Infinitesimal generator of a unitary representation

Definition

Let $\pi$ be a strongly continuous unitary representation of a Lie group $G$ on $\mathcal H$ , and let $X$ lie in the Lie algebra of $G$ . The infinitesimal generator in the direction $X$ is the unique self-adjoint operator $A_X$ supplied by Stone's theorem such that

\pi(\exp(tX))=e^{itA_X}\qquad(t\in\mathbb R).

Its domain consists exactly of vectors $v$ for which the strong limit

\lim_{t\to0}\frac{\pi(\exp(tX))v-v}{t}

exists; on that domain the limit equals $iA_Xv$ . The generator is generally unbounded and therefore includes its domain as part of its data.

Relation with the derived representation

Every smooth vector lies in $\operatorname{Dom}(A_X)$ . The derived representation operator satisfies, on $\mathcal H^\infty$ ,

d\pi(X)=iA_X, \qquad A_X=-i\,d\pi(X).

The first equality denotes equality of the two actions on the smooth-vector domain, not equality of maximal operators. The derived operator is essentially skew-adjoint there, and its closure is $iA_X$ Warner, §4.4.

Covariance and common domains

Unitary conjugation transports generators according to

A_{\operatorname{Ad}(g)X}=\pi(g)A_X\pi(g)^{-1}.

On the common domain $\mathcal H^\infty$ , the assignment $X\mapsto d\pi(X)$ is linear and preserves Lie brackets. Statements such as $A_{X+Y}=A_X+A_Y$ require care at the level of unbounded operators: the algebraic equality holds on smooth vectors, while the domains and closures of the maximal operators must still be controlled.

Sign conventions

References

Michael Reed and Barry Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Academic Press, 1972. DOI record. Relevant: §VIII.4 on one-parameter unitary groups.
Garth Warner, Harmonic Analysis on Semi-Simple Lie Groups I, Grundlehren der mathematischen Wissenschaften 188, Springer, 1972. DOI record. Relevant: §4.4 on derived representations and their generators.