Repeated-Inertia Obstructions to Spectral Mod-p Satake for GSp₄

1. Introduction and Main Results

1.1. The Comparison Problem

The mod-p Satake isomorphism identifies the spherical Hecke algebra of a Serre weight with an explicitly described monoid algebra. A different construction attaches a component of a moduli stack of Galois representations, or of étale $(\phi ,\Gamma )$-modules, to the same weight. The comparison between the Hecke algebra and global regular functions on that component is a geometric precursor to questions in the categorical p-adic Langlands program. It is important to distinguish three logically different statements: the construction of a comparison homomorphism, the existence of an abstract algebra isomorphism, and the construction of a categorical action. This paper excludes the second possibility for a concrete infinite family without requiring the first or the third.

Lee established an injection from the global functions on a non-Steinberg ${\mathrm{GL}}_n$ component to the spherical Hecke algebra, and proved surjectivity for sufficiently deep weights [2]. In the rank-two boundary example of ([2], Example 5.3.6(2)), equal adjacent inertial characters force different ordered Frobenius coordinates to represent points with identical semisimplification. An ordered coordinate that is regular on an auxiliary torus therefore need not extend globally. Lee also explicitly credits Kansal and Savoie with an earlier proof, by a different argument, that the rank-two boundary global-function ring and Hecke algebra are not abstractly isomorphic. We do not claim this rank-two mismatch as new. Our contribution is the geometric transfer to the symplectic family in Theorem 1, with a precise Laurent upper bound and verification of its full parameter interval.

For ${\mathrm{GSp}}_4$, the situation is subtler: the relevant component is not the entire ${\mathrm{GL}}_4$ stack, and a restriction to a convenient split or non-split locus is not automatically faithful. Lee’s explicit construction of symplectic Emerton–Gee components ([1], Proposition 4.1.13 and its proof) supplies a suitable schematically dense Siegel family. We exploit this geometric input before applying the rank-two repeated-inertia calculation. In particular, the complete non-split ${\mathrm{GL}}_2$ base, and not the codimension-one fixed-unramified-parameter slice, is essential.

1.2. A Systematic Non-Generic Boundary

Write $F(a,b;c)$ for a symplectic Serre weight in the conventions of [1], and let ${\mathcal{C}}_{F(a,b;c)}$ be the corresponding reduced irreducible component over k. Our family is

$${\sigma }_{p,d,c}=F(p-2+d,p-2;c),p>3,1\le d\le p-2,c\in \mathbf{Z}.$$

(1)

As always, the central integer c is interpreted modulo the appropriate central-weight equivalence; writing it as an integer is a convenient choice of representative. Both simple-root gaps d and $p-2$ lie strictly between 0 and $p-1$, so these are regular, p-restricted, non-Steinberg weights. They are not strongly generic: the second gap is at the $p-2$ boundary.

Theorem 1 

(Uniform strict Hecke mismatch). Let $p>3$, $k={\overline{\mathbf{F}}}_p$, $K={\mathbf{Q}}_p$, and let $\sigma ={\sigma }_{p,d,c}$ be as in (1). Let ${\mathcal{C}}_{\sigma }$ be the reduced symplectic Emerton–Gee component constructed in ([1], Proposition 4.1.13). There is an injective k-algebra homomorphism, induced by restriction to a split Siegel Levi family,

$$\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})⟶k[D^{\pm 1},q^{\pm 1}],$$

(2)

where D is the unramified determinant parameter on the corresponding ${\mathrm{GL}}_2$ component and q is the independent common unramified twisting parameter. Moreover

$${trdeg}_k\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\le 2<3={trdeg}_k{\mathcal{H}}_{{\mathrm{GSp}}_4}\left(\sigma \right),\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\neg {\cong }_k{\mathcal{H}}_{{\mathrm{GSp}}_4}\left(\sigma \right).$$

(3)

Here ${\mathcal{H}}_{{\mathrm{GSp}}_4}\left(\sigma \right)\simeq k[U,V,Z^{\pm 1}]$ is the spherical mod-p Hecke algebra. We do not claim that D or q individually descends to a regular function on ${\mathcal{C}}_{\sigma }$.

The proof has the following architecture. The weight dictionary produces an auxiliary rank-two weight of highest-weight gap $p-2$. On its non-split locus Lee’s Siegel extensions form a vector bundle, and their unramified twists have schematically dense image in ${\mathcal{C}}_{\sigma }$. Conjugation by the scalar Siegel Levi element contracts extension parameters without changing their moduli-stack isomorphism class. Thus functions on ${\mathcal{C}}_{\sigma }$ are detected by the split zero section. This zero section extends over the reduced rank-two component, and Lee’s rank-two spectral injection places its global functions inside a one-sided Laurent cone. Repeated inertia requires invariance under transposition of the two ordered parameters; the intersection of that cone with its reflection is just the determinant Laurent line. The remaining variable is the common unramified twist.

1.3. The Endpoint and the Original Numerical Example

The upper endpoint in Theorem 1 deserves notice. A historical uniform calculation used $1\le d\le p-3$; the extra case $d=p-2$ makes both simple-root gaps equal to $p-2$. It does not enter the très-ramifiée branch, which is controlled by the condition $a-b=p-1$. Nor does it introduce the cyclotomic Siegel obstruction: its obstruction inertial character becomes ${\omega }^{p-1}=1$, not $\omega$. The full external hypotheses relevant to this endpoint are recorded in Section 2 and Section 4; the endpoint is not included simply by extrapolation from numerical experiments.

