Induced representation and Frobenius reciprocity

Let us take a pair of compact Lie groups .

Definition:  is the space of square integrable functions taking values in and satisfying the covariance property:

Schematic notation:      

Now we can prove Eq. (5):

The space is a direct sum of finite-dimensional reps of . Therefore, can be interpreted as the multiplicity of in . (More precisely, the multiplicity of in the component of corresponding to the tensor in rep )