Compact Lie Group Case
Complex irreducible representations of compact Lie groups have been completely classified. They are always finite-dimensional, unitarizable (i.e. admit an invariant positive definite Hermitian form) and are parametrized by their highest weights, which are precisely the dominant integral weights for the group. If G is a compact semisimple Lie group with a Cartan subalgebra h then its coadjoint orbits are closed and each of them intersects the positive Weyl chamber h*+ in a single point. An orbit is integral if this point belongs to the weight lattice of G. The highest weight theory can be restated in the form of a bijection between the set of integral coadjoint orbits and the set of equivalence classes of irreducible unitary representations of G: the highest weight representation L(λ) with highest weight λ∈h*+ corresponds to the integral coadjoint orbit G·λ. The Kirillov character formula amounts to the character formula earlier proved by Harish-Chandra.
Read more about this topic: Orbit Method
Famous quotes containing the words compact, lie, group and/or case:
“The worst enemy of truth and freedom in our society is the compact majority. Yes, the damned, compact, liberal majority.”
—Henrik Ibsen (1828–1906)
“O mighty Caesar! Dost thou lie so low?
Are all thy conquests, glories, triumphs, spoils,
Shrunk to this little measure? Fare thee well.”
—William Shakespeare (1564–1616)
“It’s important to remember that feminism is no longer a group of organizations or leaders. It’s the expectations that parents have for their daughters, and their sons, too. It’s the way we talk about and treat one another. It’s who makes the money and who makes the compromises and who makes the dinner. It’s a state of mind. It’s the way we live now.”
—Anna Quindlen (20th century)
“I am absurdly fearful about this voyage. Various little omens have combined to give me a dark feeling.... Perhaps we shall live to laugh at these. But in case of mishap I should perish with my husband and child, perhaps to be transferred to some happier state.”
—Margaret Fuller (1810–1850)