\Delta({u}\cap{p}_{-}))).$
In the Tables 1 and 2, we have determined $\Phi_{q},\Gamma,Y_{q}$ for each
$\theta$-stable parabolic subalgebra containing
${h}\oplus\sum_{\alpha\in\Delta_{k}^{+}}{g}^{\alpha}.$

In the Tables 1 and 2, we can see that $Y_{q}$ is either singleton, or
$\frac{SU(k)}{S(U(1)\times U(k-1))}(k\geq 2),$ or $\frac{SO(2k+1)}{SO(2)\times
SO(2k-1)}(k\geq 1),$ or $\frac{SO(2k)}{SO(2)\times SO(2k-2)}(k\geq 2).$ We
have $P(\textrm{singleton},t)=1,$ $P(\frac{SU(k)}{S(U(1)\times
U(k-1))},t)=1+t^{2}+t^{4}+\cdots+t^{2k-2}\textrm{ for all }k\geq
2,P(\frac{SO(2k+1)}{SO(2)\times
SO(2k-1)},t)=1+t^{2}+t^{4}+\cdots+t^{4k-2}\textrm{ for all }k\geq
1,P(\frac{SO(2k)}{SO(2)\times
SO(2k-2)},t)=1+t^{2}+t^{4}+\cdots+t^{2k-4}+2t^{2k-2}+t^{2k}+\cdots+t^{4k-4}\textrm{
for all }k\geq 2.$ See [5]. Since
$H^{r}({g},K;A_{{q},K})=H^{p,q}({g},K;A_{{q},K})\cong
H^{p-R_{+}({q}),q-R_{-}({q})}(Y_{q};\mathbb{C}),$ for unique non-negative
integers $p,q$ with $p+q=r,p-q=R_{+}({q})-R_{-}({q});$ we write two variable
Poincaré polynomial $P_{q}(x,t)$ for $H^{*}({g},K;A_{{q},K}),$ and the
coefficient of the term $x^{p}t^{q}$ in $P_{q}(x,t)$ is
dim$(H^{p,q}({g},K;A_{{q},K})).$

Table 1. Poincaré Polynomials of The Irreducible Unitary Representations $A_{q}$ of $SO_{0}(2,2l-1)(l\geq 1):$ | $\Delta({u}\cap{p}_{-}),$
---
$R_{-}({q})$
$\Phi_{q}$ |  | $\Delta({u}\cap{p}_{+}),$
---
$R_{+}({q})$
$\Gamma$ | $Y_{q}$ | $P_{q}(x,t)$
| empty,
---
$0$
| $\\{\nu,\phi_{2},\ldots,\phi_{l}\\};$
---
$\nu=\phi_{1}.$
See (a).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu\\},2l-1$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots$
$+\phi_{i}\\},2l-i$
$(2\leq i\leq l)$
empty, $0$
| $\\{\nu\\}$
---
$\\{\phi_{i}\\}$
$(2\leq i\leq l)$
empty
| singleton
---
$\frac{SU(i)}{S(U(1)\times U(i-1))}$
$(2\leq i\leq l)$
$\frac{SO(2l+1)}{SO(2)\times SO(2l-1)}$
| $x^{2l-1}$
---
$x^{2l-i}+x^{2l-i+1}t$
$+\cdots+x^{2l-1}t^{i-1}$
$(2\leq i\leq l)$
$1+xt+\cdots$
$+x^{2l-1}t^{2l-1}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{i}\\},$
$i$
$(1\leq i\leq l-1)$
| $\\{\nu_{1},\nu_{2},\phi_{2},\ldots,$
---
$\phi_{i},\phi_{i+2},\ldots,\phi_{l}\\};\nu_{1}$
$=-(\phi_{1}+\cdots+\phi_{i}),$
$\nu_{2}=\phi_{1}+\cdots+\phi_{i+1}.$
See (b).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu_{2}\\},2l-1-i$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots$
$+\phi_{j}\\},2l-j$
$(i+2\leq j\leq l)$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{i}+$
$2\phi_{i+1}+\cdots$
$+2\phi_{l}\\},i$
