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\begin{document}
\today

\bigskip
\title
{Unitary representations with half-integer parameter}
\maketitle

\section{Parametrization via $\cCO$}\label{1}

\bigskip
\subsection{}\label{1.1}
There is an exact sequence
\begin{equation}
  \label{1.1}
1\longrightarrow\kappa\longrightarrow  A(^L\phi)\longrightarrow\ovl{A(^L\phi)}
\longrightarrow 1.
\end{equation}
We attach a \textit{unipotent} infinitesimal character to each
conjugacy class in $\ovl{A(^L\phi)}.$
\subsection{Type B}\label{b} The dual group is type C. A nilpotent orbit $\cCO$ is
given by a partition 
\begin{equation}
  \label{b1}
  0\le 2n_0<2n_1\le\dots<2n_{2i-1}\le 2n_{2i}<\dots \le 2n_{2p}.
\end{equation}
The centralizer is $G(^L\phi)$ is a product of $O(2)$'s, one for each
pair $n_{2i-1}=n_{2i}$ and $O(1)$'s one for each remaining $n_i>0.$

An element $s\in G(^L\phi)$ which gives rise to a halfinteger
parameter can be written as
\begin{equation}
  \label{b2}
  s=(\ep_1,\dots,\ep_p),\qquad \ep_i=\pm 1
\end{equation}
with $\ep_0=1$ if $n_0=0.$ An element $s$ is in $\kappa$  if and only
if
\begin{equation}
  \label{b3}
s\in\kappa \leftrightarrow \ep_{2i-1}\ep_{2i}=1 .
\end{equation}
The infinitesimal character of the unipotent representation is as
follows
\begin{enumerate}
\item For each $\ep_i=1,$ add $(n_i-1/2,\dots , 1/2).$
\item For $\ep_i=-1$ order the $n_i$ in increasing order and name them
  $m_0,\dots ,m_{2q}.$ Then add
\begin{alignat}{2}
  \label{b4}
 &(m_j,\dots ,1)\ &&\text{ if }\ j\ \text{ is even,}\notag\\
&&&\\
 &(m_j-1,\dots ,1,0)\ &&\text{ if }\ j\ \text{ is odd.}\notag
\end{alignat}
\end{enumerate}

\bigskip
Every coset has a representative of the form
\begin{alignat}{2}\label{b5}
  &(\eta_0,\eta_0,\dots ,\eta{p-1},\eta_{p-1},1)\ &&\text{ if }\ n_0\ne
  0,\notag\\
&\\
  &(1,\eta_0,\eta_1,\eta_1,\dots ,\eta_{p-1},\eta_{p-1},1)\ &&\text{ if }\ n_0=0.\notag
\end{alignat}
The infinitesimal character corresponding to this  representative is
called unipotent.

\subsection{Type C}\label{c} The dual group is type B. A nilpotent orbit $\cCO$ is
given by a partition 
\begin{equation}
  \label{c1}
  0< 2n_0+1\le 2n_1+1<\dots<2n_{2i}+1\le 2n_{2i+1}+1<\dots < 2n_{2p}+1.
\end{equation}
The centralizer is $G(^L\phi)$ is a product of $O(2)$'s, one for each
pair $n_{2i}=n_{2i+1}$ and $O(1)$'s one for each remaining $n_i>0.$

An element $s\in G(^L\phi)$ which gives rise to a halfinteger
parameter can be written as
\begin{equation}
  \label{c2}
  s=(\ep_0,\dots,\ep_{2p}),\qquad \ep_i=\pm 1,\qquad \ep_0\cdot\ \dots\cdot\ep_{2p}=1.
\end{equation}
An element $s$ is in $\kappa$  if and only if
\begin{equation}
  \label{c3}
s\in\kappa \leftrightarrow \ep_{2i}\ep_{2i+1}=1 .
\end{equation}
The infinitesimal character of the unipotent representation is as
follows
\begin{enumerate}
\item For $\ep_i=1,$ order the $n_i$ in increasing order and name them
  $m_0,\dots ,m_{2r}.$ Then add
\begin{alignat}{2}
  \label{b4}
 &(m_j,\dots ,1)\ &&\text{ if }\ j\ \text{ is even,}\notag\\
&&&\\
 &(m_j,\dots ,1,0)\ &&\text{ if }\ j\ \text{ is odd.}\notag
\end{alignat}
\item For $\ep_i=-1,$ order the $n_i$ in increasing order and name them
  $m_0,\dots ,m_{2q+1}.$ Then add
\begin{alignat}{2}
  \label{b4}
 &(m_j+1/2,\dots ,1/2)\ &&\text{ if }\ j\ \text{ is odd,}\notag\\
&&&\\
 &(m_j-1/2,\dots ,1/2)\ &&\text{ if }\ j\ \text{ is even.}\notag
\end{alignat}
\end{enumerate}

\bigskip
Every coset has a representative of the form
\begin{equation}\label{c5}
(1,\eta_1,\eta_1,\dots ,\eta_{p},\eta_{p}).
\end{equation}
The infinitesimal character corresponding to this  representative is
called unipotent.

