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Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups · arXivDesk

1402.1717Feb 7, 201422 pages; added Section 5 in v2 to separate H-type results and rank one results

Invariant differential operators on H-type groups and discrete components in restrictions of complementary series of rank one semisimple groups

Jan Möllers, Bent Ørsted, Genkai Zhang

Representation Theory

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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 $G_1$

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on Sobolev 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 nilpotent radical of a parabolic subgroup in a semisimple group.

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