Global character of an admissible representation

Definition

Let $G$ be a real reductive Lie group with a fixed Haar measure, and let $\pi$ be a continuous, finite-length admissible Hilbert globalization in the standard real-reductive category. For $f\in C_c^\infty(G)$ , form the integrated operator

\pi(f)=\int_G f(g)\pi(g)\,dg.

The Harish–Chandra trace-class theorem makes $\pi(f)$ trace class for this class of globalizations. The global character, or Harish–Chandra character, of $\pi$ is the distribution

\Theta_\pi(f)=\operatorname{Tr}\pi(f),\qquad f\in C_c^\infty(G).

It replaces the usually undefined pointwise trace of the individual infinite-dimensional operators $\pi(g)$ .

This definition includes nonunitary admissible globalizations; it uses no $*$ -representation property of $f\mapsto\pi(f)$ .

Invariance and additivity

The distribution $\Theta_\pi$ is invariant under conjugation: translating a test function by $g\mapsto xgx^{-1}$ does not change its value. Characters are additive in short exact sequences of finite-length admissible representations and therefore depend only on the corresponding class in the Grothendieck group. For a finite-dimensional representation, the definition recovers integration against the ordinary trace function $g\mapsto\operatorname{Tr}\pi(g)$ .

Infinitesimal eigencharacter

If $\pi$ has infinitesimal character $\chi$ , then its global character is a joint eigendistribution for the invariant differential operators arising from $Z(U(\mathfrak g_\mathbb C))$ . This connects the central algebraic invariant of the differentiated representation with a conjugation-invariant analytic object on $G$ .

Regularity

The Harish–Chandra regularity theorem represents $\Theta_\pi$ by a locally integrable function that is real analytic on the regular semisimple set. The phrase “character value at $g$ ” refers to this representing function where it is defined, not to an operator trace of $\pi(g)$ .

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986; reprint 2001. Author-maintained record. Relevant: Chapter X on global characters.
Harish-Chandra, “Invariant Eigendistributions on a Semisimple Lie Group,” Transactions of the American Mathematical Society 119 (1965), 457–508. DOI record. Relevant: invariant eigendistributions and character regularity.