An automorphic form on a classical group is said to be singular if it has only degenerate Fourier coefficients along (the center of) a certain maximal parabolic subgroup. Such singular automorphic forms were studied in the case of holomorphic forms on classical domains by Maass, Freitag and Resnikoff, among others. Their representation theoretic treatment were begun by Howe.
Suppose $k$ is a number field and $D$ is a division algebra with an involution $i$, such that the set of fixed points of $\iota$ is $k$. Let $U$ be a finite dimensional vector space over $D$ endowed with a non-degenerate Hermitian or skew-Hermitian form, and let $G$ be the corresponding group of isometries. We denote by $n$ the Witt index of $U$; this is also the split rank of $G$. Suppose $V$ is another vector space over $D$ with a non-degenerate form which is Hermitian (resp. skew-Hermitian) if $U$ is skew-Hermitian (resp. Hermitian). Let $G'$ be the isometry group of $V$. Then $(G,G')$ is a typical reductive dual pair. We assume the dimension of $V$ over $D$ to be $\leq n$. Let $v$ denote places of $k$ and ${\bf A}$ the adele ring of $k$. Let $\sigma=\otimes \sigma_v$ be an irreducible admissible unitary representation of $G'({\bf A})$. Let $\theta(\sigma_v)$ be the local theta lift of $\sigma_v$ to $G(k_v)$ and set $\theta(\sigma)=\otimes \theta(\sigma_v)$. If $\dim(V)<n$ and $\theta(\sigma)$ is automorphic (that is, it can be realized as a space of smooth functions on $G(k)\sbs G({\bf A}))$, then it is singular in the sense that all smooth functions contained in its space are singular. It can be shown that all irreducible singular unitary automorphic representations are of the form $\theta(\sigma)$.
Let ${\scr A}(G)$ be the space of automorphic forms and ${\scr A}_2(G)$ its intersection with $L^2(G(k)\sbs G({\bf A}))$. Let $m(\pi,{\scr A}(G))$ and $m(\pi,{\scr A}_2(G))$ be the multiplicity of $\pi$ in ${\scr A}(G)$ and ${\scr A}_2(G)$ respectively. Our main result amounts to two inequalities contained in the following
Theorem: In the above setting, excluding the case where $D$ is a quaternion algebra and $V$ is skew-Hermitian, one has $$m(\theta(\sigma),{\scr A}(G))\leq m(\sigma,{\scr }(G'))\tag1$$ $$m(\theta(\siogma),{\scr A}_2(G))\geq m(\sigma,{\scr A}_2(G'))\tag2$$
A number of consequences can be deduced. First assume $V$ (equivalently, $G'$) to be anisotropic over $k$; then ${\scr A}(G')={\scr A}_2(G')$. The above inequalities give $$m(\theta(\sigma),{\scr A}(G))=m(\theta(\sigma),{\scr A}_2(G))=m(\sigma,{\scr A}(G'))\quad (G'\ {\rm anisotropic})\tag3$$ The second inequality in (3) was proven by Howe, at least when $G$ is the symplectic group.
Next assume $\sigma$ is automorphic and one dimensional, and $V$ is not the two dimensional split quadratic space over $k$. Then $m(\sigma,{\scr A}(G'))=m(\sigma,{\scr A}_2(G'))=1$, and (1) and (2) amount to $$m(\theta(\sigma),{\scr A}(G))=m(\theta(\sigma),{\scr A}_2(G))=1\tag4$$ When $G$ is the symplectic group and $\sigma$ is trivial, (4) is due to Kudla, Rallis and Soudry $(n=2)$ and Kudla and Rallis ($n$ general).
The exclusion clause in the theorem is necessary, and is a result of the failure of the Hasse principle for skew-Hermitian forms over quaternion algebras. This in fact leads to some interesting phenomenon about multiplicities. The simplest case is the following. We tak e$D$ to be a quaternion algebra, $U$ the isotropic Hermitian space of dimension 2, and $V$ of dimension 1; one might write $G={\rm Sp}(,1)$, $G'=O^*(2)$. Now take $\sigma$ to be any automorphic representation. Then we have $$m(\theta(\sigma),{\scr A}(G))=m(\theta(\sigma),{\scr A}_2(G))=2^{s-2}\tag5$$ Here $s$ is the number of places of $k$ at which $D$ is ramified. The representation $\theta(\sigma)$ is cuspidal if and only if $\sigma$ is non-trivial. It would be amusing to test formula (5) against Arthur's general conjectures.