Corollary 1 

(An explicit prime-83 component). At $p=83$ and $(d,c)=(25,0)$ one obtains

$$\sigma =F(106,81;0),\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\hookrightarrow {\overline{\mathbf{F}}}_{83}[D^{\pm 1},q^{\pm 1}],trdeg\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\le 2<3.$$

This is the original single-component mismatch, now a special case of Theorem 1.

1.4. Relation with Existing Work and Limits

The strongly generic ordinary ${\mathrm{GSp}}_4$ comparison investigated in [8] concerns weights with different gap conditions. Its global-function computation and its conditional completed-local comparison have different hypotheses and should not be conflated. The present argument works at an explicit non-generic edge by a geometric obstruction rather than by constructing all would-be Hecke generators. We require neither global automorphic patching nor a ${\mathrm{GL}}_4$ generic Satake theorem. Related quotient presentations of ${\mathrm{GL}}_2$ components are developed in [5]; our use of rank-two global functions rests instead on the precise non-Steinberg theorem of [2]. In particular, the auxiliary ${\mathrm{GL}}_2$ highest-weight gap equals $p-2$, exactly the boundary excluded by hypothesis (2) of ([5], Theorem 5.0.1). That theorem does not determine this boundary component’s global functions; the restriction and symmetry argument below provides the result we need. Kansal–Savoie ([6], Theorem 1.1) study the singularities of these extremely non-generic rank-two components, including the all-$p-2$ family. Their geometric results provide further context; they are not an additional hypothesis in our symplectic argument.

Several tempting stronger claims are deliberately omitted. We do not determine the exact image of (2), and in particular do not assert equality with the two-variable Laurent algebra. A conjectural refinement from a Laurent upper bound to a polynomial–Laurent upper bound involving a separate Klingen wall needs an independent geometric factorization, not furnished by the Siegel argument. Likewise, resonance and Yoneda-obstruction calculations address a different non-generic phenomenon. These questions are separated in Section 9.

2. Weight Conventions and the Repeated-Inertia Dictionary

2.1. Serre Weights and the Ordered Inertial Exponents

Fix $k={\overline{\mathbf{F}}}_p$ with $p>3$, and write $\omega$ for the mod-p cyclotomic character of $G_K=Gal(\overline{K}/K)$, normalized to have inertial order $p-1$. Write ${ur}_t$ for an unramified character whose chosen Frobenius value is $t\in k^{\times }$; choosing geometric versus arithmetic Frobenius everywhere only replaces the parameters by their inverses. All formulas below use one consistent choice.

The weight coordinates are the integer triple $(a,b;c)$ of ([1], §2 and §4.1). The two simple-root gaps are $a-b$ and b. The p-restricted chamber is

$$0\le a-b\le p-1,0\le b\le p-1.$$

(4)

Strict inequalities give the regular range. For weights (1) the gaps are d and $p-2$, so both inequalities are strict. This excludes both Steinberg gaps $p-1$, while deliberately retaining the repeated-inertia boundary $p-2$.

The ordered four-dimensional inertial exponents in the standard representation are

$$e_1=a+b+c+3,e_2=a+c+2,e_3=b+c+1,e_4=c.$$

(5)

They satisfy $e_1+e_4=e_2+e_3=a+b+2c+3$, which is precisely the inertial similitude relation. Inserting $a=p-2+d$ and $b=p-2$ gives

$$(e_1,e_2,e_3,e_4)=(2p-1+d+c,p+d+c,p-1+c,c)\equiv (d+c+1,d+c+1,c,c)(modp-1).$$

(6)

Thus the first pair and last pair agree on inertia, but their unramified Frobenius parameters need not agree. The distinction between equality of inertial restrictions and equality of full characters is essential for both the extension calculation and the semisimplification argument.

Lemma 1 

(Character and similitude bookkeeping). Let $e_i$ be as in (5) for ${\sigma }_{p,d,c}$. Then

$$e_1\equiv e_2\equiv d+c+1,e_3\equiv e_4\equiv c(modp-1),$$

and the common similitude inertial exponent is $d+2c+1$. The symplectic Siegel radical attached to the first pair has three Jordan–Hölder inertial characters, all equal to ${\omega }^{d+1}$.

Proof. 

The first assertion follows by subtraction in (5). The value of the similitude exponent follows from $e_1+e_4$ modulo $p-1$. On the split rank-two diagonal, the Siegel radical is ${Ad}_S(\theta ,\chi )={Sym}^2\left(\theta \right)\otimes {\chi }^{-1}$, with $\theta$ having diagonal inertial characters ${\omega }^{e_1}$ and ${\omega }^{e_2}$. Its three graded inertial exponents are $2e_1-\ell$, $e_1+e_2-\ell$, and $2e_2-\ell$, where $\ell =e_1+e_4=e_2+e_3$. Their common residue is $2(d+c+1)-(d+2c+1)=d+1$.    □

2.2. The Auxiliary Rank-Two Weight

Lee associates to $F(a,b;c)$ the rank-two weight

$${\sigma }^{\prime }=F_{{\mathrm{GL}}_2}(a+b+c+2,a+c+2).$$

(7)

For the family (1) its coordinates are

$${\sigma }^{\prime }=F_{{\mathrm{GL}}_2}(2p-2+d+c,p+d+c),(2p-2+d+c)-(p+d+c)=p-2.$$

(8)

Writing its ordered rank-two inertial characters as ${\omega }^{{\mu }_1+1}$ and ${\omega }^{{\mu }_2}$ shows

$${\mu }_1+1=2p-1+d+c\equiv d+c+1\equiv p+d+c={\mu }_2(modp-1).$$

(9)

In particular, every weight in this family produces a non-Steinberg rank-two weight with repeated inertial characters. The fact that d disappears from the highest-weight difference of ${\sigma }^{\prime }$ explains the uniformity of the rank-two part of the proof.

