Several-variable SL(2)-orbit approximation for polarized nilpotent orbits

The one-variable argument needed below also applies to a polarized real variation: its logarithmic frame, curvature/cylinder extension, real nilpotent weight grading, explicit norm models and finite-dimensional polarization recognition use only the real flat bundle and Q, never lattice integrality. Repeating those displayed constructions over R proves exactly the same extension/LMHS assertion for the real monodromy combinations used here; rationality is used only to label a resulting MHS rational when a rational local system is present. Put $N(c)=\sum c_i{N}_i$ for positive $c_i$. It defines the one-variable orbit z↦exp(zN(c))F on a sufficiently high half-plane. The same construction on a face fixes the complementary coordinates in its admissible interior. (Uniform comparison of positive rays.) Fix positive c,d and y large. Join iyc to iyd by changing coordinates one at a time, keeping each imaginary coordinate between y $min(c_i,d_i)$ and y $max(c_i,d_i)$. For the i-th coordinate, with all others held fixed, this is a horizontal one-variable period map on a translated upper half-plane. The Schwarz constant depends only on the Hodge numbers, not on the held parameters. After increasing the common lower bound Y, its Higgs operator satisfies $\mathrm{\mid }\mathrm{\mid }{A}_i\mathrm{\mid }\mathrm{\mid }\le C\mathrm{/}(\mathrm{Im}{z}_i)$. A fixed translation of the half-plane changes this bound by at most a factor two here.

Along a path varying only $z_i$, write $\mathrm{\nabla }=D+{\sum }_i(A_{i}d{z}_i+A_i^{†}d{\bar{z}}_i)$ in the Hodge splitting; $D$ is metric-compatible. For a flat vector $v$, the identity $D_{z_i}v=-A_{i}v$ and its conjugate give ${\mathrm{\partial }}_{z_i}\log h(v,v)=-2h(A_{i}v,v)\mathrm{/}h(v,v)$. Cauchy–Schwarz therefore yields $\mathrm{\mid }d\log h(v,v)\mathrm{\mid }\le 4\mathrm{\parallel }{A}_i\mathrm{\parallel }\mathrm{\mid }d{z}_i\mathrm{\mid }$.

By step 1.1 and the horizontal Schwarz estimate, $\mathrm{\parallel }{A}_i\mathrm{\parallel }\le C\mathrm{/}(\mathrm{Im}{z}_i)$ once all fixed coordinates lie above a common sufficiently large bound. Join $iyc$ to $iyd$, for positive vectors $c,d$, by changing the imaginary coordinates one at a time. Integration of step 2.1 along the $i$th segment gives

$\mid \log \frac{h_{iyc}(v,v)}{h_{iyd}(v,v)}\mid \le C_0{\sum }_i\mid \log (c_i\mathrm{/}{d}_i)\mid$.

Exponentiating step 4.1 yields

$e^{-C_0{\sum }_i\mathrm{\mid }\log (c_i\mathrm{/}{d}_i)\mathrm{\mid }}{h}_{iyc}(v,v)\le h_{iyd}(v,v)\le e^{C_0{\sum }_i\mathrm{\mid }\log (c_i\mathrm{/}{d}_i)\mathrm{\mid }}{h}_{iyc}(v,v)$. These constants are independent of $y$ and $v$, and uniform when $c,d$ range over a fixed compact subset of the positive orthant. This compares the actual flat-vector metrics, not only the compact-dual period points.

The one-variable sharp growth theorem, applied to z↦exp(zN(c))F, says that the exponent of $h_{iyc}(v,v)$ is exactly the least l with $v\in W_l(N(c))$. Its proof uses only a nilpotent real marking action, the polarized Hodge data and the Schwarz estimate; no integrality of $c_i$ is needed. The two-sided comparison therefore implies

W(N(c))=W(N(d))

for every positive c,d. Denote this common zero-centred filtration by W(C). The same conclusion on any face uses its own positive coefficients. No limit $c_i\to 0$ is asserted to retain the interior filtration.

(A nilpotent horizontal generator has no uncontrolled compact part.) At a polarized Hodge filtration, decompose a real Q-skew endomorphism B satisfying $B{F}^p\subset F^{p-1}$ by its Hodge degree. It has only degrees −1,0,1: transversality excludes degrees $\le -2$ and reality gives the opposite exclusion. Write

$B=A+K+A\ast$,

where A has degree −1, K degree zero and K* =−K; the upper-degree component is A* for the positive Hodge metric. These adjoint signs follow by applying Q-skewness in each pair of Hodge blocks and using the alternating signs of the flat Hermitian polarization.