| $\\{\nu_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\phi_{j}\\}$
$(i+2\leq j\leq l)$
$\\{\nu_{1}\\}$
| singleton
---
$\frac{SU(j-i)}{S(U(1)\times U(j-i-1))}$
$(i+2\leq j\leq l)$
$\frac{SO(2l-2i+1)}{SO(2)\times SO(2l-2i-1)}$
| $x^{2l-1-i}t^{i}$
---
$x^{2l-j}t^{i}$
$+x^{2l-j+1}t^{i+1}+$
$\cdots+x^{2l-i-1}t^{j-1}$
$(i+2\leq j\leq l)$
$x^{i}t^{i}$
$+x^{i+1}t^{i+1}+\cdots$
$+x^{2l-i-1}t^{2l-i-1}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{l}\\},$
$l$
| $\\{\nu_{1},\nu_{2},\phi_{2},$
---
$\ldots,\phi_{l-1}\\};\nu_{1}$
$=-(\phi_{1}+\cdots+\phi_{l}),$
$\nu_{2}=\phi_{1}+\cdots$
$+\phi_{l-1}+2\phi_{l}.$
See (c).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu_{2}\\},l-1$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{j-1}$
$+2\phi_{j}+\cdots$
$+2\phi_{l}\\},j-1$
$(2\leq j\leq l-1)$
empty, $0$
| $\\{\nu_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\phi_{j}\\}$
$(2\leq j\leq l-1)$
$\\{\nu_{1}\\}$
| singleton
---
$\frac{SU(l-j+1)}{S(U(1)\times U(l-j))}$
$(2\leq j\leq l-1)$
$\frac{SU(l)}{S(U(1)\times U(l-1))}$
| $x^{l-1}t^{l}$
---
$x^{j-1}t^{l}$
$+x^{j}t^{l+1}+$
$\cdots+x^{l-1}t^{2l-j}$
$(2\leq j\leq l-1)$
$t^{l}$
$+xt^{l+1}+\cdots$
$+x^{l-1}t^{2l-1}$
| $\Delta({u}\cap{p}_{-}),$
---
$R_{-}({q})$
$\Phi_{q}$ |  | $\Delta({u}\cap{p}_{+}),$
---
$R_{+}({q})$
$\Gamma$ | $Y_{q}$ | $P_{q}(x,t)$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{i-1}$
$+2\phi_{i}+\cdots+2\phi_{l}\\},$
$2l-i+1$
$(3\leq i\leq l)$
| $\\{\nu_{1},\nu_{2},\phi_{2},\ldots,$
---
$\phi_{i-2},\phi_{i},\ldots,\phi_{l}\\};$
$\nu_{1}=-(\phi_{1}+\cdots+\phi_{i-1}$
$+2\phi_{i}+\cdots+2\phi_{l}),\nu_{2}$
$=\phi_{1}+\cdots+\phi_{i-2}$
$+2\phi_{i-1}+\cdots+2\phi_{l}.$
See (d).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu_{2}\\},i-2$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+$
$\phi_{j-1}+2\phi_{j}+\cdots$
$+2\phi_{l}\\},j-1$
$(2\leq j\leq i-2)$
empty, $0$
| $\\{\nu_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\phi_{j}\\}$
$(2\leq j\leq i-2)$
$\\{\nu_{1}\\}$
| singleton
---
$\frac{SU(i-j)}{S(U(1)\times U(i-j-1))}$
$(2\leq j\leq i-2)$
$\frac{SU(i-1)}{S(U(1)\times U(i-2))}$
| $x^{i-2}t^{2l-i+1}$
---
$x^{j-1}t^{2l-i+1}$
$+x^{j}t^{2l-i+2}+$
$\cdots+x^{i-2}t^{2l-j}$
$(2\leq j\leq i-2)$
$t^{2l-i+1}$
$+xt^{2l-i+2}+\cdots$
$+x^{i-2}t^{2l-1}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+2\phi_{2}$
$+\cdots+2\phi_{l}\\},$
$2l-1$
| $\\{\nu,\phi_{2},\ldots,$
---
$\phi_{l}\\};\nu=-(\phi_{1}+$
$2\phi_{2}+\cdots+2\phi_{l}).$
See (e).