Remark 1 

(Why the two different gaps matter). The condition $b=p-2$ ensures the repeated inertia in the auxiliary ${\mathrm{GL}}_2$ representation, while $d=a-b\ne p-1$ selects the full rank-two source in Lee’s Siegel construction. These are independent structural requirements. Confusing them would lead to using the wrong slice as a dense family.

2.3. Siegel Cohomology and the Extra Endpoint

The relevant obstruction group is

$$H^2\left(G_K,{Ad}_S(\theta ,{\chi }_{\sigma })\right),{Ad}_S(\theta ,{\chi }_{\sigma })\simeq {Sym}^2\left(\theta \right)\otimes {\chi }_{\sigma }^{-1}.$$

By local Tate duality its dual is a space of cyclotomic-twisted invariants. A sufficient vanishing criterion, on the rank-two family in question, is that the Jordan–Hölder inertial constituents of ${Ad}_S$ are distinct from $\omega$. In the present family these constituents are ${\omega }^{d+1}$. Equality ${\omega }^{d+1}=\omega$ would require $d\equiv 0(modp-1)$, excluded by $1\le d\le p-2$. Hence

$${\omega }^{d+1}\ne \omega (1\le d\le p-2).$$

(10)

For $d=p-2$ the left side is the trivial inertia character, so it is still distinct from $\omega$ because $p>3$. The external relative construction that turns this cohomological input into a Siegel vector bundle is recalled in Section 4. The numerical congruence alone is not being used as a replacement for that construction.

Remark 2 

(A non-admissible endpoint). At $d=0$, ${\omega }^{d+1}=\omega$ and the gap $a-b=0$ is not regular. Thus the same argument does not automatically extend to $d=0$. At $d=p-1$, the gap $a-b=p-1$ enters Lee’s très-ramifiée branch with a different source, and our use of the full ${\mathrm{GL}}_2$ base no longer follows. These exclusions are mathematical, not cosmetic.

3. The Mod-P Satake Algebra and Central Normalization

3.1. A Concrete Antidominant Cocharacter Monoid

We record the rank calculation explicitly to avoid treating the Hecke transcendence degree as an unexplained dimension count. Choose cocharacter coordinates $\nu =(m,n;s)\in {\mathbf{Z}}^3$ adapted to the diagonal torus of ${\mathrm{GSp}}_4$. In the convention used here the antidominant conditions are

$$m\le n,2n\le s.$$

(11)

The central cocharacter lattice has rank one, generated by $\zeta =(1,1;2)$. Put

$${\nu }_1=(-1,-1;-1),{\nu }_2=(-2,-1;-2),\zeta =(1,1;2).$$

Lemma 2 

(Unimodular Satake coordinates). Every integral antidominant cocharacter is uniquely expressed as

$$(m,n;s)=(s-2n){\nu }_1+(n-m){\nu }_2+(s-m)\zeta .$$

(12)

The coefficients of ${\nu }_1$ and ${\nu }_2$ are nonnegative integers, while the central coefficient is an arbitrary integer. Consequently the antidominant monoid is isomorphic to ${\mathbf{N}}^2\oplus \mathbf{Z}$ and its monoid algebra is $k[U,V,Z^{\pm 1}]$.

Proof. 

The column matrix $P=({\nu }_1,{\nu }_2,\zeta )$ has determinant $-1$. It is therefore a basis of the complete integral cocharacter lattice. Direct multiplication yields (12). The inequalities (11) say exactly that $s-2n\ge 0$ and $n-m\ge 0$; $s-m$ is unrestricted. Uniqueness follows from unimodularity, and the monoid algebra is the tensor product of two polynomial and one Laurent polynomial algebra.    □

Herzig’s Satake isomorphism [3] identifies the spherical Hecke algebra of the regular Serre weight with this monoid algebra. We only use its structure as a ring, not any assertion about global functions on a symplectic component. In particular

$${\mathcal{H}}_{{\mathrm{GSp}}_4}\left(\sigma \right)\simeq k[U,V,Z^{\pm 1}],{trdeg}_k{\mathcal{H}}_{{\mathrm{GSp}}_4}\left(\sigma \right)=3.$$

(13)

The identity holds throughout the regular family (1). A change of choice of central generator replaces Z by $Z^{-1}$ but does not change either the algebra or its dimension.

3.2. Why the Central Variable Matters

An unramified scalar twist by ${ur}_q$ multiplies all four standard Frobenius eigenvalues by q and multiplies the similitude Frobenius parameter by $q^2$. Suppressing the similitude parameter would incorrectly identify points whose line data differ. That mistaken identification can produce a spurious bound of transcendence degree one. Our argument retains q as an independent invertible parameter and therefore proves only the robust bound (3).

In a fixed set of ordered split-torus variables $(x,y,z)$, with the standard pairing of the four symplectic eigenvalues, the scalar twist is

$$(x,y,z)⟼(qx,qy,q^{2}z).$$

(14)

For the root-datum matrix P of Lemma 2, the normalized exponent vector in the character (dual) basis is $r=P^{-T}\nu$, rather than the cocharacter-basis coefficient vector $P^{-1}\nu$. Since the central column of P is $\zeta ={(1,1,2)}^{\top }$, the dual-basis identity $(1,1,2)P^{-T}\nu ={\left(P^{-1}\zeta \right)}^{\top }\nu =e_3^{\top }\nu =s$ gives $r_1+r_2+2r_3=s$. Hence the corresponding Laurent character transforms by a power $q^{\pm s}$, with the sign set by the Satake evaluation convention. The common-twist variable cannot simply be set equal to one without changing the moduli problem.