If B is nilpotent, tr(B²)=0. Degree considerations and the adjoint identities give

0=tr(K²)+2tr(AA*)=−||K||²_HS+2||A||²_HS.

Thus $\mathrm{\parallel }K{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}=\sqrt{2}\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}$ and $\mathrm{\parallel }B{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}\le (2+\sqrt{2})\mathrm{\parallel }A{\mathrm{\parallel }}_{\mathrm{H}\mathrm{S}}$. In particular nilpotency rules out a large unitary degree-zero part invisible to the horizontal differential. For $B=N_i$ at Φ(z), A is precisely the horizontal Higgs coefficient in the i-th direction. Consequently

$\mathrm{\mid }\mathrm{\mid }{N}_i\mathrm{\mid }{\mathrm{\mid }}_{h_z}\le C\mathrm{/}(\mathrm{Im}{z}_i)$.

This is the needed bound on the whole constant endomorphism $N_i$, not merely on its period-map tangent class.

(Every generator lowers the common interior weight by two.) Choose a real Q-skew grading H of W(C) with $[H,N(c)]=-2N(c)$, as constructed in the one-variable proof. Along the c-ray, that proof gives uniform equivalence of $h_{iyc}$ with the explicit weight metric $y^H$. Decompose the fixed operator $N_i$ into ad H eigencomponents $(N_i{)}_a$. the whole-generator trace estimate above implies

$\mathrm{\mid }\mathrm{\mid }{y}^{H\mathrm{/}2}{N}_i{y}^{-H\mathrm{/}2}\mathrm{\mid }{\mathrm{\mid }}_0\le C\mathrm{/}y$.

Projection to the finite eigenspaces gives $y^{a\mathrm{/}2}\mathrm{\mid }\mathrm{\mid }(N_i{)}_a\mathrm{\mid }{\mathrm{\mid }}_0\le C_a\mathrm{/}y$. Let $y\to \mathrm{\infty }$. Every component with $a>-2$ vanishes. Therefore

$N_i{W}_l(C)\subset W_{l-2}(C)$.

The one-variable theorem for each c shows that (W(C)[−k],F) is an LMHS polarized by N(c). The given transversality and the displayed weight lowering show that each $N_i$ is a morphism from this MHS to its Tate twist by −1, hence has type (−1,−1). Every positive combination polarizes the same MHS, with its own primitive Lefschetz forms.

(Relative face goal and simultaneous real split replacement.) Fix nonempty faces I⊂J, let T be a positive combination in I and S a positive combination in J\I, and put $U=T+S$. Write $W=W(I)$, $M=W(J)$. We prove $M=M(S,W)$: S lowers M by two and $S^a$ identifies M-degrees w+a,w−a on $G{r}_w^W$. Fix the interior face J and first work with no other variables. By the cone root, (M[−k],F), $M=W(J)$, is polarized by every positive combination U of the $N_i$ in J, and every $N_i$ is a (−1,−1) morphism of that MHS. The finite intrinsic formula for W(I) uses only sums, intersections, kernels and images of powers of T. These are MHS subobjects: view $T^b$ as a morphism $H(b)\to H$ for its image, and $H\to H(-b)$ for its kernel. The strictness/abelian-category theorem for MHS therefore shows that each $W(I{)}_w$ is a sub-MHS of (M[−k],F).

Let Z₀ act by p+q−k on the Deligne summand $I^{p,q}$ of this MHS. It is a complex grading of M, preserves F, and $[Z_0,N_i]=-2N_i$. The last identity is functoriality of the explicit Deligne splitting applied to $N_i:H\to H(-1)$; the shift of the splitting under a Tate twist follows immediately from its displayed filtration formula.

We may make Z₀ Q-skew while preserving all these properties. Its negative Q-adjoint −Z₀† is another grading of M, since M is Q-self-dual. It preserves F by Q-isotropy, and has the same commutators with the $N_i$. The finite negative-weight conjugation construction from the one-variable proof gives a unique exp B carrying Z₀ to −Z₀†, where B strictly lowers M and commutes with every $N_i$. Its weight-by-weight construction also preserves F, because both gradings preserve F and the ad Z₀ weight projections preserve F. Adjoint uniqueness gives $B^{†}=B$. Replacing Z₀ by exp(B/2)Z₀exp(−B/2) therefore gives a complex Q-skew grading Z with the same F and $N_i$ properties.

