Classification of spherical nilpotent orbits for \(U(p,p)\) (Q1766219)

The author gives a characterization of spherical nilpotent \(K_{\mathbb C}\)-orbits for the symmetric pair \((G,K)=(U(p,q),U(p)\times U(q))\). If a nilpotent \(K_{\mathbb C}\) -orbit \({\mathcal O}_D\) with a signed Young diagram \(D\) is a theta lift, then it is spherical if and only if \(D\) is of the form \([3^\epsilon\cdot2^k\cdot1^l]\) with \(\epsilon=0,1,\;k, l\geq0,\;3\epsilon+2k+l=p+q\). To obtain this characterization, he noticed that \({\mathcal O}\) is spherical if and only if the regular function ring \(\mathbb C[\overline{\mathcal O}]\) has a multiplicity-free decomposition as a \(K_{\mathbb C}\)-module. Furthermore, if \({\mathcal O}\) is a theta lift such as \({\mathcal O}=\theta({\mathcal O}')\), then the \(K_{\mathbb C}\)-module structure of \(\mathbb C[\overline{\mathcal O}]\) is completely described via that of \(\overline{{\mathcal O}'}\) [see the previous paper by the author, \textit{H. Ochiai} and \textit{C.-B. Zhu} math.RT/0312453, 2003]. Using these facts, when \(D\) is of the above form, we can obtain a multiplicity-free decomposition of \(\mathbb C[\overline{{\mathcal O}}_D]\) and thus, \({\mathcal O}_D\) is spherical. When \(D\) is not of the listed form, constructing non multiplicity-free decompositions of \(\mathbb C[\overline{{\mathcal O}}_D]\) for five minimal orbits \({\mathcal O}_D\) with non-listed form \(D\), we can deduce that \({\mathcal O}_D\) with non-listed form \(D\) is not spherical. When \(p=q\), each spherical orbit is a theta lift. Hence, a complete classification of the spherical \(K_{\mathbb C}\)-orbits follows.