Remark 3 

(Independence of the rank-two determinant). The symbol D in our target algebra belongs to the defining rank-two Levi base, not to a separately constructed symplectic Hecke operator. Likewise q is a parameter on the source of a family. Injection into $k[D^{\pm 1},q^{\pm 1}]$ does not supply a reverse map or prove that either generator extends from the family to ${\mathcal{C}}_{\sigma }$.

4. Lee’s Siegel Extension Family and Schematic Density

4.1. The Precise External Geometric Input

We isolate the geometric statement used in the proof. This is a specialization of the construction in ([1], proof of Proposition 4.1.13, §4.1.15), not an original theorem about all symplectic deformation spaces. For $\sigma =F(a,b;c)$ the rank-two label is (7). Let ${\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }\subset {\mathcal{X}}_{2,\mathrm{red}}$ denote its reduced irreducible component and let $U_{{\sigma }^{\prime }}\subset {\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }$ be Lee’s non-split open locus.

Lee introduces a fixed-parameter codimension-one substack ${\mathcal{C}}_{{\sigma }^{\prime }}^{\mathrm{fixed}}$ defined by fixing one unramified diagonal parameter. The source of the symplectic family is a smooth cover of this fixed locus only in the très-ramifiée branch $a-b=p-1$. The non-regular wall $a-b=0$ also requires the open excision $t_2^2\ne 1$ in Lee’s construction ([1], §4.1.15). In the otherwise branch with $1\le a-b\le p-2$ it is a smooth cover of all$U_{{\sigma }^{\prime }}$ without that excision. Since $a-b=d$ and $1\le d\le p-2$, we are always in the full-source branch.

Choose a smooth surjective cover $T\to U_{{\sigma }^{\prime }}$ carrying the universal rank-two family ${\theta }_T$. Let ${\chi }_{\sigma }$ denote the fixed tame similitude character prescribed by the weight. In our range the obstruction group $H^2(G_K,{Ad}_S({\theta }_T,{\chi }_{\sigma }))$ vanishes, and the construction supplies a vector bundle of symplectic Siegel cocycles/extension data $\pi :E_T\to T$. The immediate input is Lee’s Siegel extension-bundle construction ([1], §4.1.15, construction preceding Lemma 4.1.17), which permits both Siegel terms to vary. Its underlying relative Herr-complex machinery is ([4], Corollary 5.1.25, Remark 5.1.30, §5.1.33, Lemma 5.3.2, and the discussion preceding Proposition 5.4.4); the fixed-representation statement of that proposition alone is not being applied to two varying terms. Its family points have the form

$$0⟶{\theta }_T⟶{\rho }_e⟶{\theta }_T^{\vee }\otimes {\chi }_{\sigma }⟶0.$$

(15)

Taking the common unramified twists gives a map

$$h:E_T\times {\mathbf{G}}_m⟶{\mathcal{C}}_{\sigma },(e,q)⟼\left({\rho }_e\otimes {ur}_q,{\chi }_{\sigma }{ur}_q^2\right).$$

(16)

By the Definition of Lee’s component in the proof of Proposition 4.1.13, the scheme-theoretic image of this family is ${\mathcal{C}}_{\sigma }$. Thus h is schematically dense. The strong conclusion of schematic density, rather than merely a set-theoretic statement for closed points, is what will make pullback on global sections injective.

4.2. Compatibility with the Uniform Weights

Proposition 1 

(Verification of the Siegel branch). For every $p>3$, every $1\le d\le p-2$, and every allowed central representative c, the weight ${\sigma }_{p,d,c}$ satisfies the following properties in Lee’s construction:

(i) 

${\sigma }^{\prime }$ is a non-Steinberg rank-two weight with highest-weight gap $p-2$;

(ii) 

the symplectic source belongs to the otherwise branch and therefore covers the full non-split $U_{{\sigma }^{\prime }}$;

(iii) 

the Siegel obstruction has no cyclotomic inertial constituent, including at $d=p-2$;

(iv) 

the unramified twisting family has scheme-theoretically dense image in ${\mathcal{C}}_{\sigma }$.

Proof. 

Part (i) is (8). Since $d\ne p-1$, Lee’s explicit case distinction gives (ii). Part (iii) follows from Lemma 1 and the non-equality (10); the existence of the relative bundle, rather than just its pointwise obstruction calculation, uses Lee’s construction. Finally (iv) is exactly the scheme-theoretic-image assertion for the corresponding unramified twists in ([1], proof of Proposition 4.1.13). Each criterion holds verbatim when $d=p-2$: the otherwise branch condition is still satisfied and ${\omega }^{d+1}=1\ne \omega$.    □

