Indecomposable projectives correspond to indecomposable Soergel modules

At a height-one root prime $p$ with $T=R_p$, a two-label component has the upper projective $P_a={\mathrm{\Delta }}_T(a)$ and the pair projective $P_b$; $\mathrm{Hom}(P_a,P_b)$ and $\mathrm{Hom}(P_b,P_a)$ are free of rank one, the canonical inclusion $i$ generates the first, and choosing $j$ in the second lifting its nonzero residual map gives a nonzero nilpotent $ij$ on the residue while $ji=h$ vanishes modulo $p$ by the nonsplitness of the residual standard extension; $1$ and $ij$ are residually independent and form a $T$-basis of the order $\mathrm{End}(P_b)=Z=\{(u_a,u_b):u_a\equiv u_b<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>\tau ({\alpha }^{\vee })\}$ of [F3], in generic tuple coordinates $ij=(h,0)$ and the order has basis $(1,1),(\alpha ,0)$ with $\alpha =\tau ({\alpha }^{\vee })$, so $h$ is a unit times $\alpha$ and rescaling $j$ makes $ji=\alpha$; then $V_T=\mathrm{Hom}(P_b,-)$ sends $P_b$ to $Z$ and $P_a$ to the rank-one upper branch $T$, with ${\mathrm{Hom}}_Z(Z,Z)=Z$, ${\mathrm{Hom}}_Z(T,T)=T$, ${\mathrm{Hom}}_Z(Z,T)=T$ generated by the projection and ${\mathrm{Hom}}_Z(T,Z)=T(\alpha ,0)$, and $V(j),V(i)$ are these generators.

Here is the actual deformed wall map missing from the ordinary formula [F2]. Write $A=R{\otimes }_{R^W}R$ and $A^s=R^s{\otimes }_{R^W}R$, with the common right parameter. For the wall target, adjunction and $T_{\mathrm{o}\mathrm{u}\mathrm{t}}{P}_R^{\nu }=P_R^{\lambda }$ give $V_R^{\nu }{T}_{\mathrm{o}\mathrm{n}}\cong {\mathrm{Res}}_{A^s}^A{V}_R^{\lambda }$, including the $A^s$ action by the central inclusion in [F3]. The closed fiber of $T_{\mathrm{o}\mathrm{n}}{P}_R^{\lambda }$ is two copies of the singular antidominant projective: its projective-injective summands have antidominant socle, and its Verma flag has every singular standard twice. Lifting both maps of this decomposition by [F3] gives $T_{\mathrm{o}\mathrm{n}}{P}_R^{\lambda }\in \mathrm{add}(P_R^{\nu })$. Its canonical evaluation by $V_R^{\nu }$ equals $A$ as an $(A^s,A)$-bimodule, by adjunction with $T_{\mathrm{o}\mathrm{u}\mathrm{t}}$; thus $T_{\mathrm{o}\mathrm{n}}{P}_R^{\lambda }=P_R^{\nu }{\otimes }_{A^s}A$ by the evaluation isomorphism on $\mathrm{add}(P_R^{\nu })$. The opposite adjunction now gives $V_R^{\lambda }{T}_{\mathrm{o}\mathrm{u}\mathrm{t}}N\cong {\mathrm{Hom}}_{A^s}(A,V_R^{\nu }N)$. Choose a root coordinate $\alpha$ with $s\alpha =-\alpha$. Since $R=R^s\oplus \alpha R^s$, the coefficient-of-$\alpha$ map $\mathrm{\ell }:R\to R^s$ has pairing matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$ on $1,\alpha$, hence identifies $R$ with its $R^s$-dual. Extending the right parameter identifies ${\mathrm{Hom}}_{A^s}(A,-)$ with $A{\otimes }_{A^s}-$, including the left $A$ action. Consequently $V_R{\theta }_s\cong A{\otimes }_{A^s}{V}_R\cong R{\otimes }_{R^s}{V}_R$ by actual action-compatible maps. This argument uses adjunction and relative Frobenius duality, not an isomorphism of residues.

For a polynomial Bott-Samelson bimodule $B$, $M=B{\otimes }_{S}R$ is finite free over the right $R$. A left denominator nonzero at the origin acts invertibly: modulo the right maximal ideal its constant plus positive-degree part is invertible, since positive left invariants annihilate the finite-dimensional coinvariant reduction and positive-degree operators there are nilpotent; Nakayama makes its matrix invertible over $R$. Both left and right actions therefore extend to $R$. Weyl invariants balance across every tensor factor, so these actions factor through $A$. Iterating the map in step 1.2 identifies the localized polynomial tensor word with $V_R$ of the deformed wall word, with empty word the identity graph $R=V_R{\mathrm{\Delta }}_R(\lambda )$. All identifications preserve the generic tuple branches and the right parameter. The fixed average coordinate in the type-A permutation model is split off before this construction.

