Definition of $(\mathfrak{g}, H)$-modules used in Cap and Slovak's parabolic geometry text?

Question

On page 68 of Cap and Slovak's parabolic geometries text, they mention $(\mathfrak{g}, H)$-modules for $\mathfrak{g}$ the Lie algebra of a Lie group $G$ and $H \hookrightarrow G$ a closed subgroup. Though they are interested in the case where $G$ is semisimple and $H$ is a parabolic subgroup, they imply that the method they use works in general. Unfortunately, I can't seem to find the definition of a $(\mathfrak{g}, H)$-module in their text. I would guess that it is a vector space $V$ which is both a $\mathfrak{g}$-rep and an ${H}$-rep such that $$h.(X.v) = (\operatorname{Ad}(h)X).(h.v)$$ for all $h \in H$, $X \in\mathfrak{g}$, and $v \in V$. I would also guess that a homomorphism of $(\mathfrak{g}, H)$-modules should just be a linear map which is a homomorphism of $\mathfrak{g}$-reps and $H$-reps. Is this correct?

There is a wikipedia page for $(\mathfrak{g}, K)$-modules where $K$ is a maximal compact subgroup of a real reductive group $G$, but it seems to me that conditions 2 and 3 listed in the wiki definition may not be relevant to the situation in Cap and Slovak.

Answer 1

In the book, we don't work with a general notion of $(\mathfrak g,H)$-modules, but only with the explicit examples $J^\infty_o E$ for a homogeneous vector bundle $E\to G/H$ and $\mathcal U(\mathfrak g)\otimes_{\mathcal U(\mathfrak h)}V$ for a representation $V$ of $H$. As you thought, a homomorphism of $(\mathfrak g,H)$-modules is just a linear map which is equivariant for the actions of both $\mathfrak g$ and $H$.

The general definition of a $(\mathfrak g,H)$-module $W$ that I would use, would certainly use condition 3 from the web page you link to. This says that the restriction of the $\mathfrak g$-action to $\mathfrak h\subset\mathfrak g$ coincides with the derivative of the $H$-action. Second you have to use the condition on compatibility with the adjoint action, which I would write as $Ad(h)(X)\cdot w=h\cdot X\cdot h^{-1}\cdot w$ for $h\in H$, $X\in\mathfrak g$ and $w\in W$.

The subtle point about a general definition is which assumptions on the topology of $W$ and continuity/differentiability of the action of $H$ one should include. This depends on the applications you have in mind. In representation theory, you usually would usually look at object like Harish-Chandra modules, i.e.~$K$-finite vectors in a unitary representation of $G$. The definition in the web page you link looks kind of taylored to this situation to me. The restriction to finite dimensions makes sure that continuity of the group action implies differentiability so that the condition on the infinitesimal representation of $\mathfrak h$ makes sense.