Infinitesimal equivalence of admissible representations

Definition

Let $G$ be a real reductive group, $K$ a maximal compact subgroup, and $(\pi,E)$ , $(\sigma,F)$ two admissible continuous representations whose $K$ -finite parts are Harish–Chandra modules. They are infinitesimally equivalent if there is a linear isomorphism

T:E_K\longrightarrow F_K

that intertwines both the action of $K$ and the differentiated action of the complexified Lie algebra $\mathfrak g$ . Equivalently, $E_K$ and $F_K$ are isomorphic as $(\mathfrak g,K)$ -modules. No continuity of $T$ on the ambient completed spaces is part of this definition.

Data retained and forgotten

Infinitesimal equivalence retains every $K$ -type and its multiplicity, the action of the universal enveloping algebra, infinitesimal characters, and the submodule structure of the Harish–Chandra module. It forgets the topology and norm of the global representation. Consequently, topological equivalence implies infinitesimal equivalence, but the converse requires a globalization theorem or additional hypotheses. This distinction is part of the standard passage between admissible group representations and $(\mathfrak g,K)$ -modules Wallach, Chapter 4, §4.5.

Relation to globalization

Different Banach or distribution globalizations of one Harish–Chandra module are infinitesimally equivalent even when their ambient topologies are unlike. Inside the Casselman–Wallach category, however, the globalization theorem upgrades infinitesimal equivalence uniquely to a continuous $G$ -isomorphism. Thus the relation is strictly weaker in a broad topological category but coincides with isomorphism in the smooth admissible moderate-growth Fréchet category.

Example and near-miss

The smooth and distribution globalizations attached to the same irreducible Harish–Chandra module have the same $K$ -finite vectors and are therefore infinitesimally equivalent. By contrast, two representations with the same infinitesimal character need not be infinitesimally equivalent: equality of the central character records only the action of $Z(U(\mathfrak g))$ , not the entire $(\mathfrak g,K)$ -module.

Conventions and scope

Some authors define infinitesimal equivalence first for admissible Hilbert representations and then extend it to other globalizations. The invariant criterion is the $(\mathfrak g,K)$ -module isomorphism in the core. Merely intertwining the $\mathfrak g$ -actions is insufficient because it can lose the action of the possibly disconnected compact group $K$ .

References

Nolan R. Wallach, Real Reductive Groups I, Academic Press, 1988. Publisher record. Relevant: Chapter 4, §4.5 on infinitesimal equivalence.
Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VIII on admissible representations and their underlying $(\mathfrak g,K)$ -modules.