Now compare Z with its complex conjugate. The same unique negative-weight conjugation gives $\bar{Z}=\exp CZ\exp (-C)$, with C strictly lowering M and commuting with every $N_i$. Since both gradings are Q-skew, uniqueness gives $C^{†}=-C$; conjugating the equation and using uniqueness gives $\bar{C}=-C$. Put

$H=\exp (C\mathrm{/}2)Z\exp (-C\mathrm{/}2)$, $F_s=\exp (C\mathrm{/}2)F$.

Then H is real and Q-skew, grades M, preserves $F_s$, and $[H,N_i]=-2N_i$. The pair $(M[-k],F_s)$ is R-split: on every H eigenspace its filtration is the pure real Hodge filtration on the corresponding $G{r}^M$, unchanged because exp(C/2) acts trivially there. Q and all primitive forms are unchanged on these graded pieces. Consequently every positive U still polarizes this R-split MHS.

All face weight filtrations are preserved in this construction. Initially they are preserved by Z₀ as MHS subobjects. Alternatively, once a grading has $[Z,N_i]=-2N_i$, it preserves the intrinsic kernel/image formula for each face weight filtration. The correcting B,C commute with every $N_i$ and thus preserve those same filtrations. Hence each face filtration remains a sub-MHS after the split replacement: it is H-stable, and on $G{r}^M$ its induced pure substructures are unchanged.

For completeness, a split polarized MHS gives $\exp (iU)F_s\in D$ for every positive polarizing U. To see this directly, its real grading H and U form an sl₂ representation by the primitive string formulas. The primitive spaces carry the polarized pure Hodge structures specified by the primitive forms; on each string exp(iU) is the explicit rank-two symmetric-power model of the completed one-variable proof. Tensoring with the primitive pure structure gives a polarized pure structure, and taking their finite orthogonal direct sum proves membership. Thus $\exp (zT+wU)F_s$ lies in D whenever Im z,Im $w>0$ and $U=T+S$: its imaginary generator is (Im z+Im w)T+(Im w)S, an interior positive combination.

(A polarized one-variable variation on every face-graded piece.) For each fixed w in the upper half-plane apply the completed one-variable theorem to

$z\mapsto \exp (zT+wU)F_s$.

Its untwisted limiting filtration is exactly $\exp (wU)F_s$, and its weight filtration is $W=W(I)$. Thus $(W[-k],\exp (wU)F_s)$ is polarized by T. U preserves W since it commutes with T. On every W-graded piece, and on its T-primitive subspaces, $\exp (wU)F_s$ is therefore a polarized pure Hodge filtration. It is holomorphic in w and U-transverse. The polarization is the fixed real form $Q_l(u,v)=Q(u,T^{l}v)$ on a highest primitive piece $P_l\subset G{r}_l^W$. U preserves that primitive piece, commutes with T and is $Q_l$-skew.

H preserves W, hence acts on all these graded spaces. It preserves their induced $F_s$, preserves $P_l$, and satisfies $[H,U]=-2U$. On $P_l$ the operator $H_l=H-l\cdot 1$ is $Q_l$-skew. Indeed

$Q_l(Hu,v)+Q_l(u,Hv)=2l{Q}_l(u,v)$,

by Q-skewness of H and $[H,T^l]=-2l{T}^l$. Subtracting l on each side gives the assertion. $H_l$ is real and semisimple with integral eigenvalues.

Because $H_l$ preserves the induced $F_s$ and rescales U, the real $Q_l$-isometry $y^{H_l\mathrm{/}2}$ carries $\exp (iyU)F_s$ to $\exp (iU)F_s$. Hence the actual Hodge metric on $P_l$ satisfies the exact formula

$h_y(v,v)=h_1(y^{H_l\mathrm{/}2}v,y^{H_l\mathrm{/}2}v)$.

In a fixed eigenbasis this is uniformly equivalent to ${\sum }_a{y}^a\mathrm{\mid }\mathrm{\mid }{v}_a\mathrm{\mid }{\mathrm{\mid }}^2$. The completed sharp one-variable growth theorem, applied to this pure variation on $P_l$, says that the same growth exponent identifies $W(U\mathrm{\mid }{P}_l)$. Therefore that monodromy filtration is exactly the filtration by $H_l$ eigenvalues. In particular $U^a$ identifies $H_l$ eigenvalue a and −a spaces. This conclusion is proved from the metric formula; a relative hard-Lefschetz statement was not assumed.

