Topological Group Case
For a topological group G, the regular representation in the above sense should be replaced by a suitable space of functions on G, with G acting by translation. See Peter-Weyl theorem for the compact case. If G is a Lie group but not compact nor abelian, this is a difficult matter of harmonic analysis. The locally compact abelian case is part of the Pontryagin duality theory.
Read more about this topic: Regular Representation
Famous quotes containing the words group and/or case:
“The trouble with tea is that originally it was quite a good drink. So a group of the most eminent British scientists put their heads together, and made complicated biological experiments to find a way of spoiling it. To the eternal glory of British science their labour bore fruit.”
—George Mikes (b. 1912)
“And the case of butterflies so rich it looks
As if all summer settled there and died.”
—Philip Larkin (1922–1986)