| empty, $0$
---
$\\{\nu\\}$ | singleton | $t^{2l-1}$
$\nu$$\phi_{2}$$\phi_{i}$$\phi_{l-1}$$\phi_{l}$(a) ${b}_{l}:$$\nu_{1}$$\nu_{2}$$\phi_{3}$$\phi_{l-1}$$\phi_{l}$(b) ${b}_{l}:$$(i=1)$$\phi_{2}$$\phi_{i}$$\nu_{1}$$\nu_{2}$$\phi_{i+2}$$\phi_{l-1}$$\phi_{l}$$(2\leq i\leq l-2)$$\phi_{2}$$\phi_{l-1}$$\nu_{1}$$\nu_{2}$$(i=l-1)$$\phi_{2}$$\phi_{l-1}$$\nu_{2}$$\nu_{1}$(c) ${b}_{l}:$$\phi_{2}$$\phi_{i-2}$$\nu_{2}$$\nu_{1}$$\phi_{i}$$\phi_{l-1}$$\phi_{l}$(d) ${b}_{l}:$$\nu$$\phi_{2}$$\phi_{l-1}$$\phi_{l}$(e) ${b}_{l}:$ Table 2. Poincaré Polynomials of The Irreducible Unitary Representations $A_{q}$ of $SO_{0}(2,2l-2)(l\geq 3):$ | $\Delta({u}\cap{p}_{-}),$
---
$R_{-}({q})$
$\Phi_{q}$ |  | $\Delta({u}\cap{p}_{+}),$
---
$R_{+}({q})$
$\Gamma$ | $Y_{q}$ | $P_{q}(x,t)$
| empty,
---
$0$
| $\\{\nu,\phi_{2},\ldots,\phi_{l}\\};$
---
$\nu=\phi_{1}.$
See (a).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu\\},2l-2$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{i}\\},2l-$
$i-1(2\leq i\leq l-2)$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\},$
$l-1(a=l-1,l)$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\xi_{1}\textrm{ or }\xi_{2}\\},l$
empty, $0$
| $\\{\nu\\}$
---
$\\{\phi_{i}\\}$
$(2\leq$
$i\leq l-2)$
$\\{\phi_{a}\\}$
$(a=l-1,l)$
$\\{\phi_{l-1},\phi_{l}\\}$
empty
| singleton
---
$\frac{SU(i)}{S(U(1)\times U(i-1))}$
$(2\leq i\leq l-2)$
$\frac{SU(l)}{S(U(1)\times U(l-1))}$
$\frac{SU(l-1)}{S(U(1)\times U(l-2))}$
$\frac{SO(2l)}{SO(2)\times SO(2l-2)}$
| $x^{2l-2}$
---
$x^{2l-i-1}$
$+x^{2l-i}t+$
$\cdots+x^{2l-2}t^{i-1}$
$x^{l-1}+x^{l}t+$
$\cdots+x^{2l-2}t^{l-1}$
$x^{l}+x^{l+1}t+$
$\cdots+x^{2l-2}t^{l-2}$
$1+xt+\cdots$
$+2x^{l-1}t^{l-1}+$
$\cdots+x^{2l-2}t^{2l-2}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{i}\\},$
$i$
$(1\leq i\leq l-3)$
$(l\geq 4)$
| $\\{\nu_{1},\nu_{2},\phi_{2},\ldots,$
---
$\phi_{i},\phi_{i+2},\ldots,\phi_{l}\\};\nu_{1}$
$=-(\phi_{1}+\cdots+\phi_{i}),$
$\nu_{2}=\phi_{1}+\cdots+\phi_{i+1}.$
See (b).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu_{2}\\},2l-2-i$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{j}\\},2l-j$