The primitive decomposition of $G{r}^W$ V is a direct sum of strings $T^j{P}_l$, at W-degree l−2j. U preserves $P_l$ and commutes with T. H has eigenvalues shifted down by 2j on $T^j{P}_l$. Thus the corresponding H−(l−2j) eigenvalues and U-string structures are the same as on $P_l$. We have proved on every W-degree w that $U^a$ identifies H-eigenvalue w+a and w−a. Since H splits M, these are precisely the required M-graded isomorphisms on $G{r}_w^W$.

On the whole space U lowers M by two. Thus M is the relative monodromy filtration of U with respect to W. On a fixed $G{r}_w^W$ the induced action of T is zero, because T lowers W by two; hence U and $S=U-T$ induce the same action there. S also lowers M by two. Consequently the same M is the relative monodromy filtration of S with respect to W, proving the stated identity.

(Restriction to earlier face steps.) For K⊂I, every $N_i$ with $i\in I$ is a (−1,−1) morphism of the T-limiting MHS just constructed, by the simultaneous-lowering cone root applied to face I. In particular the intrinsic kernel/image construction of $W(K{)}_b$ gives sub-MHSs of $(W(I)[-k],\exp (wU)F_s)$ for every w in the upper half-plane. Thus $G{r}_a^{W(I)}(W(K{)}_b)$ is a pure Hodge subvariation of $G{r}_a^{W(I)}(V)$, stable under U and H.

The entire graded space $G{r}_a^{W(I)}(V)$ has a fixed polarization preserved by U and by H−a. Use its primitive string decomposition at degree $a=l-2j$ and transfer $Q_l$ from $P_l$ to $T^j{P}_l$ by the isomorphism $T^j$, then take the orthogonal direct sum. This is a polarization in the repository convention $h(u,v)=Q(Cu,\bar{v})$: shifting both Hodge indices down by j leaves the Weil operator C unchanged, and the new weight k+a differs from k+l by the even integer 2j, so its parity also agrees. Both U and H preserve each primitive string summand. Under $T^j$ the operator H−a pulls back to $H-a-2j=H-l$; the displayed adjoint calculation thus gives skewness for this transferred $Q_l$. Hence the constructed whole-graded polarization has the asserted invariances.

Restrict its positive Hodge metric to the pure Hodge subvariation $G{r}_a^{W(I)}(W(K{)}_b)$. The exact rescaling identity and the sharp one-variable growth theorem apply there as well. They prove the same relative-Lefschetz isomorphisms on that subspace. Therefore $W(J){\mathrm{\mid }}_{W(K{)}_b}$ is the relative monodromy filtration of $S{\mathrm{\mid }}_{W(K{)}_b}$ with respect to $W(I){\mathrm{\mid }}_{W(K{)}_b}$. This supplies the earlier-face restriction clause used by a Deligne system; it is not inferred merely from existence on V.

For J smaller than the full original index set, fix the remaining imaginary coordinates sufficiently far inside the original period domain and apply the same construction to the J-variable nilpotent orbit with that fixed limiting filtration. The monodromy filtrations and their relative identities are properties of the operators alone. This proves the assertion for every nested pair of nonempty faces. For the initial pure filtration, the first relative identity is the definition of W(I)[−k].

For the next finite construction, let M be the relative monodromy filtration of N with respect to W, and let Z grade M, preserve W and satisfy [Z,N]=−2N. We construct the unique grading Y of W commuting with Z for which the negative Y-degree components of N are highest-primitive under the leading sl₂ triple. The preceding face argument supplies this exact hypothesis when the construction is applied below.

(Initial grading and the leading sl₂ representation.) Since Z preserves W and is semisimple, split W separately inside its finitely many Z eigenspaces; this gives some grading Y₀ of W commuting with Z. Put $H_0=Z-Y_0$. On the Y₀-weight-w summand, the degree-zero component $L=N_0$ of N is the operator induced on $G{r}_w^W$. The relative-monodromy isomorphisms say that $L^r$ identifies the H₀-weight-r and H₀-weight-(-r) summands. The finite primitive/string construction therefore gives the unique raising operator R with $[H_0,R]=2R$ and $[R,L]=H_0$. On a highest string e,L $e,\dots ,L^m$ e it is R $L^{a}e=a(m-a+1)L^{a-1}e$. Thus (L,H₀,R) is an actual sl₂ representation, not an assumed relative splitting theorem.

Use the adjoint representation on End(V). If End(V)_s denotes H₀-weight s, its elementary string decomposition gives, for j≥1,

End(V)_{j−2} = [L,End(V)j] ⊕ (ker ad R ∩End(V){j−2}).

