Intertwining operator

A continuous linear operator $ T: E _ {1} \rightarrow E _ {2} $ such that $ T \pi _ {1} ( x) = \pi _ {2} ( x) T $, where $ \pi _ {1} $ and $ \pi _ {2} $ are mappings of a set $ X $ into two topological vector spaces $ E _ {1} $ and $ E _ {2} $ and $ x \in X $. This concept is especially fruitful in the case when $ X $ is a group or an algebra and $ \pi _ {1} , \pi _ {2} $ are representations of $ X $. The set of intertwining operators forms the space $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $, which is a subspace of the space of all continuous linear mappings from $ E _ {1} $ to $ E _ {2} $. If $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) = ( 0) $ and $ \mathop{\rm Hom} ( \pi _ {2} , \pi _ {1} ) = ( 0) $, then $ \pi _ {1} $ and $ \pi _ {2} $ are called disjoint representations. If $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $ contains an operator that defines an isomorphism of $ E _ {1} $ and $ E _ {2} $, then $ \pi _ {1} $ and $ \pi _ {2} $ are equivalent. If $ E _ {1} , E _ {2} $ are locally convex spaces, if $ E _ {1} ^ {*} $ and $ E _ {2} ^ {*} $ are their adjoints, and if $ \pi _ {1} ^ {*} $ and $ \pi _ {2} ^ {*} $ are the representations contragredient to $ \pi _ {1} $ and $ \pi _ {2} $, respectively (cf. Contragredient representation), then for any $ T \in \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) $, the operator $ T ^ {*} $ is contained in $ \mathop{\rm Hom} ( \pi _ {2} ^ {*} , \pi _ {1} ^ {*} ) $. If $ \pi _ {1} $ and $ \pi _ {2} $ are finite-dimensional or unitary representations and $ \pi _ {1} $ is irreducible, then $ \pi _ {2} $ admits a subrepresentation equivalent to $ \pi _ {1} $ if and only if $ \mathop{\rm Hom} ( \pi _ {1} , \pi _ {2} ) \neq ( 0) $. See also Intertwining number.

References

[1]	A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian)
[2]	A.I. Shtern, "Theory of group representations" , Springer (1982) (Translated from Russian)