Remark 4 

(Relative versus pointwise cohomology). On a fixed k-point, $H^2=0$ gives a familiar unobstructedness calculation for extension classes. A family over a base requires a cohomology-and-base-change statement to produce a vector bundle and a morphism to a stack. We invoke the bundle explicitly constructed in Lee’s proof; we do not manufacture a universal continuous Galois character over a discrete polynomial ring. This distinction avoids replacing a moduli-stack statement by a collection of unrelated numerical fibers. At the new endpoint $d=p-2$, the Siegel radical has trivial inertial character, so $H^0(G_K,{Ad}_S({\theta }_t,{\chi }_{\sigma }))$, and consequently $H^1$, may jump along special unramified fibers. This does not affect the required vector bundle: Lee constructs $E_T=\underline{Spec}Sym\left({\left(Z_T^1\right)}^{\vee }\right)$ from the sheaf of 1-cocycles $Z_T^1$, not from $H^1$. Cohomology and base change identify the fibers of the coherent relative obstruction sheaf with $H^2$; their vanishing makes that sheaf zero by Nakayama’s lemma. Thus the relative obstruction sheaf has constant rank zero, so the cocycle module remains locally free of constant rank by the relative construction in ([4], Proposition 5.4.4) and ([1], §4.1.15). No vanishing or constancy assumption on $H^0$ is used. More concretely, in a local perfect Herr representative $C_T^0\to C_T^1\stackrel{d^1}{\to }{C}_T^2$, the vanishing of the relative $H^2=coker\left(d^1\right)$ implies that $d^1$ is surjective; since both terms are finite locally free, its kernel $Z_T^1$ is finite locally free and compatible with base change. This is a statement about actual cocycles, not the quotient $H^1=Z_T^1/im\left(d^0\right)$.

4.3. What the Dense Family Does and Does Not Assert

The family need not be a dense open immersion. Its source may have nontrivial automorphisms, and extension classes can have different presentations. The only property needed below is that if a global regular function on ${\mathcal{C}}_{\sigma }$ pulls back to zero along h, then that function vanishes identically. Schematic density supplies precisely this implication. We do not infer a presentation of $\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})$ as a polynomial ring on extension coordinates, nor do we use a fixed-$t_2$ slice as a surrogate for T.

For the reader’s convenience, the dependencies can be summarized by the sequence

where the lower horizontal arrow is the intrinsic split Levi construction of Section 6. The diagram commutes up to the canonical 2-isomorphism of the zero-section family. The next section explains why every global function pulled back by h is constant along the vertical vector-bundle fibers.

5. Contraction Along the Siegel Radical

5.1. A General Lemma for Stacks

Lemma 3 

(Contraction of a schematically dense vector-bundle family). Let k be an infinite field, B an algebraic stack over k, $\pi :E\to B$ a vector bundle, and X an algebraic stack. Suppose $h:E\to X$ is schematically dense and, for each $r\in k^{\times }$, $h\circ \left[r\right]$ is 2-isomorphic to h, where $\left[r\right]$ multiplies vectors by r. Then the map on global sections induced by the zero section $s:B\to E$,

$$\Gamma (X,\mathcal{O})\stackrel{h^{*}}{\to }\Gamma (E,\mathcal{O})\stackrel{s^{*}}{\to }\Gamma (B,\mathcal{O}),$$

is injective. The assertion remains valid for $E\times {\mathbf{G}}_m\to B\times {\mathbf{G}}_m$ when scaling fixes the last factor.

Proof. 

Schematic density means that the morphism of structure sheaves ${\mathcal{O}}_X\to h_{*}{\mathcal{O}}_E$ is injective. Passing to global sections therefore makes $h^{*}$ injective. A 2-isomorphism between maps of stacks induces equality of pullbacks of functions, so $h^{*}f$ is invariant under every scalar $r\in k^{\times }$.

To prove fiberwise constancy, pull back along a smooth affine chart $W\to B$ on which the bundle is trivial, say $E_W\simeq W\times {\mathbf{A}}^m$. A regular function on this chart has a finite decomposition into homogeneous terms in $x_1,\dots ,x_m$. If $F={\sum }_{j=0}^N{F}_j$ with $F_j$ homogeneous of total degree j, invariance gives ${\sum }_j(r^j-1)F_j=0$ for every $r\in k^{\times }$. Because k is infinite, polynomial identity in r forces $F_j=0$ for $j>0$. Thus F is the pullback of $s^{*}F$ and has no fiber dependence. The same identity holds on overlapping charts, hence globally $h^{*}f={\pi }^{*}{s}^{*}{h}^{*}f$. If $s^{*}{h}^{*}f=0$, then $h^{*}f=0$, so $f=0$. The product statement follows by repeating the argument with $B\times {\mathbf{G}}_m$ as base.    □

Remark 5 

(What the lemma does not require). No quotient-stack presentation of X is assumed, and we do not require h to be representable or injective on geometric points. The argument uses invariance of global functions, not freeness of the scalar action. Even when the extension family is highly redundant as a moduli parameterization, its schematic image can detect functions.

5.2. Symplectic Conjugation Supplies the Required Invariance

Lemma 4 

(Scalar Siegel extensions are isomorphic). For the map (16), let $\left[r\right]$ scale the cocycle coordinate e by $r\in k^{\times }$ and leave q fixed. Then $h\circ \left(\right[r]\times 1)$ and h are 2-isomorphic as maps to the unframed symplectic Emerton–Gee stack, including the similitude line.

Proof. 

