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\title{Some small unipotent representations of indefinite 
orthogonal groups and the theta correspondence}

%    Information for first author
\author{Annegret Paul}
%    Address of record for the research reported here
\address{Department of Mathematics, Western Michigan University,
Kalamazoo, MI 49008}
\email{annegret.paul@wmich.edu}
\author{Peter Trapa}
\address{Department of Mathematics, University of Utah, Salt Lake City,
UT 84112}
\email{ptrapa@math.utah.edu}
%\thanks{Support information for the first author.}

%    General info
%\subjclass{Primary 54C40, 14E20; Secondary 46E25, 20C20}

% \date{January 1, 1994 and, in revised form, June 22, 1994.}

%\dedicatory{This paper is dedicated to our authors.}

%\keywords{Differential geometry, algebraic geometry}



\begin{abstract}
\end{abstract}

\maketitle



\section{Introduction}
\label{sec:intro}


\section{Preliminaries and notation}
\label{sec:prel}

Recall that each $K$-type $\mu$ which occurs in $\caF$ has associated to it a degree
(the minimum degree of polymomials in the $\mu$-isotypic subspace), and that if
$\pi$ and $\pi'$ are representations of $\widetilde G$ and $\widetilde G'$, respectively, 
which correspond to each other,
then each $K$-type for $\widetilde G$ which is of minimal degree in $\pi$ will occur in $\caH$
and correspond to a $K$-type for $\widetilde G'$ of minimal degree in $\pi'$. Since
for given choice of $n$, $\spn$ is a member of many dual pairs, we refer to
the degree of a $K$-type $\sigma$ for $\meta(2n,\bbR)$ for the dual pair $\spopq$ as
the \it $(p,q)$-degree \rm of $\sigma$. As we will see below in Proposition
\ref{prop:harm}, The degree of a $K$-type for $\opq$ depends on the difference
$p-q$ only, and the degree of a $K$-type for $\opq$ is independent of $n$.
\bigskip

\begin{prop}
\label{prop:harm}
Let $p$, $q$, and $n$ be non-negative integers, 
$p_0=\lbrack\frac p2\rbrack$, and $q_0=\lbrack \frac q2\rbrack$. The 
correspondence of $K$-types in the space of joint harmonics $\caH$ for the dual
pair $\bigl(\spn,\opq\bigr)$ is given as follows.

\begin{enumerate}
\item 
Let 
\beq
\label{eq:mu}
\mu=(a_1,a_2,\dots,a_x,0,\dots,0;\e)\otimes(b_1,b_2,\dots,b_y,0,\dots,0;\eta)
\eeq

\noindent
be a $K$-type for $\opq$, with $a_x>0$ and $b_y>0$. Then $\mu$ occurs in $\caH$ 
if and only if $n\ge x+\frac{1-\e}2(p-2x)+\frac{1-\eta}2(q-2y)+y$. In that case, 
$\mu$ corresponds to

\beq
\label{eq:mucor}
\left(\tfrac{p-q}2,\tfrac{p-q}2,\dots,\tfrac{p-q}2\right)
+(a_1,\dots,a_x,\underbrace{1,\dots,1}_{\frac{1-\e}2(p-2x)},0,\dots,0,
\underbrace{-1,\dots,-1}_{\frac{1-\eta}2(q-2y)},
-b_y,\dots,-b_1).
\eeq
 
\item
If a $K$-type $\mu$ for $\opq$ as in (\ref{eq:mu}) occurs in the Fock space,
then the degree of $\mu$ is 

\beq
\label{eq:mudeg}
\sum_{i=1}^xa_i+\sum_{i=1}^yb_i+\frac{1-\e}2(p-2x)+\frac{1-\eta}2(q-2y).
\eeq

\noindent
For a $K$-type for $\meta(2n,\bbR)$ $\xi$ which occurs in $\caF$, write

\beq
\xi=\left(\tfrac{p-q}2,\tfrac{p-q}2,\dots,\tfrac{p-q}2\right)
+(a_1,a_2,\dots,a_n).
\eeq

\noindent
Then the $(p,q)$-degree of $\xi$ is $\sum_{i=1}^n|a_i|$.

