Super Harish–Chandra pair

Definition

A super Harish–Chandra pair over $\mathbb R$ is a pair $(G_0,\mathfrak g)$ consisting of:

1. a finite-dimensional real Lie group $G_0$;
2. a finite-dimensional real Lie superalgebra $\mathfrak g=\mathfrak g_{\bar0}\oplus\mathfrak g_{\bar1}$;
3. an identification $\mathfrak g_{\bar0}\cong\operatorname{Lie}(G_0)$;
4. a smooth action $\operatorname{Ad}:G_0\to\operatorname{Aut}_{\bar0}(\mathfrak g)$ by even Lie-superalgebra automorphisms.

The action restricts on $\mathfrak g_{\bar0}$ to the ordinary adjoint action of $G_0$, and its differential at the identity is the adjoint action of $\mathfrak g_{\bar0}$ on all of $\mathfrak g$.

A morphism $(G_0,\mathfrak g)\to(H_0,\mathfrak h)$ is a Lie-group homomorphism $\Phi_0:G_0\to H_0$ and an even Lie-superalgebra homomorphism $\phi:\mathfrak g\to\mathfrak h$, compatible with the even-part identifications and the two group actions.

The adjective “super” is important: this object is not a Harish–Chandra module from the representation theory of real reductive groups.