A bounded operator T is said to be homogeneous if σ(T) ⊆ D and φ(T) is unitarily equivalent to T for all φ ∈ M¨ob. The intertwining unitary U(φ) between φ(T) and T for any homogeneous operator in the Cowen-Douglas class Bn(D) can be chosen to ensure the map φ 7→ U(φ) is a representation of the group M¨ob. Moreover, this intertwining relationship is the Mackey imprimitivity restricted to the function algebra A(D) consisting of functions holomorphic ina neigbourhood of the closed disc D instead of the C ∗ algebra C(T) of continuous functions. The definition of homogeneous operators has a natural generalization to the commuting tuple of operators. A classification of all the irreducible tuples in the Cowen-Douglas class Br(D n ) except r = 1, 2, 3, is not known. However, a new family of irreducible tuples in Br(D n ) which are homogeneous with respect to M¨obn have been obtained whose associated representation is multiplicity free.
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