\end{enumerate}

\end{prop}

\bigskip
\begin{prop}
\label{prop:lktmin}
Let $p$, $q$, and $n$ be non-negative integers, and suppose $\pi$ and $\pi'$ are
genuine irreducible admissible representations of $\meta(2n,\bbR)$ and $\opq$
which correspond to each other in the correspondence for the dual pair $\spopq$.
If $p+q\le 2n+1$ then every LKT of $\pi$ is of minimal $(p,q)$-degree in $\pi$,
and if $2n\le p+q$ then every LKT of $\pi'$ is of minimal degree in $\pi'$. 

\end{prop}

\bigskip\noindent
{\bf Proof.}

\qed

\bigskip
\begin{prop}
\label{prop:dllkt}
Let $p$, $q$, $m$, and $n$ be non-negative integers such that $p+q\le m\le\frac n2$.
Let $\pi=\theta_m(\one_{p,q})$, and $\pi'=\theta_{2m,n}(\pi)$.

\begin{enumerate}

\item
The $K$-types of $\pi$ are precisely those of the form
\beq
\left(\frac{p-q}2,\dots,\frac{p-q}2\right)+(a_1,a_2,\dots,a_p,0,\dots,0,-b_q,\dots,-b_1),
\eeq

\noindent
where $a_i$ and $b_i$ are non-negative even integers for all $i$, and each $K$-type occurs
with multiplicity one.

\item
Let $a=\frac{p-q+n}2-m$. Then $\pi'$ has a unique LKT $\Lambda$, given by

\beq
\label{eq:o2m2nlkt}
\Lambda=\left\{\begin{aligned}
         &(a,a,\dots,a,\underbrace{0,\dots,0}_q;0,\dots,0)\quad&\text{if $a\ge 0$ and even};\\
         &(a,a,\dots,a,\underbrace{1,\dots,1}_q;0,\dots,0)\quad&\text{if $a> 0$ and odd};\\
         &(0,\dots,0;\underbrace{-a,\dots,-a}_{m-p},0,\dots,0)
                                                         \quad&\text{if $a\le 0$ and even};\\
         &(0,\dots,0;\underbrace{-a,\dots,-a}_{m-p},\underbrace{1,\dots,1}_p,0,\dots,0)
                                                         \quad&\text{if $a< 0$ and odd}.                                        
            \end{aligned}
            \right.
\eeq

\end{enumerate}

\end{prop}

\bigskip\noindent
{\bf Proof.}
The first part of the proposition is Proposition 2.1, together with Corollary 2.7(c),
of \cite{kr}. Let

\beq
\sigma=\left(\tfrac{p-q}2,\dots,\tfrac{p-q}2\right)+(a_1,a_2,\dots,a_p,0,\dots,0,-b_q,\dots,-b_1),
\eeq

\noindent
be a $K$-type of $\pi$. Then

\beq
\sigma=\left(\tfrac{2m-n}2,\dots,\tfrac{2m-n}2\right)
+(a+a_1,\dots,a+a_p,a,\dots,a,a-b_q,\dots,a-b_1),
\eeq

\noindent
and the $(2m,n)$-degree of $\sigma$ is

\beq
d=\sum_{i=1}^p|a+a_i|+\sum_{i=1}^q|a-b_q|+(m-p-q)|a|.
\eeq

Suppose $a$ is even. Then the degree of $\sigma$ is minimized if
$a_i=0$ and $b_i=a$ for all $i$ if $a\ge 0$, and by choosing $a_i=-a$ and $b_i=0$
for all $i$ if $a$ is negative. So $\pi$ has a unique $K$-type $\sigma_0$ of minimal
degree. The $K$-type $\Lam$ corresponding to $\sigma_0$ in the space of joint harmonics 
(see Proposition \ref{prop:harm}) must
then be the unique LKT of $\pi'$ by Proposition \ref{prop:lktmin}.