Let $Q,Q^{\mathrm{\prime }}$ be deformed block projectives. Their canonical comparison under $V_R$ is an isomorphism at every height-one prime. At nonroot primes, the distinct simple Verma components give one-label Morita equivalences; at root primes, each projective decomposes into the two elementary projectives of step 1.1 by lifting its actual residue decomposition with the projective Hom base-change maps in [F3], so that step verifies every pair of summands. The source Hom module is finite free over $R$ by [F3]. The target ${\mathrm{Hom}}_A(V_{R}Q,V_R{Q}^{\mathrm{\prime }})$ need not be assumed free in advance: it is the submodule of finite free ${\mathrm{Hom}}_R(V_{R}Q,V_R{Q}^{\mathrm{\prime }})$ satisfying the finitely many linear commutation equations for a finite $R$-basis of $A$. Inside the generic Hom space it is the intersection of its height-one localizations, since a rational matrix belongs to the free ambient matrix lattice precisely when all its height-one localizations do, and the $A$-linearity equations already hold generically. The coefficientwise UFD intersection argument of the central-action lemma gives the same intersection property for the source Hom lattice. Their height-one equalities therefore prove the comparison is an isomorphism over $R$; in particular the target is now free. This is full faithfulness on deformed projectives.

For polynomial Bott-Samelson modules $M,M^{\mathrm{\prime }}$ the module ${\mathrm{Hom}}_A(M,M^{\mathrm{\prime }})$ is the localization of the polynomial Hom module, and the composite ${\mathrm{Hom}}_A(M,M^{\mathrm{\prime }}){\otimes }_R\mathbb{C}\cong {\mathrm{Hom}}_{{\mathcal{O}}_R}(Q_R,Q_R^{\mathrm{\prime }}){\otimes }_R\mathbb{C}\cong {\mathrm{Hom}}_{\mathcal{O}}(Q,Q^{\mathrm{\prime }})\cong {\mathrm{Hom}}_C(VQ,V{Q}^{\mathrm{\prime }})$ is the natural reduction map by step 3.1, projective Hom base change and the ordinary Struktursatz, all maps being the canonical composition maps; hence reduction-of-Hom holds for Bott-Samelson modules and, by taking summands, for all polynomial Soergel bimodules, the degree components giving the graded statement.

Absolute complex extension follows by the actual same equations as real extension, not by a real-only assertion. For the rational $B_w$, write $E_{\mathbb{Q}}^0$ for its finite-dimensional local degree-zero endomorphism algebra and $J_{\mathbb{Q}}$ for its nilpotent radical; [F4] gives $E_{\mathbb{Q}}^0\mathrm{/}{J}_{\mathbb{Q}}=\mathbb{Q}$. Fixed-degree maps are kernels of finite rational linear commutation equations on homogeneous generator images. Tensoring these equations with the flat field $\mathbb{C}$ identifies $E_{\mathbb{C}}^0=E_{\mathbb{Q}}^0{\otimes }_{\mathbb{Q}}\mathbb{C}$. Its ideal $J_{\mathbb{Q}}\otimes \mathbb{C}$ is nilpotent and its quotient is $\mathbb{C}$, so every element with nonzero scalar residue is invertible by a finite geometric series; hence this is a local algebra with residue $\mathbb{C}$. The same flat equations preserve its top graph and degree. Extending each rational word decomposition gives a complex decomposition into these absolutely indecomposable labels; finite Krull--Schmidt uniqueness then classifies every summand of every complex word. Thus the normalized polynomial $B_w$ has the required local degree-zero algebra over $\mathbb{C}$. Step 4.1 makes the degree-zero algebra of ${\bar{B}}_w$ its quotient, hence local. Here is the needed passage to ungraded endomorphisms. For the finite-dimensional graded algebra $E=\mathrm{End}({\bar{B}}_w)$, grading scalings preserve its Jacobson radical $J$; interpolation in finitely many distinct scaling parameters extracts every homogeneous component, so $J$ is graded. The semisimple quotient $E\mathrm{/}J$ is a product of matrix algebras by [F6]. Its grading derivation preserves each factor, since a derivation kills central idempotents. Every derivation $D$ of ${\mathrm{Mat}}_n(\mathbb{C})$ is inner: with matrix units, $H={\sum }_{i}D(e_{i1})e_{1i}$ satisfies $D(a)=Ha-aH$, by applying the product rule to $e_{ij}{e}_{k1}$ and summing in $k$. The grading derivation is diagonalizable; writing $H=H_s+H_n$ shows $\mathrm{ad}{H}_n=0$, because it is the commuting nilpotent part of $\mathrm{ad}H$. A scalar nilpotent matrix is zero, so $H$ is semisimple up to an irrelevant scalar. Its degree-zero centralizer is consequently a product of matrix algebras on its eigenspaces. Thus $(E\mathrm{/}J{)}_0$ is semisimple. It is a quotient of the local $E_0$, so it equals $\mathbb{C}$; a product of such centralizers can have dimension one only if there is a single matrix factor of size one. Hence $E\mathrm{/}J=\mathbb{C}$, and $E$ is local. This proves that ${\bar{B}}_w$ remains indecomposable after forgetting grading; it supplies the graded-algebra argument rather than attributing innerness to Wedderburn-Artin.