\newpage

\subsection{Type D}\label{d} The dual group is type D. A nilpotent orbit $\cCO$ is
given by a partition 
\begin{equation}
  \label{c1}
  0< 2n_0+1\le 2n_1+1<\dots<2n_{2i}+1\le 2n_{2i+1}+1<\dots < 2n_{2p+1}+1.
\end{equation}
The centralizer is $G(^L\phi)$ is a product of $O(2)$'s, one for each
pair $n_{2i}=n_{2i+1}$ and $O(1)$'s one for each remaining $n_i>0.$

An element $s\in G(^L\phi)$ which gives rise to a halfinteger
parameter can be written as
\begin{equation}
  \label{d1}
  s=(\ep_1,\dots,\ep_{2p+1}),\qquad \ep_i=\pm 1.
\end{equation}
An element $s$ is in $\kappa$  if and only if
\begin{equation}
  \label{d2}
s\in\kappa \leftrightarrow \ep_{2i}\ep_{2i+1}=1 .
\end{equation}
The infinitesimal character of the unipotent representation is as
follows
\begin{enumerate}
\item For $\ep_i=1,$ order the $n_i$ in increasing order and name them
  $m_0,\dots ,m_{2r+1}.$ Then add
\begin{alignat}{2}
  \label{d3}
 &(m_j,\dots ,1)\ &&\text{ if }\ j\ \text{ is even,}\notag\\
&&&\\
 &(m_j,\dots ,1,0)\ &&\text{ if }\ j\ \text{ is odd.}\notag
\end{alignat}
\item For $\ep_i=-1,$ order the $n_i$ in increasing order and name them
  $m_0,\dots ,m_{2q+1}.$ Then add
\begin{alignat}{2}
  \label{d4}
 &(m_j+1/2,\dots ,1/2)\ &&\text{ if }\ j\ \text{ is odd,}\notag\\
&&&\\
 &(m_j-1/2,\dots ,1/2)\ &&\text{ if }\ j\ \text{ is even.}\notag
\end{alignat}
\end{enumerate}

\bigskip
Every coset has a representative of the form
\begin{equation}\label{d5}
(1,\eta_1,\eta_1,\dots ,\eta_{p},\eta_{p},1).
\end{equation}
The infinitesimal character corresponding to this  representative is
called unipotent.

\section{Parametrization via $\CO$}\label{2}

We rely on \cite{BV2} and \cite{B1}. First, we prescribe the
infinitesimal character $\la_\CO.$  The main property will be that the
unipotent representations (irreducible $(\fg,K)$ modules whose
annihilator in the universal enveloping algebra is maximal with the
given infinitesimal character) are unitary and in 1-1 correspondence
with the irreducible characters of the component group $A(\CO).$ 
The notation is as in \cite{B1}. An orbit is called cuspidal if it is
not induced from any proper Levi component. For special orbits whose
dual is even the infinitesimal character is one half the semisimple
element of the Lie triple corressponding to the dual orbit. For the
other orbits we need the case-by-case analysis.

\noindent{\bf Type A.}\quad A nilpotent orbit is determined by its
Jordan canonical form. It is given by a partition \ie a 
sequence of numbers in decreasing order $(n_1,\dots ,n_k)$ that add up
to $n.$  Let $(m_1,\dots ,m_l)$ be the dual partition. Then the
infinitesimal character is 
$$
(\frac{m_1-1}{2},\dots ,-\frac{m_1-1}{2},\dots
,\frac{m_l-1}{2},\dots, -\frac{m_l-1}{2})
$$
The orbit is induced from the trivial orbit on the Levi component
$GL(m_1)\times \dots \times GL(m_l).$ The corresponding
unipotent representation is spherical and induced irreducible from the
trivial representation on the same Levi component. All orbits are
{\it stably trivial}.

\noindent{\bf Type B.}\quad A nilpotent orbit is determined by its
Jordan canonical form (in the standard representation). It is
parametrized by a partition $(n_1,\dots ,n_k)$ of $2n+1$ such that
every even part occurs an even number of times. 
Let $(m'_0,\dots ,m'_{2p'})$ be the dual partition (add an $m'_{2p'}=0$ if
necessary, in order to have an odd number of terms).  
If there are any $m'_{2j}=m'_{2j+1}$ then pair them
together and remove them from the partition. 
Then relabel the remaining columns and pair them up, the rest of the
columns $(m_0)(m_1,m_2)\dots (m_{2p-1}m_{2p}).$ The members of each
pair have the same parity and $m_0$ is odd. Then form a parameter
$$
\aligned
(m_0)&\leftrightarrow (\frac{m_0-2}{2},\dots , 1/2),\\
(m'_{2j}=m'_{2j+1})&\leftrightarrow (\frac{m_{2j}-1}{2},\dots , -\frac{m_{2j}-1}{2})\\
(m_{2i-1}m_{2i})&\leftrightarrow (\frac{m_{2i-1}}{2},\dots , -\frac{m_{2i}-2}{2}) 
\endaligned$$
In case $m'_{2j}=m'_{2j+1},$ the nilpotent orbit is induced from a
parabolic subalgebra $\fp$ with Levi component $so(*)\times gl(m'_{2j})$ with
the trivial nilpotent on the $gl$ factor. The
component groups in $G$ and $P$ are equal. The unipotent representations are
unitarily induced irreducible from similar parameters on the Levi
component.  Similarly if some $m_{2i-1}=m_{2i},$ then the nilpotent is
induced irreducible from a nilpotent on an $so(*)\times
gl(\frac{m_{2i-1}+m_{2i}}{2})$ with the trivial nilpotent on the $gl$
factor. The component groups of the centralizers in $G$ and $P$ coincide. 
The unipotent representations are again induced irreducible from the
nilpotent orbit on $so(*)$ with partition the one for $\CO$ but with
$m_{2i-1},m_{2i}$ removed. The {\it stably trivial} orbits are the
ones such that every odd sized part appears an even number of
times except for the largest size. An orbit is trinagular if it has
partition $(1,1,3,3,\dots,2m-1,2m-1,2m+1).$ It is induced from the
trivial nilpotent orbit on $\fm=gl(2)\times gl(4)\dots gl(2m).$ The
component group $A_P$ is trivial.