If $a$ is odd then $\pi$ has more than one $K$-type of minimal $(p,q)$-degree:
for $a< 0$ choose $b_i=0$ for all $i$, and $a_i=-a+1$ or $-a-1$ in such a way that
the resulting weight is dominant. The corresponding $K$-types for $\opq$
are then those of the form

\beq
\Lam_{k,l}=\bigl(\underbrace{1,\dots,1}_{k},0,\dots,0;\underbrace{-a,\dots,-a}_{m-p},
     \underbrace{1,\dots,1}_l,0,\dots,0\bigr)
\eeq

\noindent
with $k+l=p$. By Proposition \ref{prop:lktmin}, the LKT's of $\pi'$ are precisely those
$\Lam_{k,l}$ for which the
Vogan-norm (\cite{vgb} Definition 5.4.18) of $\Lam_{k,l}$, 

\beq
\label{eq:norm}
||\Lam_{k,l}||=<\Lam_{k,l}+2\rho_c,\Lam_{k,l}+2\rho_c>
\eeq

\noindent
is minimal. Since the quantity $<\Lam_{k,l},\Lam_{k,l}>$ only depends on
$k+l=p$, we can minimize the quantity 
$n_{k,l}=\hf\bigl(||\Lam_{k,l}||-<2\rho_c,2\rho_c>-<\Lam_{k,l},\Lam_{k,l}>\bigr)$ instead.
If $r=\lbrack\frac n2\rbrack$ then 
$2\rho_c=(2m-2,2m-4,\dots,2,0;n-2,n-4,\dots,n-2r)$. So

\beq
n_{k,l}=\sum_{i=1}^k(2m-2i)+\sum_{i=1}^{m-p}(-a)(n-2i)+\sum_{i=1}^l(n-2m+2p-2i).
\eeq

\noindent
The second sum is independent of $k$ and $l$, and it is clear that $n_{k,l}$ will
be minimized by $k=p$, $l=0$ if $2m\le n-2m+2p$, and by $k=0$, $l=p$ if
$2m\ge n-2m+2p$. It remains to show that we must have $4m>n+2p$.
We know that $a=\frac{p-q+n}2-m<0$, so $p-q+n<2m$, Moreover, we have assumed that
$p+q\le m$, so we get $2p+n<3m\le 4m$, and we are done.

The case $a>0$ is much easier and left to the reader.

\qed



\begin{thebibliography}{ABMW}
\ifx\undefined\bysame
\newcommand{\bfsame}{\leavevmod\hbox to3em{\hrulefill}\,}
\fi

\bibitem{hwt} J. Adams, \textit{Unitary Highest Weight Modules},
Adv. Math. {\bf 63} (1987), 113--137.

\bibitem{abc} J. Adams and D. Barbasch, \textit{Reductive dual pair 
correspondence for complex groups}, J. Funct. Anal. {\bf 132} No. 1
(1995), 1--42.

\bibitem{abgen} \bysame, \textit{Genuine representations of the metaplectic
group}, Comp. Math. {\bf 113} (1998), 23--66.

\bibitem{abv} J. Adams, D. Barbasch, and D. Vogan, \textit{The Langlands 
Classification and Characters for Real Reductive Groups}, Progress
in Mathematics vol. 104, Birkh\"auser, Boston-Basel-Berlin, 1992.

\bibitem{ar} J. Arthur, \textit{On Some Problems Suggested by the Trace
Formula}, Lecture Notes in Mathematics vol. 1024, Springer Verlag,
Berlin-Heidelberg-New York, 1983.

\bibitem{bv} D. Barbasch and D. Vogan, \textit{Unipotent representations
of complex semisimple groups}, Ann. Math. {\bf 121} (1985), pp. 41--110.

\bibitem{cmc}
D.~H.~Collingwood and W.~M.~McGovern, {\em Nilpotent orbits in
semisimple Lie algebras}, Chapman and Hall (London), 1994.


\bibitem{ehw} T. Enright, R. Howe, and N. Wallach, \textit{Classification
of unitary highest weight modules}, {\it in} ``Representation Theory
of Reductive Groups'' (P.C. Trombi, Ed.), pp. 97--144,
Birkh\"auser, Boston, 1983.