Locally on T write the four-dimensional representation as a Siegel-block matrix

$${\rho }_e\left(g\right)=\left(\begin{array}{cc}\theta \left(g\right)& E_e\left(g\right)\\ 0& {\chi }_{\sigma }\left(g\right)\theta {\left(g\right)}^{-\top }\end{array}\right),$$

where the cocycle $E_e\left(g\right)$ satisfies the appropriate symplectic relations. Let

$$J=\left(\begin{array}{cc}0& I_2\\ -I_2& 0\end{array}\right),s_r=\left(\begin{array}{cc}r{I}_2& 0\\ 0& I_2\end{array}\right).$$

A direct calculation gives

$$s_r^{\top }J{s}_r=rJ,s_r{\rho }_e\left(g\right)s_r^{-1}={\rho }_{re}\left(g\right).$$

(17)

Thus $s_r\in {\mathrm{GSp}}_4\left(k\right)$ with multiplier r. The moduli object records a symplectic pairing valued in a rank-one similitude module. In the groupoid of ([1], Definition 4.1.1), an isomorphism is a compatible pair of maps on the rank-four module and on the pairing-valued line. Here that pair is $(s_r,r{\mathrm{id}}_N)$, with $N={\chi }_{\sigma }{ur}_q^2$. It is defined intrinsically on the global direct sum ${\theta }_T\oplus ({\theta }_T^{\vee }\otimes {\chi }_{\sigma })$; no local basis of ${\theta }_T$ is needed. Multiplication by r on the line is the accompanying isomorphism; it does not replace the Galois character ${\chi }_{\sigma }$ by a different one. The same pair of maps gives an isomorphism after tensoring the standard module by ${ur}_q$, with similitude module ${\chi }_{\sigma }{ur}_q^2$. Since scalar matrices in the two Levi blocks commute with changes of trivialization, the construction glues over T and is natural in the family. The resulting natural isomorphism is the required 2-isomorphism.    □

This is the point where working on the unframed stack is essential. On a framed deformation space conjugation need not identify points with different frames, and the same argument cannot be copied verbatim. The scalar multiplier r on the similitude line also explains why it would be incorrect to replace $s_r$ by a strict ${\mathrm{Sp}}_4$ gauge transformation without further work.

Proposition 2 

(Detection by the Siegel zero section). Let $s:T\times {\mathbf{G}}_m\to E_T\times {\mathbf{G}}_m$ be the zero section. The composite

$$\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\stackrel{h^{*}}{\to }\Gamma (E_T\times {\mathbf{G}}_m,\mathcal{O})\stackrel{s^{*}}{\to }\Gamma (T\times {\mathbf{G}}_m,\mathcal{O})$$

is injective for every weight (1).

Proof. 

By Proposition 1, h is schematically dense and $E_T$ is a vector bundle. By Lemma 4, pullback of any global function is invariant under all fiber scalings. Lemma 3, applied after adjoining the unramified twist parameter, gives the result.    □

6. The Split Levi Map and Descent from the Atlas

6.1. The Zero Section Is Intrinsically Defined

The split extension of (15) is not tied to a choice of cocycles or a basis of $\theta$. Given an object $\theta$ of the rank-two étale stack and the fixed similitude character ${\chi }_{\sigma }$, there is a canonical symplectic object

$$L(\theta ,q)=\left((\theta \oplus {\theta }^{\vee }\otimes {\chi }_{\sigma })\otimes {ur}_q,{\chi }_{\sigma }{ur}_q^2,{\langle ,\rangle }_{\mathrm{can}}\right),$$

(18)

where ${\langle ,\rangle }_{\mathrm{can}}$ is the perfect alternating pairing between the two summands. It is functorial in isomorphisms of $\theta$ and in base change. Consequently it defines a morphism of stacks without first choosing a smooth atlas.

Lemma 5 

(Descent along the rank-two atlas). The zero-section map $T\times {\mathbf{G}}_m\to {\mathcal{C}}_{\sigma }$ in Proposition 2 descends to a morphism

$${\iota }_U:U_{{\sigma }^{\prime }}\times {\mathbf{G}}_m⟶{\mathcal{C}}_{\sigma }.$$

Pullback along ${\iota }_U$ is injective on global functions.

Proof. 

Formula (18) makes sense for the universal object over any test scheme of $U_{{\sigma }^{\prime }}$. Its pullback to T is exactly $e=0$ in the family (16), so it factors through ${\mathcal{C}}_{\sigma }$ on this open. Functoriality supplies the descent isomorphisms on $T{\times }_{U_{{\sigma }^{\prime }}}T$ with the cocycle condition. As a closed-target factorization, this also follows by faithfully flat descent along the smooth atlas (cf. [7], Lemma 101.38.2). The resulting map is ${\iota }_U$. If a global function restricts to zero on $U_{{\sigma }^{\prime }}\times {\mathbf{G}}_m$, then its restriction on $T\times {\mathbf{G}}_m$ is zero. Injectivity from Proposition 2 gives the claim.    □

6.2. Extension from the Open to the Whole Rank-Two Component

We need an upper bound involving $\Gamma ({\mathcal{C}}_{{\sigma }^{\prime }}^{\prime },\mathcal{O})$, not merely functions on an open subset. The reverse direction from the algebra of functions on an open to that on a closed component is generally false, so an explicit map of stacks on the entire component is required.

Lemma 6 

(A reduced-source closed-target principle). Let Y be a reduced algebraic stack, $W\subset Y$ a schematically dense open substack, and $Z\hookrightarrow X$ a closed immersion of algebraic stacks. If $f:Y\to X$ is a morphism whose restriction to W factors through Z, then f factors through Z.

Proof. 