\noindent{\bf Type C.}\quad A nilpotent orbit is determined by its
Jordan canonical form (in the standard representation). It is
parametrized by a partition $(n_1,\dots ,n_k)$ of $2n$ such that
every odd part occurs an even number of times. 
Let $(m'_0,\dots ,m'_{2p'})$ be the dual partition (add a $m'_{2p'}= 0$ if
necessary in order to have an odd number of terms). 
If there are any $m'_{2j-1}=m'_{2j}$ pair them up and
remove them from the partition. Then relabel and pair
up the remaining columns $(m_0m_1)\dots
(m_{2p-2}m_{2p-1})(m_{2p}).$ The members of each pair have the same
parity. The last one, $m_{2p},$ is always even. Then
form a parameter
$$
\aligned
(m'_{2j-1}=m'_{2j})&\leftrightarrow (\frac{m_{2j}-1}{2},\dots , 
-\frac{m_{2j}-1}{2})\\
(m_{2i}m_{2i+1})&\leftrightarrow (\frac{m_{2i}}{2},\dots , 
-\frac{m_{2i+1}-2}{2}),\\
m_{2p}&\leftrightarrow (\frac{m_{2p}}{2},\dots , 1).
\endaligned$$
The nilpotent orbits and the unipotent representations have the same
properties with respect to these pairs as the corresponding ones in
type B. The {\it stably trivial} orbits are the
ones such that every even sized part appears an even number of
times. An orbit is called triangular if it corresponds to the
partition $(2,2,4,4,\dots ,2m,2m).$ It is induced from the trivial
orbit on $\fm=sp(2m)\times gl(1)\times \dots\times gl(2m-1).$ The
component group $A_P$ is trivial.

\noindent{\bf Type D.}\quad A nilpotent orbit is determined by its
Jordan canonical form (in the standard representation). It is
parametrized by a partition $(n_1,\dots ,n_k)$ of $2n$ such that
every even part occurs an even number of times. 
Let $(m'_0,\dots ,m'_{2p'-1})$ be the dual partition (add a $m'_{2p-1}=0$ if
necessary). If there are any $m'_{2j}=m'_{2j+1}$ pair them up and
remove from the partition. 
Then pair up the remaining columns $(m_0m_{2p})(m_1,m_2)\dots
(m_{2p-2}m_{2p-1}).$ The members of each pair have the same parity and
$m_0, m_{2p-1}$ are even. Then form a parameter
$$
\aligned
(m'_{2j}=m'_{2j+1})&\leftrightarrow (\frac{m'_{2j}-1}{2}\dots , -\frac{m'_{2j}-1}{2})\\
(m_0m_{2p-1})&\leftrightarrow (\frac{m_0-2}{2},\dots ,-\frac{m_{2p-1}}{2}),\\
(m_{2i-1}m_{2i})&\leftrightarrow (\frac{m_{2i-1}}{2}\dots , -\frac{m_{2i}-2}{2}) 
\endaligned$$
The nilpotent orbits and the unipotent representations have the same
properties with respect to these pairs as the corresponding ones in
type B. An exception occurs when the partition is formed of pairs
$(m'_{2j}=m'_{2j+1})$ only. In this case there are two nilpotent
orbits corresponding to the partition. There are also two nonconjugate
Levi components of the form $gl(m'_0)\times gl(m'_2)\times \dots
gl(m'_{2p'-2})$ of parabolic subalgebras. There are two unipotent
representations each induced irreducible from the trivial
representation on the corresponding Levi component. The {\it stably
trivial} orbits are the ones such that every even sized part appears
an even number of times. A nilpotent orbit is triangular if it
corresponds to the partition $(1,1,3,3,\dots,2m-1,2m-1).$ It is
induced from the trivial orbit in the Levi component
$\fm=gl(2)\times\dots\times gl(2m-2).$ The component group $A_P$ is trivial.


\newpage
\ifx\undefined\bysame
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\fi

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\end{document}


