Statement
A distribution on a group G or its Lie algebra is called invariant if it is invariant under conjugation by G.
A distribution on a group G or its Lie algebra is called an eigendistribution if it is an eigenvector of the center of the universal enveloping algebra of G (identified with the left and right invariant differential operators of G.
Harish-Chandra's regularity theorem states that any invariant eigendistribution on a semisimple group or Lie algebra is a locally integrable function. The condition that it is an eigendistribution can be relaxed slightly to the condition that its image under the center of the universal enveloping algebra is finite-dimensional. The regularity theorem also implies that on each Cartan subalgebra the distribution can be written as a finite sum of exponentials divided by a function Δ that closely resembles the denominator of the Weyl character formula.
Read more about this topic: Harish-Chandra's Regularity Theorem
Famous quotes containing the word statement:
“Eloquence must be grounded on the plainest narrative. Afterwards, it may warm itself until it exhales symbols of every kind and color, speaks only through the most poetic forms; but first and last, it must still be at bottom a biblical statement of fact.”
—Ralph Waldo Emerson (1803–1882)
“The parent is the strongest statement that the child hears regarding what it means to be alive and real. More than what we say or do, the way we are expresses what we think it means to be alive. So the articulate parent is less a telling than a listening individual.”
—Polly Berrien Berends (20th century)
“He that writes to himself writes to an eternal public. That statement only is fit to be made public, which you have come at in attempting to satisfy your own curiosity.”
—Ralph Waldo Emerson (1803–1882)