Moreover ad L is injective on $End(V{)}_j$. To verify both assertions, check each irreducible string: at positive weight j the lowering map is injective; its target at j−2 is missed exactly by a highest vector of weight j−2. When $j=1$ the target has negative weight −1 and no highest vector occurs, so the lowering map is bijective. This proves the decomposition without a semisimple-group theorem.

The same decomposition holds inside every ad Y₀ eigenspace, since L,H₀,R commute with Y₀. In particular, inside ad Y₀ weight −j it identifies the ad Z weight −2 target with the image under ad L of the ad Z weight zero source, plus its highest-primitive complement.

(Eliminate successively, with no infinite series.) Hold Y₀,Z fixed as coordinate gradings and conjugate N only. Since N preserves W, its ad Y₀ weights are nonpositive. Its weight-zero term is L. Every negative term $N_{-j}$ has ad Z weight −2, hence ad H₀ weight j−2.

Suppose weights −1,…,−j+1 have already been normalized to lie in ker ad R. Decompose the weight −j term uniquely as

$N_{-j}=[L,U_{-j}]+P_{-j}$,

where $U_{-j}$ has ad Y₀ weight −j and ad Z weight zero, and $P_{-j}$ lies in ker ad R. Conjugate the current N by $\exp (U_{-j})$. Its term of weight −j changes by $[U_{-j},L]=-[L,U_{-j}]$, leaving exactly $P_{-j}$; all other new commutators have more negative W degree. The previously normalized terms stay fixed. The conjugation commutes with Z because $[Z,U_{-j}]=0$.

Continue through the finitely many possible negative ad Y₀ weights. All $U_{-j}$ strictly lower W, so their exponentials and products are finite nilpotent polynomials. Let u be the product, so $uN{u}^{-1}$ is normalized relative to Y₀. Set $Y=u^{-1}{Y}_{0}u$. It splits W, commutes with Z, and transporting the leading triple and the primitive conditions gives exactly the claimed normalization of N relative to Y. This is an actual finite algorithm, requiring no convergence estimate.

(Uniqueness and naturality.) Any two gradings of W commuting with Z differ by a unique unipotent conjugation lowering W and commuting with Z. Indeed, on each Z eigenspace send the first splitting's lift of every $G{r}_w^W$ vector to the second splitting's lift; this determines the unipotent map, and its finite logarithm strictly lowers W.

Suppose two normalized gradings differ. In coordinates for the first, let $U_{-j}$ be the highest nonzero W-degree component of the logarithm of the conjugation. Its change to the $N_{-j}$ term lies in im ad L, whereas both normalized terms lie in ker ad R. The direct-sum highest-primitive direct-sum decomposition above forces $[L,U_{-j}]=0$. Injectivity of ad L on the positive H₀ weight j then forces $U_{-j}=0$, a contradiction. This proves uniqueness, and hence independence of the initial splitting Y₀.

Replacing N by λN replaces L by λL and R by λ^{-1}R and leaves the normalization condition unchanged; uniqueness gives the same Y. Direct sums of the unique gradings satisfy the same normalization and are therefore the unique grading of the direct sum.

For a map f:V→V′ preserving W and commuting with N and Z, consider the shear (v,v′)↦(v,v′+f(v)) on V⊕V′. It is an automorphism preserving W,N,Z. Uniqueness makes it commute with the canonical direct-sum grading, which says exactly fY=Y′f. Thus naturality holds even when f is not invertible.

Duality follows from the same uniqueness: the negative transpose of Y grades the dual W, the negative transpose of Z grades the dual M, and transposing the triple and primitive normalization gives the corresponding dual construction. Consequently a preserved perfect pairing identifies Y with its negative adjoint whenever W,M are self-dual in the zero-centred convention; in a shifted convention subtract the common central scalar first.

(Construct the commuting gradings.) Order the generators as $N_1,\dots ,N_d$. Set $W_j=W(N_1+\cdots +N_j)$, and let $W_0$ be the pure filtration concentrated in degree zero. The preceding steps establish all successive relative identities and earlier-face restriction clauses. Use the simultaneous split replacement of the relative-face proof for the full cone. It gives a real Q-skew grading $H_d$ of $W_d$ and an R-split filtration $F_s$, with $H_d{F}_s=F_s$, $[H_d,N_i]=-2N_i$, and every $W_j$ a sub-MHS. Moreover

$F=\exp (\eta )F_s$,

where η is complex Q-skew, commutes with every $N_i$, preserves every $W_j$, and strictly lowers $W_d$. Explicitly $\eta =-C\mathrm{/}2$ in that construction; its first Q-skew correction preserved F.