\bibitem{hel} S. Helgason, \textit{Differential Geometry and Symmetric
Spaces}, Academic Press, New York, 1962.

\bibitem{hnotes} R. Howe, \textit{Elements of the oscillator representation}, 
Manuscript based on a course on dual pairs given at Yale University (19?),
unpublished.

\bibitem{l2dual} R. Howe, \textit{$L^2$-duality for stable dual pairs}, preprint.

\bibitem{theta} \bysame, \textit{$\theta$-series and invariant theory},
{\it in}  Proc. Sympos. Pure Math. Vol. 33, Amer. Math. Soc.,
Providence, RI (1979), 275--285.

\bibitem{rank} \bysame, \textit{On a notion of rank for unitary representations
of the classical groups}, Harmonic Analysis and Group Representations,
C.I.M.E. II Ciclo 1980, Cortona-Arezzo, A. Figa-Talamanca coord., Liquori
Editore, Naples (1982), 223--331.

\bibitem{howe} \bysame, \textit{Small unitary representations of 
classical groups}, in Group representations, ergodic theory, operator 
algebras and math. physics, C.C. Moore, ed. Springer 1986, pp. 121-150.

\bibitem{transc} \bysame, \textit{Transcending classical invariant theory},
J. Am. Math. Soc. {\bf 2} (1989), 535--552.

\bibitem{howezhu} R. Howe and C. Zhu, \textit{Eigendistributions for
orthogonal groups and representations of symplectic groups}, 
J. reine angew. Math., to appear.

\bibitem{huangzhu} J. Huang and C. Zhu, \textit{On certain small
representations of indefinite orthogonal groups}, Representation
Theory {\bf 1} (1997), 190--206.

\bibitem{knapp} A. Knapp, \textit{Representation Theory of Semisimple Groups:
An Overview Based on Examples}, Princeton University Press, Princeton,
NJ, 1986.

\bibitem{kv} A. Knapp and D. Vogan, \textit{Cohomological Induction and
Unitary Representations}, Princeton University Press, Princeton, NJ, 1995.

\bibitem{kave} M. Kashiwara and M. Vergne, \textit{On the Segal-Shale-Weil
representations and harmonic polynomials}, Inven. Math. 44 (1978), 1--47.

\bibitem{kudla} S. Kudla, \textit{On the local theta correspondence},
Invent. Math. {\bf 83} (1986), 229--255.

\bibitem{kr} S. Kudla and S. Rallis, \textit{Degenerate Principal Series
and Invariant Distributions}, Isr. J. Math. {\bf 69} (1990), 25-45.

\bibitem{leeloke} S. Lee and H. Loke, \textit{Degenerate principal series
representations of $U(p,q)$ and $Spin_0(p,q)$}, preprint.

\bibitem{leezhu} 
S.~Lee and C.~Zhu, \textit{Degenerate principal series 
and local theta correspondence}, Trans.~Amer.~Math.~Soc. {\bf 350}
(1998), no.~12, 5017--5046.

\bibitem{leezhu2}
\bysame, \textit{Degenerate principal series and local theta correspondence II},
Israel J. Math. {\bf 100} (1997), 29--59.

\bibitem{li-singular} 
J.-S.~Li, \textit{Singular unitary representations
of classical groups}, Invent.~Math.~{\bf 97} (1989), 237--255.

\bibitem{li-classify}
\bysame, \textit{On the classification of irreducible low rank
unitary representations of classical groups}, Comp. Math.~{\bf 71} (1989),
29--48.

\bibitem{li-theta} 
\bysame, \textit{Local theta lifting for unitary representations
with nonzero cohomology}, Duke Math.~J.~{\bf 61} (1990), 913--937.

\bibitem{lptz} J.-S.~Li, A.~Paul, E.-C.~Tan, C.~Zhu, {\it
The explicit duality correspondince of $(Sp(p,q),O^*(2n))$}, preprint.