$-1(i+2\leq j\leq l-2)$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{l-2}+\phi_{a}\\},$
$l-1(a=l-1,l)$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\xi_{1}\textrm{ or }\xi_{2}\\},l$
| $\\{\nu_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\phi_{j}\\}$
$(i+2\leq$
$j\leq l-2)$
$\\{\nu_{1},\phi_{a}\\}$
$(a=l-1,l)$
$\\{\nu_{1},\phi_{l-1},\phi_{l}\\}$
| singleton
---
$\frac{SU(j-i)}{S(U(1)\times U(j-i-1))}$
$(i+2\leq$
$j\leq l-2)$
$\frac{SU(l-i)}{S(U(1)\times U(l-i-1))}$
$\frac{SU(l-1-i)}{S(U(1)\times U(l-2-i))}$
| $x^{2l-2-i}t^{i}$
---
$x^{2l-j-1}t^{i}$
$+x^{2l-j}t^{i+1}+$
$\cdots+x^{2l-2-i}t^{j-1}$
$x^{l-1}t^{i}+x^{l}t^{i+1}+$
$\cdots+x^{2l-2-i}t^{l-1}$
$x^{l}t^{i}+x^{l+1}t^{i+1}+$
$\cdots+x^{2l-2-i}t^{l-2}$
| $\Delta({u}\cap{p}_{-}),$
---
$R_{-}({q})$
$\Phi_{q}$ |  | $\Delta({u}\cap{p}_{+}),$
---
$R_{+}({q})$
$\Gamma$ | $Y_{q}$ | $P_{q}(x,t)$
,, | ,, |  | $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\phi_{1}+\cdots+\phi_{i}+$
$2\phi_{i+1}+\cdots+2\phi_{l-2}$
$+\phi_{l-1}+\phi_{l}\\},i$
$\\{\nu_{1}\\}$ | $\frac{SO(2l-2i)}{SO(2)\times SO(2l-2i-2)}$ |  | $x^{i}t^{i}+x^{i+1}t^{i+1}$
---
$+\cdots+2x^{l-1}t^{l-1}$
$+\cdots+$
$x^{2l-i-2}t^{2l-i-2}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{l-2}\\},$
$l-2$
| $\\{\nu_{1},\nu_{2},\nu^{\prime}_{2},$
---
$\phi_{2},\ldots,$
$\phi_{l-2}\\};\nu_{1}=-$
$(\phi_{1}+\cdots+\phi_{l-2}),$
$\nu_{2}=\xi_{1},\nu^{\prime}_{2}=\xi_{2}.$
See (c).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu_{2},\textrm{or }\nu^{\prime}_{2}\\},l$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq\xi_{i}\\},$
$l-1(i=1,2)$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+$
$\cdots+\phi_{l}\\},l-2$
| $\\{\nu_{1},\nu_{2},\nu^{\prime}_{2}\\}$
---
$\\{\nu_{1},\xi_{i}\\}$
$(i=1,2)$
$\\{\nu_{1}\\}$
| singleton
---
$\frac{SU(2)}{S(U(1)\times U(1))}$
$\frac{SO(4)}{SO(2)\times SO(2)}$
| $x^{l}t^{l-2}$
---
$x^{l-1}t^{l-2}$
$+x^{l}t^{l-1}$
$x^{l-2}t^{l-2}+$
$2x^{l-1}t^{l-1}$
$+x^{l}t^{l}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{l-2}$
$+\phi_{a}\\},l-1$
$(a=l-1,l)$
| $\\{\nu_{1},\nu_{2},\phi_{2},$
---
$\ldots,\phi_{l-2},\phi_{a}\\};\nu_{1}$
