Invariant differential operators on the Schrödinger model for the minimal representation of the conformal group (Q2902461)

Consider three differential operators on \(\mathbb{R}^{2n-1}\) NEWLINE\[NEWLINE\begin{aligned} D_1&:=\left(x_1+\frac{x_1}{4}\,\Delta-\frac{1}{2}\,E\,\frac{\partial}{\partial x_1}-\frac{n-1}{2}\,\frac{\partial}{\partial x_1}\right),\\ D_2&:=\sqrt{-1}\sum_{j=1}^{n-1}\left(x_{2j}\,\frac{\partial}{\partial x_{2j+1}}-x_{2j+1} \,\frac{\partial}{\partial x_{2j}}\right),\\ D_3&:=|x|\left(\frac{1}{4}\,\Delta-1\right), \end{aligned}NEWLINE\]NEWLINE where \(|x|=\big(\sum_{j=1}^{2n-1}x_j^2\big)^{1/2}\) is the norm of \(x=(x_1,\dotsc,x_{2n-1})\in\mathbb{R}^{2n-1}\), \(\Delta=\sum_{j=1}^{2n-1}\partial^2/\partial x_j^2\) and \(E=\sum_{j=1}^{2n-1}x_j\partial/\partial x_j\).NEWLINENEWLINEApplying representation theory, the authors prove the following theorem. {\parindent=6mm \begin{itemize}\item[1.] The differential operators \(D_1, D_2, D_3\) extend to self-adjoint operators on the Hilbert space \(L^2(\mathbb{R}^{2n-1},|x|^{-1}dx)\), where \(dx=dx_1\dotsm dx_{2n-1}\). \item[2.] \(D_1, D_2, D_3\) mutually commute. \item[3.] \(D_1, D_2, D_3\) have only discrete spectra on \(L^2(\mathbb{R}^{2n-1},|x|^{-1}dx)\), respectively. \item[4.] The set of the joint eigenvalues of \((D_1, D_2, D_3)\) is given by NEWLINE\[NEWLINE \big\{(x,y,z)\in\mathbb{Z}^3:\;x+y+z-n+1\equiv 0 \pmod 2,\;|x|+|y|\leq-z-n+1\big\}.NEWLINE\]NEWLINE \item[5.] \(D_1, D_2, D_3\) are algebraically independent. NEWLINENEWLINE\end{itemize}} Given the identity component \(SO_0(2n,2)\) of the conformal orthogonal group \(O(2n,2)\) (\(n>1\)), let \(\pi_+\) be the irreducible representation of \(SO_0(2n,2)\) on \(L^2(\mathbb{R}^{2n-1},|x|^{-1}dx)\) through the Schrödinger model of \(O(2n,2)\) constructed by \textit{T. Kobayashi} and \textit{B. Ørsted} [Adv. Math. 180, No. 2, 486--512 (2003; Zbl 1046.22004); 180, No. 2, ibid. 513--550 (2003; Zbl 1049.22006); 180, No. 2, ibid. 551--595 (2003; Zbl 1039.22005)] and enriched by Kobayashi-Mano. Consider a compact group NEWLINE\[NEWLINEU:=U(n_1)\times U(n_2)\times U(1)\;\;(n_1,n_2\geq 1,\;n_1+n_2=n).NEWLINE\]NEWLINE For any subgroup \(H\) of \(SO_0(2n,2)\), let \({\mathcal D}(\mathbb{R}^{2n-1} \setminus\{0\})^H\) consist of all differential operators on \(\mathbb{R}^{2n-1} \setminus\{0\}\) that are invariant under the action of \(\pi_+(H)\). It is proved that, for any subgroup \(H\) of \(SO_0(2n,2)\) which contains \(U\), the restriction \(\pi_+|H\) is multiplicity-free, and the algebra \({\mathcal D}(\mathbb{R}^{2n-1} \setminus\{0\})^H\) is commutative.NEWLINENEWLINEThe generators of the algebra \({\mathcal D}(\mathbb{R}^{2n-1}\setminus \{0\})^H\) for the two compact subgroups \(H=I_1\times SO(2n-1)\times SO(2)\) and \(H=SO(2n)\times SO(2)\) of the group \(O(2n,2)\) are also obtained.