Split formal deformations preserve simple-block labels; the trace determinant bounds exceptional parameters

By [F1], $A\cong {\prod }_{\lambda \in \mathrm{\Lambda }}{\mathrm{Mat}}_{d_{\lambda }}(R)$. Reducing this isomorphism modulo $h$ and using $R\mathrm{/}hR\cong k$ from [F4] gives the stated product for the special fiber. Tensoring the matrix-unit basis with the field $K$ gives $K{\otimes }_{R}A\cong {\prod }_{\lambda \in \mathrm{\Lambda }}{\mathrm{Mat}}_{d_{\lambda }}(K)$. The lifted block identities $p_{\lambda }={\sum }_{a=1}^{d_{\lambda }}{E}_{\lambda ,aa}$ reduce to the specified block identities and extend to the generic fiber.

Let $L$ be any field and $d\ge 1$. If $v\in L^d$ is nonzero, choose a nonzero coordinate $v_j$; for each $i$, the matrix unit $E_{ij}$ sends $v$ to $v_j{e}_i$, and scalar multiplication by $v_j^{-1}$ produces $e_i$. Thus the natural left module $L^d$ is simple. Conversely, let $S$ be a simple left ${\mathrm{Mat}}_d(L)$-module. Since $1={\sum }_i{E}_{ii}$ and $S\mathrm{\ne }0$, choose $i$ and $s\in S$ with $E_{ii}s\mathrm{\ne }0$. Define the $L$-linear map $L^d\to S$ by $e_j\mapsto E_{ji}s$. It intertwines the action of each matrix unit because $E_{pq}{E}_{ji}={\delta }_{qj}{E}_{pi}$; since matrix units form an $L$-basis of the algebra, it is a module homomorphism. It is nonzero, so its image is all of $S$ by simplicity; its kernel is a submodule of the simple natural module and is not the whole module, so it is zero. Hence every simple module for one matrix factor is its natural module and is unique up to isomorphism.

In a finite product of matrix algebras, the central block identities are pairwise orthogonal and sum to $1$. On a nonzero simple module their images are submodules whose sum is direct and equals the module. At least one image is nonzero; if two were nonzero, each would be a proper nonzero submodule, contradicting simplicity. Therefore exactly one block identity has nonzero image, that identity acts as the identity, and all others act as zero. Step 1.2 classifies the simple module for the unique acting factor. The same lifted $p_{\lambda }$ labels the corresponding factors in the special and generic fibers, and each natural module has dimension $d_{\lambda }$.

For any finite Hecke specialization with $D_W\mathrm{\ne }0$, [F3] says its regular-trace Gram determinant is nonzero, so its regular-trace pairing is nondegenerate; [F2] then gives semisimplicity. If $D_W=0$, [F2] gives no conclusion. This proves the stated scope of the exceptional parameter set and does not infer labels or splitness for these general fibers.