The factorization is equivalent to the vanishing of the image of the defining quasi-coherent ideal ${\mathcal{I}}_Z\subset {\mathcal{O}}_X$ under $f^{*}:f^{-1}{\mathcal{O}}_X\to {\mathcal{O}}_Y$. This ideal vanishes on W. To check that it vanishes on Y, work locally on a smooth affine cover of Y with reduced coordinate rings. On each component, a regular function vanishing on a schematically dense open is zero, so the ideal image vanishes identically. The conclusion descends along the cover. See also the scheme-theoretic image and smooth descent formalism of ([7], Section 101.38). Equivalently, the restriction map ${\mathcal{O}}_Y\to j_{*}{\mathcal{O}}_W$ is injective and detects the ideal.    □

Proposition 3 

(Split Levi factorization through the selected component). The functor (18) defines a morphism of reduced stacks

$$\iota :{\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }\times {\mathbf{G}}_m⟶{\mathcal{C}}_{\sigma }$$

(19)

whose restriction to $U_{{\sigma }^{\prime }}\times {\mathbf{G}}_m$ is ${\iota }_U$. In particular

$$\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\hookrightarrow \Gamma ({\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }\times {\mathbf{G}}_m,\mathcal{O}).$$

(20)

Proof. 

The split construction (18) first gives a map ${\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }\times {\mathbf{G}}_m\to {\mathcal{X}}_{\mathrm{Sym},\mathrm{red}}$ into the ambient reduced symplectic stack, without imposing any relative extension obstruction. On $U_{{\sigma }^{\prime }}\times {\mathbf{G}}_m$ it is the zero section of Lee’s defining extension family and therefore factors through the closed substack ${\mathcal{C}}_{\sigma }\hookrightarrow {\mathcal{X}}_{\mathrm{Sym},\mathrm{red}}$. Since ${\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }$ is reduced and its non-split open is schematically dense (see [4], Theorems 5.5.12 and 6.5.1 and [2], §5.1), so is ${\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }\times {\mathbf{G}}_m$, and Lemma 6 makes the map factor globally through ${\mathcal{C}}_{\sigma }$. The induced pullback on global sections is injective, because its further restriction to $U_{{\sigma }^{\prime }}\times {\mathbf{G}}_m$ is injective by Lemma 5.    □

Remark 6 

(Two independent uses of density). We have used schematic density twice and for different purposes. Density of the extension family in ${\mathcal{C}}_{\sigma }$ makes the zero section detect every global function on the target. Density of $U_{{\sigma }^{\prime }}$ in the rank-two component allows the split map to factor through the chosen closed symplectic component on the entire rank-two base. Neither inference follows merely from knowing that the split points belong to the right component individually.

6.3. Products with a Torus and Global Sections

Lemma 7 

(An affine-flat product formula). Let Y be a quasi-compact quasi-separated algebraic stack over a field k, and let A be a flat k-algebra. There is a natural isomorphism

$$\Gamma (Y\times SpecA,\mathcal{O})\simeq \Gamma (Y,\mathcal{O}){\otimes }_{k}A.$$

In particular, it applies to $A=k\left[q^{\pm 1}\right]$ and the finite-type reduced rank-two Emerton–Gee component.

Proof. 

Choose a finite disjoint union of affine schemes giving a smooth surjective presentation of Y; its relation algebraic space is quasi-compact and quasi-separated and has a finite affine étale cover. Degree-zero descent expresses global sections as an equalizer. Flat tensor product commutes with these finite products and equalizers, both on the relation space and on the presentation. This gives the formula. The rank-two component is of finite presentation over k by ([4], Theorem 5.5.12), so these hypotheses apply; $k\left[q^{\pm 1}\right]$ is flat over k.    □

7. Repeated-inertia ${\mathbf{GL}}_{\mathsf{2}}$ Geometry and Laurent Support

7.1. The non-Steinberg Spectral Injection

For the rank-two weight ${\sigma }^{\prime }$ in (8), put ${\mathcal{C}}^{\prime }={\mathcal{C}}_{{\sigma }^{\prime }}^{{\mathrm{GL}}_2}$. The mod-p Satake map of ([2], Proposition 5.3.3), combined with the rank-two torus coordinates and the corresponding antidominant monoid, gives an injection

$$\Gamma ({\mathcal{C}}^{\prime },\mathcal{O})\hookrightarrow k[t_1,{\left(t_1{t}_2\right)}^{\pm 1}]\subset k[t_1^{\pm 1},t_2^{\pm 1}].$$

(21)

Here $t_1,t_2$ are ordered unramified Frobenius parameters on the diagonal semisimple locus. Set $D=t_1{t}_2$. Only an injection is used; Lee’s stronger isomorphism theorem in sufficiently deep weight range does not apply at the $p-2$ boundary.

The cone in (21) is conveniently described by lattice support. A Laurent monomial $t_1^A{t}_2^B$ can be written as

$$t_1^A{t}_2^B=t_1^{A-B}{D}^B.$$

(22)

It belongs to $k[t_1,D^{\pm 1}]$ if and only if $A-B\ge 0$. Thus the spectral image has support in the closed half-space $A\ge B$.

7.2. Why Global Functions Cannot Distinguish the Two Orders

The diagonal inertia in (9) has two identical characters, even though its two unramified factors are free. Lee’s repeated-inertia example ([2], Example 5.3.6(2)) observes that a non-split representation with ordered Frobenius parameters $(t_1,t_2)$ can be paired with a non-split representation in the same component with ordered parameters $(t_2,t_1)$. The two representations have the same semisimplification. In the reduced Emerton–Gee stack, semisimplification lies in the closure of the point by ([4], Theorem 6.6.3(3)); regular functions have the same value on that specialization. Hence a global function has identical Laurent restrictions at the two ordered points.