$=-(\phi_{1}+\cdots+$
$\phi_{l-2}+\phi_{a}),$
$\nu_{2}=\phi_{1}$
$+\cdots+\phi_{l-2}+\phi_{b}$
$(b\in\\{l-1,l\\}$
$\setminus\\{a\\}).$
See (d).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq\nu_{2}\\},$
---
$l-1$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{l}\\},l-2$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+$
$\cdots+\phi_{j-1}+2\phi_{j}+\cdots$
$+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},$
$j-1$
$(2\leq j\leq l-2,l\geq 4)$
empty, $0$
| $\\{\nu_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\phi_{a}\\}$
$\\{\nu_{1},\phi_{j}\\}$
$(2\leq j\leq l-2,$
$l\geq 4)$
$\\{\nu_{1}\\}$
| singleton
---
$\frac{SU(2)}{S(U(1)\times U(1))}$
$\frac{SU(l-j+1)}{S(U(1)\times U(l-j))}$
$(2\leq j\leq l-2,$
$l\geq 4)$
$\frac{SU(l)}{S(U(1)\times U(l-1))}$
| $x^{l-1}t^{l-1}$
---
$x^{l-2}t^{l-1}$
$+x^{l-1}t^{l}$
$x^{j-1}t^{l-1}+$
$x^{j}t^{l}+\cdots+$
$x^{l-1}t^{2l-j-1}$
$t^{l-1}+$
$xt^{l}+\cdots$
$+x^{l-1}t^{2l-2}$
| $\Delta({u}\cap{p}_{-}),$
---
$R_{-}({q})$
$\Phi_{q}$ |  | $\Delta({u}\cap{p}_{+}),$
---
$R_{+}({q})$
$\Gamma$ | $Y_{q}$ | $P_{q}(x,t)$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\xi_{1},\textrm{or }\xi_{2}\\},$
$l$
| $\\{\nu_{1},\nu^{\prime}_{1},\nu_{2},\phi_{2},$
---
$\ldots,\phi_{l-2},\\};\nu_{1}$
$=-\xi_{1},\nu^{\prime}_{1}=$
$-\xi_{2},\nu_{2}=$
$\phi_{1}+\cdots+\phi_{l}.$
See (e).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq\nu_{2}\\},$
---
$l-2$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+$
$\cdots+\phi_{j-1}+2\phi_{j}+\cdots$
$+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},$
$j-1$
$(2\leq j\leq l-2,l\geq 4)$
empty, $0$
| $\\{\nu_{1},\nu^{\prime}_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\nu^{\prime}_{1},\phi_{j}\\}$
$(2\leq j\leq$
$l-2,$
$l\geq 4)$
$\\{\nu_{1},\nu^{\prime}_{1}\\}$
| singleton
---
$\frac{SU(l-j)}{S(U(1)\times U(l-j-1))}$
$(2\leq j\leq l-2,$
$l\geq 4)$
$\frac{SU(l-1)}{S(U(1)\times U(l-2))}$
| $x^{l-2}t^{l}$
---
$x^{j-1}t^{l}+$
$x^{j}t^{l+1}+\cdots+$
$x^{l-2}t^{2l-j-1}$
$t^{l}+$
$xt^{l+1}+\cdots$
$+x^{l-2}t^{2l-2}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{l}\\},$