The normalized top summand is self-dual without assuming an unproved indecomposable-form assertion. In the unshifted tensor ring of a length-$k$ word, each added reflection factor is a quadratic finite free extension with basis $1,{\alpha }_s$ and relation ${\alpha }_s^2\in S^s$. The coefficient-of-${\alpha }_s$ functional has perfect pairing matrix $\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)$; composing these functionals along the tensor tower gives a perfect right-$S$ pairing on its rank-$2^k$ basis. It is left-invariant by commutativity of the tensor ring. The total trace has degree $-2k$, so shifting the word by $(k)$ gives a degree-zero isomorphism from the Bott-Samelson bimodule to its invariant right dual $DB={\mathrm{Hom}}_S(B,S)$. For a reduced word of $w$, $B_w$ is the unique summand with nonzero top graph $w$, of generic right rank one there; all other summands have strictly lower graph support by [F4]. Duality preserves generic graph branches: a left scalar on a rank-one branch acts by the same scalar on its right dual. Therefore $D{B}_w$ is an indecomposable summand of the very same self-dual word, has its $w$ branch, and must be the identical unique top summand, with no shift. Thus $B_w\cong D{B}_w$. Finite right projectivity makes duality commute with right origin reduction, so ${\bar{B}}_w$ has a graded self-duality centered at zero. Any two finite graded lifts with the same indecomposable underlying $C$-module differ by a shift: homogeneous components of an ungraded isomorphism and its inverse give $1={\sum }_d{g}_{-d}{f}_d$ in the local degree-zero endomorphism algebra; one summand is a unit, so $f_d$ is an isomorphism after a shift (its source and target have the same finite dimension). A nonzero shift cannot preserve centered self-duality, by symmetry of the extreme degrees. This proves unique centered normalization; rank one has degrees $-1,1$, and an uncentered shift gives $0,2$.

Induction on Bruhat length gives the label dictionary: for a reduced word $w=s_1\cdots s_k$, $P(w\cdot \lambda )$ is the unique new top-label summand of ${\theta }_{s_k}\cdots {\theta }_{s_1}\mathrm{\Delta }(\lambda )$ by [F2], and applying $V$ and the iterated wall-crossing formula gives the new summand of the reduced left tensor word $\overline{B_{s_k}\cdots B_{s_1}}$, whose top graph label is the inverse $w^{-1}$, while every proper subexpression contributes already-matched smaller labels; full faithfulness of step 3.1 for the deformed side and the indecomposability of step 5.1 match the two new summands uniquely, so $VP(w\cdot \lambda )\cong {\bar{B}}_{w^{-1}}$, and distinct labels remain non-isomorphic because $V$ is fully faithful on projectives.

The type-A equivalence and normalized dictionary follow from steps 4.1 through 6.2. For a semisimple product of type-A factors, reflection words reorder into the independent factor words and their reductions are tensor products of the factor reductions. Their endomorphism algebra is the tensor product of the finite factor endomorphism algebras: in finite bases, the commutation equations for the two independent coefficient algebras give this tensor-Hom identity. Each factor algebra is local with residue $\mathbb{C}$ by step 5.1; the sum of its tensor-extended radicals is nilpotent and its quotient is $\mathbb{C}$, so the product reduction is indecomposable. The labels, inverse convention and centered self-duality therefore combine factorwise; a rank-zero factor contributes the unit. The ungraded equivalence for every complex semisimple type needs no unprovided diagrammatic classification: define its Weyl-group Soergel-module category as the additive category of summands of polynomial Bott-Samelson right reductions. For every reflection realization, $S=S^s\oplus {\alpha }_s{S}^s$, and Weyl invariants balance across the tensor factors; the elementary quotient computation gives $B_s{\otimes }_{S}C=(C{\otimes }_{C^s}C)(1)$ exactly as in the wall-crossing theorem. Thus forgetting shifts identifies all right-reduced words with the ordinary $V$-images of wall words starting at $V\mathrm{\Delta }(\lambda )=\mathbb{C}$. Ordinary full faithfulness [F1] lifts every idempotent between these images uniquely to an idempotent of their projective words. Since all projectives occur as summands of such words by [F2], this is a fully faithful and essentially surjective equivalence in every type. Its indecomposable labels can be read from the unique new projective summands themselves. The additional type-A polynomial/diagrammatic label and centered self-dual normalization are exactly those established in steps 5.1 through 6.2; no type-A classification is substituted for a general-type classification.