Recursively define

$H_{j-1}=Y(N_j,H_j;W_{j-1},W_j)$,

using the actual finite normalized relative-grading algorithm. Its hypotheses hold by the preceding relative-face proof. We prove, downwards in j, the properties needed to continue and the resulting commutations.

(a) $H_{j-1}$ preserves each earlier filtration $W_a$. Write $W_{a,b}:=(W_a{)}_b$ for its b-th subspace, where $W_a=W(N_1+\cdots +N_a)$. For each $W_{a,b}$ with $a<j-1$, the relative-filtration identity restricts to that subspace, $H_j$ preserves it, and $N_j$ preserves it. Apply naturality of the normalized grading to the inclusion of this restricted relative system. It gives preservation of $W_{a,b}$, hence of $W_a$. Later $W_a$ are also preserved, since $H_{j-1}$ commutes with their grading operators, as shown in (c).

(b) For $i\le j-1$, $[H_{j-1},N_i]=-2N_i$. Regard $N_i$ as a map from this relative system to its shift by two: it lowers both $W_{j-1}$ and $W_j$ by two, commutes with $N_j$, and has $H_j$-degree −2. The shifted target has filtrations $W{\mathrm{\prime }}_a=W_{a-2}$ and grading $H_j+2$, so the normalized construction on that target is $H_{j-1}+2$. Naturality therefore gives $N_i{H}_{j-1}=(H_{j-1}+2)N_i$, equivalently $[H_{j-1},N_i]=-2N_i$. This is a use of the actual shifted filtrations, not a claim that $N_i$ preserves a grading automatically.

(c) $H_{j-1}$ commutes with every later $H_a$, $a\ge j$. The case $a=j$ is part of the relative-grading construction. For $a>j$, conjugation by $\exp (s{H}_a)$ preserves $W_{j-1}$, fixes $H_j$ by induction, and rescales $N_j$ by exp(−2s). The normalized relative grading is invariant under this nonzero rescaling, so uniqueness forces conjugation to fix $H_{j-1}$. This proves commutation. In particular all $H_j$ commute, and $H_0=0$ because it grades the pure zero-centred $W_0$.

(d) Each $H_j$ is real and Q-skew. Reality follows from uniqueness under conjugation. All $W_j$ are self-dual, and duality of the relative-grading algorithm gives Q-skewness. To see preservation of $F_s$, let the two Hodge tori of its split bigrading act after complexification. They preserve every $W_j$ and commute with $H_d$, and rescale every $N_i$ by the same nonzero scalar. Uniqueness and rescaling invariance, down the recursion, make their actions commute with every $H_j$. Therefore all $H_j$ preserve the full split bigrading and $F_s$.

(Construct the actual commuting triples.) Let $L_i$ be the component of $N_i$ of $H_{i-1}$-weight zero. All other $H_{i-1}$ weights are negative, since $N_i$ preserves $W_{i-1}$. The relative-grading construction gives an sl₂ triple

$(L_i,Y_i=H_i-H_{i-1},X_i)$.

We must prove that $L_i$ has weight zero for every earlier $H_a$, not merely $H_{i-1}$. For $a<i$, $N_a$ has $H_{i-1}$-weight −2. Projecting $[N_a,N_i]=0$ to $H_{i-1}$-weight −2 shows $[N_a,L_i]=0$. The projection defining $L_i$ preserves every earlier $W_a$ because all their gradings commute. Thus $L_i$ preserves each $W_a$ and commutes with $H_{i-1}$ and $N_{i-1}$. Naturality of $H_{i-2}=Y(N_{i-1},H_{i-1})$ makes it commute with $H_{i-2}$. Repeat this argument through the earlier recursion: $L_i$ commutes with every $H_a$ for $a<i$.

For $a\ge i$, $N_i$ has $H_a$-weight −2; the commuting projections show that $L_i$ has the same weight. Consequently

$[Y_a,L_i]=-2L_i$ if $a=i$, and $[Y_a,L_i]=0$ otherwise.

For $i<j$, projecting $[N_i,N_j]=0$ to $H_{j-1}$-weight −2 gives $[N_i,L_j]=0$. Since $L_j$ commutes with $H_{i-1}$, its commutator also commutes with the projection defining $L_i$; hence $[L_i,L_j]=0$. Each other $L_j,Y_j$ therefore commutes with $L_i,Y_i$. The raising operator $X_i$ is uniquely determined by its lowering operator and grading, so conjugating by $\exp (s{L}_j)$ or $\exp (s{Y}_j)$ fixes $X_i$. Repeating the uniqueness argument for $X_j$ gives mutual commutation of the full triples.