$l+1$
$(l\geq 4)$
| $\\{\nu_{1},\nu_{2},\phi_{2},$
---
$\ldots,\phi_{l-3},\phi_{l-1},\phi_{l}\\};$
$\nu_{1}=-(\phi_{1}+\cdots+\phi_{l}),$
$\nu_{2}=\phi_{1}+\cdots+\phi_{l-3}$
$+2\phi_{l-2}+\phi_{l-1}+\phi_{l}.$
See (f).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq\nu_{2}\\},$
---
$l-3$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq\phi_{1}+$
$\cdots+\phi_{j-1}+2\phi_{j}+\cdots$
$+2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},$
$j-1$
$(2\leq j\leq l-3)$
empty, $0$
| $\\{\nu_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\phi_{j}\\}$
$(2\leq j\leq$
$l-3)$
$\\{\nu_{1}\\}$
| singleton
---
$\frac{SU(l-j-1)}{S(U(1)\times U(l-j-2))}$
$(2\leq j\leq l-3)$
$\frac{SU(l-2)}{S(U(1)\times U(l-3))}$
| $x^{l-3}t^{l+1}$
---
$x^{j-1}t^{l+1}+$
$x^{j}t^{l+2}+\cdots+$
$x^{l-3}t^{2l-j-1}$
$t^{l+1}+$
$xt^{l+2}+\cdots$
$+x^{l-3}t^{2l-2}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+\cdots+\phi_{i-1}$
$+2\phi_{i}+\cdots+2\phi_{l-2}$
$+\phi_{l-1}+\phi_{l}\\},$
$2l-i$
$(3\leq i\leq l-2)$
$(l\geq 4)$
| $\\{\nu_{1},\nu_{2},\phi_{2},\ldots,$
---
$\phi_{i-2},\phi_{i},\ldots,\phi_{l}\\};$
$\nu_{1}=-(\phi_{1}+\cdots+\phi_{i-1}$
$+2\phi_{i}+\cdots+2\phi_{l-2}$
$+\phi_{l-1}+\phi_{l}),\nu_{2}$
$=\phi_{1}+\cdots+\phi_{i-2}$
$+2\phi_{i-1}+\cdots+$
$2\phi_{l-2}+\phi_{l-1}+\phi_{l}.$
See (g).
| $\\{\beta\in\Delta_{n}^{+}:\beta\geq$
---
$\nu_{2}\\},i-2$
$\\{\beta\in\Delta_{n}^{+}:\beta\geq$
$\phi_{1}+\cdots+\phi_{j-1}$
$+2\phi_{j}+\cdots+2\phi_{l}\\},$
$j-1$
$(2\leq j\leq i-2)$
empty, $0$
| $\\{\nu_{1},\nu_{2}\\}$
---
$\\{\nu_{1},\phi_{j}\\}$
$(2\leq j\leq$
$i-2)$
$\\{\nu_{1}\\}$
| singleton
---
$\frac{SU(i-j)}{S(U(1)\times U(i-j-1))}$
$(2\leq j\leq i-2)$
$\frac{SU(i-1)}{S(U(1)\times U(i-2))}$
| $x^{i-2}t^{2l-i}$
---
$x^{j-1}t^{2l-i}+$
$x^{j}t^{2l-i+1}+\cdots$
$+x^{i-2}t^{2l-j-1}$
$t^{2l-i}+$
$xt^{2l-i+1}+\cdots$
$+x^{i-2}t^{2l-2}$
| $-\\{\beta\in\Delta_{n}^{+}:\beta\leq$
---
$\phi_{1}+2\phi_{2}+\cdots+$
$2\phi_{l-2}+\phi_{l-1}+\phi_{l}\\},$
$2l-2$
| $\\{\nu,\phi_{2},\ldots,\phi_{l}\\};\nu=$
---
$-(\phi_{1}+2\phi_{2}+\cdots+$
$2\phi_{l-2}+\phi_{l-1}+\phi_{l}).$
See (h).
