Characteristic Varieties and Associated Varieties

Question

Two notions that occur often in representation theory seem to be that of a "characteristic variety" and that of an "associated variety". The former term seems exclusive to D-module theory while the latter is common in the study of primitive ideals. While I am still a beginner in terms of learning these ideas, it would help me greatly to know if there is a relationship between the two.

Answer 1

A possible way to describe the relation is as follows.

Let $G$ be a complex semi-simple group with the Lie algebra $\mathfrak{g}$, $B\subset G$ be a Borel subgroup, $X=G/B$ be the flag variety. Let $\mathcal{D}_X$ be the sheaf of rings of differential operators on $X$.

We have the $G$-equivariant moment map $\pi\colon T^*X\to \mathfrak{g}^*$. Let $M$ be a finitely generated $\mathfrak{g}$-module with trivial central character. Let $\mathcal{M}$ be the Beilinson-Bernstein localization of $M$; thus $\mathcal{M}$ is a sheaf of $\mathcal{D}_X$-modules. Then the characteristic variety $Ch(\mathcal{M})$ of $\mathcal{M}$ is a subvariety of $T^*X$, and the associated variety $V(M)$ of $M$ is a subvariety of $\mathfrak{g}^*$. The claim is that $$\pi(Ch(\mathcal{M}))=V(M).$$

This result can be found in Corollary in Section 1.9 of the paper Borho, W.; Brylinski, J.-L.; Differential operators on homogeneous spaces. III. Characteristic varieties of Harish-Chandra modules and of primitive ideals. Invent. Math. 80 (1985), no. 1, 1–68.

This paper contains also other relations between the two notions.