All triples are real and Q-skew, by their construction and uniqueness of the raising operator. Integrate using $d\rho (E_i)=L_i$, $d\rho (H_{\mathrm{s}\mathrm{t}\mathrm{d},i})=-Y_i$, $d\rho (F_{\mathrm{s}\mathrm{t}\mathrm{d},i})=X_i$. These assignments preserve $[H_{\mathrm{s}\mathrm{t}\mathrm{d}},E]=2E$, $[H_{\mathrm{s}\mathrm{t}\mathrm{d}},F_{\mathrm{s}\mathrm{t}\mathrm{d}}]=-2F_{\mathrm{s}\mathrm{t}\mathrm{d}}$ and $[E,F_{\mathrm{s}\mathrm{t}\mathrm{d}}]=H_{\mathrm{s}\mathrm{t}\mathrm{d}}$ because $[Y_i,L_i]=-2L_i$ and $[X_i,L_i]=Y_i$. The finite string decomposition therefore integrates them to a genuine representation

$\rho :\mathrm{SL}(2,R{)}^d\to \mathrm{Aut}(V_R,Q)$.

No covering-group issue is hidden: on each irreducible factor it is the ordinary integral symmetric-power representation, whose matrix entries are polynomials in the entries of an SL₂ matrix. Their commuting direct sum/tensor decomposition gives the displayed group representation.

The diagonal triple $(\sum L_i,\sum Y_i=H_d,\sum X_i)$ shows that $L=\sum L_i$ is Lefschetz for $W_d$. The filtration $F_s$ is preserved by every $Y_i$ and each $L_i$ has split Hodge type (−1,−1).

(The limiting comparison point lies in D.) Set $t_j=y_j\mathrm{/}{y}_{j+1}$ for $j<d$, $t_d=y_d$, and $u_j=t_j^{-1\mathrm{/}2}$. Define the real Q-isometry

$\beta (t)={\prod }_{j=1}^d{t}_j^{H_j\mathrm{/}2}={\prod }_{i=1}^d{y}_i^{Y_i\mathrm{/}2}$.

All factors commute. Since $N_i$ preserves $W_j$ for $j<i$ and has $H_j$-weight −2 for $j\ge i$, every joint H-weight component of

$A_i(u)=Ad(\beta (t))(y_i{N}_i)$

is a monomial in the nonnegative integral powers of $u_1,\dots ,u_{i-1}$. Its constant term is $L_i$; the equality follows from the earlier-zero-weight property proved above. Thus $A_i$ is a finite matrix polynomial, $A_i(0)=L_i$, and $A=\sum A_i$ tends to L.

For positive u near zero, $A=Ad(\beta )(\sum y_i{N}_i)$ polarizes $(W_d[-k],F_s)$, because the interior generator does and β preserves Q, $W_d$ and $F_s$. Since the limiting L is Lefschetz, its primitive kernels have their expected dimensions and vary continuously near zero; this follows directly by inverting the nonzero Lefschetz minors and writing the primitive kernel as a graph. Their primitive Hermitian forms are positive for positive u, hence nonnegative in the limit. They are nondegenerate at L by Lefschetz and the perfect Q pairing between opposite $W_d$ weights. Nonnegative plus nondegenerate implies positive definite. Therefore L polarizes the split MHS.

The elementary split-string argument in the relative-face proof now gives

$F_0=\exp (iL)F_s\in D$.

Together with the commuting triples and their Hodge bigrading, this defines a polarized SL₂^d orbit: its map on the product of upper half-planes is

${\mathrm{\Phi }}_{\mathrm{SL}}(w)=\rho (g_w)F_0$,

where $g_w$ sends $(i,\dots ,i)$ to $w$ by the stated upper-triangular matrices. With the integration convention of step 78.1, $\rho (g_{a+ib})=\exp (aL)b^{-Y\mathrm{/}2}$ on each factor; since $Y{F}_s=F_s$ and $\mathrm{Ad}(b^{-Y\mathrm{/}2})L=bL$, it follows that $\rho (g_{a+ib})\exp (iL)F_s=\exp ((a+ib)L)F_s$. Here the required polarized pure multiplicity spaces can also be identified directly. Decompose the commuting representations into tensor strings $\otimes {\mathrm{Sym}}^{m_i}(R^2)\otimes P_m$. The joint highest space $P_m$ is Hodge-stable because all the raising operators have split Hodge degree (1,1), by their uniqueness under the split Hodge-torus action. Its highest total $H_d$ degree is $\sum m_i$, and it lies in the primitive kernel of $L^{1+\sum m_i}$. The form $Q(-,L^{\sum m_i}-)$ there is a positive multinomial/factorial multiple of the product standard-string pairing times the multiplicity pairing. Its already-proved primitive polarization therefore gives the pure polarized multiplicity structure, of weight $k-\sum m_i$ after removing the highest tensor-string types $(\sum m_i,\sum m_i)$. This also proves that the representation and filtration are the tensor products used below.