| empty, $0$
---
$\\{\nu\\}$ | singleton | $t^{2l-2}$

$\nu$$\phi_{2}$$\phi_{i}$$\phi_{l-2}$$\phi_{l-1}$$\phi_{l}$(a)
${\delta}_{l}:$$(l\geq
3)$$\phi_{2}$$\phi_{i}$$\nu_{1}$$\nu_{2}$$\phi_{i+2}$$\phi_{l-2}$$\phi_{l-1}$$\phi_{l}$(b)
${\delta}_{l}:$$(l\geq
4)$$\phi_{2}$$\phi_{l-2}$$\nu_{1}$$\nu_{2}$$\nu^{\prime}_{2}$(c)
${\delta}_{l}:$$(l\geq
3)$$\phi_{2}$$\phi_{l-2}$$\phi_{a}$$\nu_{2}$$\nu_{1}$(d)
${\delta}_{l}:$$(l\geq
3)$$\phi_{2}$$\phi_{l-2}$$\nu_{2}$$\nu_{1}$$\nu^{\prime}_{1}$(e)
${\delta}_{l}:$$(l\geq
3)$$\phi_{2}$$\phi_{l-3}$$\nu_{2}$$\nu_{1}$$\phi_{l-1}$$\phi_{l}$(f)
${\delta}_{l}:$$(l\geq
4)$$\phi_{2}$$\phi_{i-2}$$\nu_{2}$$\nu_{1}$$\phi_{i}$$\phi_{l-2}$$\phi_{l-1}$$\phi_{l}$(g)
${\delta}_{l}:$$(l\geq 4)$$\nu$$\phi_{2}$$\phi_{l-2}$$\phi_{l-1}$$\phi_{l}$(h)
${\delta}_{l}:$$(l\geq 3)$

###  5.1. Poincaré polynomials of cohomologies of the irreducible unitary
representations $A_{q}$ of $SO_{0}(2,2)$

Let ${g}={\delta}_{2}.$
Note that $\Delta_{n}^{+}=\\{\phi_{1},\phi_{2}\\}.$

$\phi_{1}$$\phi_{2}$${\delta}_{2}:$

Now $\Delta({u}\cap{p}_{-})=\phi,\Delta({u}\cap{p}_{+})=\phi\implies
Y_{q}=\frac{SO(4)}{SO(2)\times SO(2)},P_{q}(x,t)=1+2xt+x^{2}t^{2};$
$\Delta({u}\cap{p}_{-})=\phi,\Delta({u}\cap{p}_{+})=\\{\phi_{i}\\}(i=1,2)\implies
Y_{q}=\frac{SU(2)}{S(U(1)\times U(1))},P_{q}(x,t)=x+x^{2}t;$
$\Delta({u}\cap{p}_{-})=\phi,\Delta({u}\cap{p}_{+})=\\{\phi_{1},\phi_{2}\\}\implies
Y_{q}=$ singleton, $P_{q}(x,t)=x^{2};$
$\Delta({u}\cap{p}_{-})=\\{-\phi_{i}\\}(i=1,2),\Delta({u}\cap{p}_{+})=\phi\implies
Y_{q}=\frac{SU(2)}{S(U(1)\times U(1))},P_{q}(x,t)=t+xt^{2};$
$\Delta({u}\cap{p}_{-})=\\{-\phi_{i}\\}(i=1,2),\Delta({u}\cap{p}_{+})=\\{\phi_{j}\\}(j=1,2,j\neq
i)\implies Y_{q}=$ singleton, $P_{q}(x,t)=xt;$
$\Delta({u}\cap{p}_{-})=\\{-\phi_{1},-\phi_{2}\\},\Delta({u}\cap{p}_{+})=\phi\implies
Y_{q}=$ singleton, $P_{q}(x,t)=t^{2}.$

## acknowledgement

Both authors acknowledge the financial support from the Department of Science
and Technology (DST), Govt. of India under the Scheme ”Fund for Improvement of
S&T Infrastructure (FIST)” [File No. SR/FST/MS-I/2019/41]. Ankita Pal
acknowledges the financial support from Council of Scientific and Industrial
Research (CSIR) [File No. 08/155(0091)/2021-EMR-I].

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