Base change, freeness and specialization in the deformed family

Work first over the polynomial ring $S$, before localizing. Put $G=F\otimes E^{\ast }$. Tensor-Hom adjunction identifies the stated tensor-object Hom module with ${\mathrm{Hom}}_{\mathfrak{g}\otimes S}(M_S(\lambda ),G\otimes M_S(\lambda ))$. A map is determined by a singular vector in the finite free deformed weight space of relative weight $\lambda$; the simple raising equations give a finite polynomial matrix $A$. The PBW model [F1] supplies the domain and codomain bases.

Every complex point of $\mathrm{Spec}S$ gives kernel dimension $\dim G_0$ by [F2], so $A$ has the same rank $s$ at every such point. Every $(s+1)$-minor therefore vanishes as a polynomial; the $s$-minors generate the unit ideal, because otherwise a maximal ideal containing them would be an evaluation ideal by Over an algebraically closed field, every maximal ideal is an evaluation ideal, contradicting rank $s$ there. On an open where an $s$-minor is invertible, row and column elimination gives $\mathrm{diag}(I_s,0)$. Thus the kernel is finite projective of rank $\dim G_0$ and commutes with arbitrary base change. This argument is over $S$, so it applies to every localization $T$, even one whose closed points are not complex evaluation points.

For each finite-dimensional simple $L$, the adjoint $L$-multiplicity module of $U{\otimes }_{Z}S$ is $S{\otimes }_{\mathbb{C}}{\mathrm{Hom}}_{\mathfrak{g}}(L,\mathcal{H})$ by the harmonic decomposition [F3] and [F2]; it is finite free of rank $\dim L_0$. The action maps it to ${\mathrm{Hom}}_{\mathfrak{g}}(L,{\mathrm{End}}_S{M}_S(\lambda ))$, which by tensor-Hom adjunction is the kernel from step 2.1 for $G=L^{\ast }$. At every complex point this map is an isomorphism by Duflo–Joseph [F2]. Its cokernel is finite and vanishes by local Nakayama at every maximal ideal of $S$; the resulting surjection splits onto the projective target, and its kernel, a projective module of rank zero, vanishes on each local free chart. Hence it is an isomorphism. Localizing gives the same result over $T$. Finite-dimensional complete reducibility decomposes every finite adjoint submodule, so these isotypic isomorphisms prove (1).

For $G=F\otimes E^{\ast }$, tensor-Hom adjunction identifies the tensor-object Hom module with the adjoint $G^{\ast }$-multiplicity module of the endomorphism algebra in (1). Step 3.1 makes it free of rank $\dim G_0=\dim (E\otimes F^{\ast }{)}_0$; equality of the last dimensions follows by dualizing $G$. Step 2.1 proves its base-change assertion. Finite sums distribute over Hom, and for summands a Hom module is the image of the split idempotent $f\mapsto e_{N}f{e}_M$ on an ambient Hom module. Such images are finite projective and commute with every base change. This proves (2) and (3) without claiming that arbitrary projective modules over arbitrary localizations are free.

Over the local ring $R$ at the origin, the deformed weight spaces of objects of ${\mathcal{D}}_R(\lambda )$ are finite projective, hence finite free: lift a residue basis, use Nakayama for a surjection from a free module, split it by projectivity, and apply Nakayama to the complementary kernel. A map whose residue fiber is an isomorphism has invertible determinant on each such weight space, so it has a weightwise inverse. The inverses commute with the $\mathfrak{g}$-action because the original map does. This proves (4). AC is inherited from the named suppliers.