\bibitem{moeglin} 
C.~Moeglin, \textit{Correspondance the Howe pour les
paires reductives duales; quelques calculs dans le cas archim\'edian},
J.~Funct.~Anal.~{\bf 85} (1989), 1--85.

\bibitem{part} 
R.~Parthasarathy, \textit{Criteria for the unitarizability of
some highest weight modules}, Proc.~Indian Acad.~Sci.~{\bf 89} (1980), 1--24.

\bibitem{p1} 
A.~Paul, \textit{Howe correspondence for real unitary groups},
J.~Funct.~Anal.~{\bf 159} (1998), 384--431.

\bibitem{p2} 
\bysame, \textit{Howe correspondence for real unitary groups II},
Proc.~Amer.~Math.~Soc.~{\bf 128}(2000), 3129--3136.

\bibitem{pmoeglin}
\bysame, \textit{On the Howe correspondence for symplectic-orthogonal
dual pairs}, preprint.

\bibitem{paultan} A. Paul and E.~C. Tan, \textit{On the dual pairs 
$(\opq,SL(2,\bbR))$, $(U(p,q),U(1,1))$, and $(Sp(p,q),O^*(4))$},
Pac. J. Math. {\bf 187} no. 2 (1999), 349--378.

\bibitem{pt}
A.~Paul and P.~Trapa, \textit{One-dimensional representations of $U(p,q)$
and the Howe correspondence}, J.~Funct.~Anal.~{\bf 195} no.1 (2002), 129--166.

\bibitem{ptknapp}
\bysame, \textit{Some small unipotent representations of indefinite 
orthogonal groups and the theta correspondence}, preprint.

\bibitem{prz} 
T.~Przebinda, \textit{On Howe's duality theorem}, 
J.~Funct.~Anal.~{\bf 81}(1988), 160--183.

\bibitem{prz2}
 \bysame, \textit{The duality correspondence of infinitesimal
characters}, Colloq.~Math.~{\bf 70}(1996), no.~1, 93--102.

\bibitem{prz:av}
\bysame, {\it Characters, dual pairs, and unitary representations}. 
Duke Math.~J., {\bf 69}(1993), no.~3, 547--592. 

\bibitem{rao} 
R.~Rao, \textit{On some explicit formulas in the theory of
the Weil representation}, Pac.~J.~Math.~{\bf 157} (1993), no.~2, 335--371.

\bibitem{brooks} 
B.~Roberts, \textit{The Weil representation and dual pairs}, 
Notes from a workshop on the theta correspondence at the University of
Maryland, May 1994.

\bibitem{ssr} S.~Salamanca-Riba, \textit{On the unitary dual of real reductive
Lie groups and the $A_{\frq}(\lambda)$-modules: the strongly regular case},
Duke Math.~J.~{\bf 96}(1999), no.~3, 521--546. 

\bibitem{spehvogan} 
B.~Speh and D.~Vogan, \textit{Reducibility
of Generalized Principal Series Representations},
Acta Math.~{\bf 145}(1980), 227--299.

\bibitem{trapa1} 
P.~Trapa, \textit{Annihilators and associated varieties of
$A_{\frq}(\lambda)$ modules for $U(p,q)$}, Compositio Math., to appear.

\bibitem{vgb} 
D.~Vogan, \textit{Representations of Real Reductive Lie Groups},
Progess in Math.~{\bf 15}(1981), Birkh\"auser(Boston).

\bibitem{vunit} 
D.~Vogan, \textit{Unitarizability of certain series of 
representations}, Ann.~Math.~{\bf 120}(1984), 141--187. 

\bibitem{vor} 
D.~Vogan, {\em Unitary representations of reductive Lie groups},
Annals of Mathematical Studies~{\bf 118}(1987), Princeton University
Press (Princeton).

\bibitem{vz} 
D.~Vogan and G.~Zuckerman, \textit{Unitary representations with
non-zero cohomology}, Compositio Math.~{\bf 53}(1984), 51--90.

\bibitem{zhuscalar}
C.~Zhu, \textit{Representations with scalar $K$-types and applications},
preprint.

\end{thebibliography}

\end{document}

