Geometric criterion for subrepresentation and irreducibility of parabolic induction

Let $\mathfrak{m}$ and $\mathfrak{m}'$ be multisegments satisfying the condition referred to in the source as

, and let $\operatorname{SA}(\mathfrak{m},\mathfrak{m}')$, $\operatorname{SG}(\mathfrak{m},\mathfrak{m}')$, and $\operatorname{IG}(\mathfrak{m},\mathfrak{m}')$ denote the source-defined algebraic and geometric conditions. **Geometric parabolic-induction conjecture.** Under condition

, (1) $\operatorname{SA}(\mathfrak{m},\mathfrak{m}')$ and $\operatorname{SG}(\mathfrak{m},\mathfrak{m}')$ are equivalent; and (2) $Z(\mathfrak{m})\times Z(\mathfrak{m}')$ is irreducible if and only if $\operatorname{IG}(\mathfrak{m},\mathfrak{m}')$. The conjecture connects representation-theoretic conditions for subrepresentation and irreducibility with geometric density conditions on commuting-variety components. Its general status is unresolved in the supplied source.