Invariant Differential Operators on H-Type Groups and Discrete Components in Restrictions of Complementary Series of Rank One Semisimple Groups

Abstract

We explicitly construct a finite number of discrete components in the restriction of complementary series representations of rank one semisimple groups $G$ to rank one subgroups $G_1$. For this we use the realizations of complementary series representations of $G$ and $G_1$ on Sobolev-type spaces on the nilpotent radicals $N$ and $N_1$ of the minimal parabolics in $G$ and $G_1$, respectively. The groups $N$ and $N_1$ are of H-type and we construct explicitly invariant differential operators between $N$ and $N_1$. These operators induce the projections onto the discrete components.Our construction of the invariant differential operators is carried out uniformly in the framework of H-type groups and also works for those H-type groups which do not occur as a nilpotent radical of a parabolic subgroup in a semisimple group.

Appendix: Jacobi Polynomials

The classical Jacobi polynomials $P_n^{(\alpha ,\beta )}(z)$ can be defined by (see [10, Eq. 10.8 (12)])

They satisfy the following parity identity (see [10, Eq. 10.8 (13)]):

$$\begin{aligned} P_n^{(\alpha ,\beta )}(-z) = (-1)^nP_n^{(\beta ,\alpha )}(z). \end{aligned}$$

The following recurrence relation in $n$ holds (see [10, Eq. 10.8 (11)]):

$$\begin{aligned}&(2n+\alpha +\beta +1)\left[ (2n+\alpha +\beta )(2n+\alpha +\beta +2)z+\alpha ^2-\beta ^2\right] P_n^{(\alpha ,\beta )}(z)\\&\quad = 2(n+\alpha )(n+\beta )(2n+\alpha +\beta +2)P_{n-1}^{(\alpha ,\beta )}(z)\\&\qquad +2(n+1)(n+\alpha +\beta +1)(2n+\alpha +\beta )P_{n+1}^{(\alpha ,\beta )}(z). \end{aligned}$$

(6.1)

We further have the following functional relation (see [10, Eq. 10.8 (33)]):

$$\begin{aligned}&(2n+\alpha +\beta +2)(1+z)P_n^{(\alpha ,\beta +1)}(z)\\&\quad = 2(n+\beta +1)P_n^{(\alpha ,\beta )}(z)+2(n+1)P_{n+1}^{(\alpha ,\beta )}(z). \end{aligned}$$

(6.2)

Identities (6.1) and (6.2) together yield

$$\begin{aligned}&2(n+\beta +1)\left[ (2n+\alpha -\beta )+(2n+\alpha +\beta +2)z\right] P_n^{(\alpha ,\beta )}(z)\\&\qquad =(n+\alpha )(2n+\alpha +\beta +2)(1+z)^2P_{n-1}^{(\alpha ,\beta +2)}(z)+4(n+1)(\beta +1)P_{n+1}^{(\alpha ,\beta )}(z). \end{aligned}$$

(6.3)

We further remark that for $\alpha =-\frac{1}{2}$ or $\beta =-\frac{1}{2}$ the Jacobi polynomials degenerate to Gegenbauer polynomials (see [10, Eq. 10.9 (21)]):

$$\begin{aligned} C_{2n}^\lambda (z) = \frac{(\lambda )_n}{(\frac{1}{2})_n}P_n^{(\lambda -\frac{1}{2},-\frac{1}{2})}(2z^2-1) = (-1)^n\frac{(\lambda )_n}{(\frac{1}{2})_n}P_n^{(-\frac{1}{2},\lambda -\frac{1}{2})}(1-2z^2). \end{aligned}$$

(6.4)