The stabilizers SO(2)^d fix F₀. This last assertion can be checked on each elementary rank-two factor: exp(iL_i) sends its split (1,1)/(0,0) filtration to the standard polarized weight-one filtration, fixed by the rotations. On tensor products with the polarized pure multiplicity spaces the same check gives invariance and horizontality. Equivalently,

${\mathrm{\Phi }}_{\mathrm{SL}}(w)=\exp (\sum w_i{L}_i)F_s$,

and its membership for all $\mathrm{Im}{w}_i>0$ follows by the commuting real scalings $\prod (\mathrm{Im}{w}_i{)}^{-Y_i\mathrm{/}2}$ and real translations. The representation and map are thus genuinely ${\mathrm{SL}}_2^d$-equivariant, including trivial factors when $L_i=0$.

(Convergent normalized expansion and uniform estimates.) Since η preserves every $W_j$ and strictly lowers $W_d$, every joint H-weight of η is nonpositive in each index and strictly negative in its last index. Consequently

B(u)=Ad(β(t))η

is a finite matrix polynomial in u, with $B(0)=0$. β fixes $F_s$, so the exact normalized original filtration is

$\beta (t)\exp (i\sum y_i{N}_i)F=\exp (iA(u))\exp (B(u))F_s=:R(u)$.

The right side is a holomorphic map of u in a complex neighborhood of the origin into the compact dual, $R(0)=F_0$. Because $F_0\in D$, the real u values near zero lie in D. Choose a real analytic local section of $G_R\to D$ at F₀: select a vector-space complement to its compact stabilizer Lie algebra and apply the ordinary finite-dimensional real analytic inverse-function theorem to its exponential map. It gives a real analytic $g(u)\in G_R$ with

g(0)=1, R(u)=g(u)F₀.

This construction proves a convergent Taylor expansion in the $u_j$, with real Lie-group values on real u. On a fixed smaller closed polydisc, it and its inverse satisfy

$\mathrm{\mid }\mathrm{\mid }g(u)-1\mathrm{\mid }\mathrm{\mid }+\mathrm{\mid }\mathrm{\mid }g(u{)}^{-1}-1\mathrm{\mid }\mathrm{\mid }\le C{\sum }_j\mathrm{\mid }{u}_j\mathrm{\mid }$,

and every fixed partial derivative has a uniform bound. Cauchy's inequalities give explicit derivative bounds C_α on any still smaller polydisc; these constants depend on the fixed orbit and sector and are independent of the ratios within that polydisc. This is an actual convergent local comparison, not an asymptotic formal series.

For bounded real parts $x_i$, normalizing the translation gives $\exp (\sum x_i{A}_i(u)\mathrm{/}{y}_i)$. Its exponent is $O(\sum \mathrm{\mid }{x}_i\mathrm{\mid }\mathrm{/}{y}_i)$, with all $A_i$ bounded. The analogous normalized SL₂ translation is $\exp (\sum x_i{L}_i\mathrm{/}{y}_i)$. Both extend analytically in u because $1\mathrm{/}{y}_i={\prod }_{j\ge i}{u}_j^2$. Hence the same comparison holds uniformly for bounded x. In particular, with $\epsilon ={\sum }_j{t}_j^{-1\mathrm{/}2}$,

$d_D(\mathrm{\Phi }(x+iy),{\mathrm{\Phi }}_{\mathrm{SL}}(x+iy))\le C\epsilon$,

for any fixed $G_R$-invariant Riemannian metric $d_D$ on the period domain, and

$\mathrm{\mid }{h}_{\mathrm{\Phi }}(v,w)-h_{\mathrm{SL}}(v,w)\mathrm{\mid }\le C\epsilon \cdot h_{\mathrm{SL}}(v,v{)}^{1\mathrm{/}2}{h}_{\mathrm{SL}}(w,w{)}^{1\mathrm{/}2}$