There is also a direct stack-level explanation which works on the entire split torus, not just on the locus of distinct Frobenius eigenvalues. By ([2], Definition 5.3.2), the map in (21) is the pullback along the split sum map ${\iota }_{\mathrm{sp}}:{\mathcal{C}}_{\mu }\to {\mathcal{C}}^{\prime }$, up to Lee’s standard cyclotomic normalization. Its two rank-one summands have the same tame inertial character since ${\mu }_1+1\equiv {\mu }_2(modp-1)$. Let $\tau$ exchange their unramified parameters $(t_1,t_2)\mapsto (t_2,t_1)$. On every test scheme, the isomorphism exchanging the two direct-sum factors defines a functorial 2-isomorphism

$${\iota }_{\mathrm{sp}}\circ \tau \simeq {\iota }_{\mathrm{sp}}.$$

The map on global functions is therefore $\tau$-invariant as an identity of Laurent functions on ${\mathcal{C}}_{\mu }\simeq [{\mathbf{G}}_m^2/{\mathbf{G}}_m^2]$. This argument strengthens, rather than replaces, the non-split semisimplification explanation of ([2], Example 5.3.6(2)).

Because the ordered non-split parameter locus is dense, the preceding equality is an equality of Laurent polynomials:

$$f(t_1,t_2)=f(t_2,t_1)\mathrm{for}\mathrm{each}f\in im\left(\Gamma ({\mathcal{C}}^{\prime },\mathcal{O})\to k[t_1^{\pm 1},t_2^{\pm 1}]\right).$$

(23)

This is a necessary symmetry of global functions; it does not assert that the component itself is a categorical quotient by ${\mathfrak{S}}_2$.

Lemma 8 

(Reflection of a one-sided Laurent cone). Let k be any field and $D=t_1{t}_2$. Then

$$k[t_1,D^{\pm 1}]\cap k{[t_1^{\pm 1},t_2^{\pm 1}]}^{{\mathfrak{S}}_2}=k\left[D^{\pm 1}\right].$$

(24)

More generally, the intersection of a Laurent monoid algebra with the subring invariant under a reflection is supported on the intersection of the monoid and its reflected monoid, provided the Laurent monomials are linearly independent.

Proof. 

Write $f={\sum }_{i\ge 0,j\in \mathbf{Z}}{a}_{ij}{t}_1^i{D}^j$ with finite support. Applying the transposition $t_1\leftrightarrow t_2$ sends $t_1$ to $D/t_1$, so symmetry gives

$$\sum _{i,j}{a}_{ij}{t}_1^i{D}^j=\sum _{i,j}{a}_{ij}{t}_1^{-i}{D}^{j+i}.$$

On the left, $t_1$ has only nonnegative powers, while on the right it has only nonpositive powers. Algebraic independence of $t_1$ and D and linear independence of their Laurent monomials force all coefficients with $i>0$ to vanish. Thus $f\in k\left[D^{\pm 1}\right]$. The reverse inclusion is immediate because $D=t_1{t}_2$ is symmetric and invertible.

For the geometric support formulation, a monomial $t_1^A{t}_2^B$ in the first ring satisfies $A\ge B$, and its reflection requires $B\ge A$. The intersection is the diagonal $A=B$, generated by $D^{\pm 1}$.    □

Proposition 4 

(The determinant is the only possible Laurent direction). For every auxiliary rank-two weight (8), the global-function ring satisfies

$$\Gamma ({\mathcal{C}}_{{\sigma }^{\prime }}^{\prime },\mathcal{O})\hookrightarrow k\left[D^{\pm 1}\right].$$

(25)

In fact, with the standard determinant morphism on the rank-two stack, this upper bound is an equality.

Proof. 

The injection (21) places the global functions in $k[t_1,D^{\pm 1}]$. Equality on semisimplifications gives the symmetry (23). Lemma 8 proves the injection (25). For the equality assertion, the determinant of a rank-two étale family is functorial. On the reduced irreducible component in question, the determinant morphism factors through the rank-one reduced component with the prescribed tame character. This rank-one component has presentation $[{\mathbf{G}}_m/{\mathbf{G}}_m]$ with trivial action and Laurent coordinate given by the unramified Frobenius value ([2], §5.2). Its pullback is D up to a fixed nonzero scalar depending only on the chosen normalization. This gives a global invertible function whose restriction is D up to that scalar; its inverse is global as well. Thus $k\left[D^{\pm 1}\right]\subseteq \Gamma ({\mathcal{C}}^{\prime },\mathcal{O})$ under the injection. Only the upper bound will be used for the symplectic theorem.    □

Remark 7 

(A stronger rank-two conclusion, not a stronger symplectic conclusion). The equality in Proposition 4 concerns the auxiliary ${\mathrm{GL}}_2$ component. The map ${\mathcal{C}}^{\prime }\times {\mathbf{G}}_m\to {\mathcal{C}}_{\sigma }$ need not be surjective on global functions after pullback, so it does not imply $\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})=k[D^{\pm 1},q^{\pm 1}]$. The direction of the ring map is crucial.

7.3. Negative Controls for the Support Argument

The mechanism is specific to the coexistence of a one-sided cone and a transposition forced by repeated inertia. If the inertial characters are distinct, the reflection symmetry is not justified by the same geometric argument. If one replaces the one-sided cone by the entire Laurent ring, symmetric Laurent functions include $t_1+t_2$ and have two algebraically independent directions. Finally, replacing the full rank-two component by a fixed-$t_2$ slice changes the function ring and invalidates the density input. These observations make clear which hypotheses are structural and which are merely notational.

8. Proof of the Uniform Theorem and Numerical Illustrations

Proof of Theorem 1 

Fix $p>3$, $1\le d\le p-2$, and c as in the statement, and let ${\sigma }^{\prime }$ be given by (8). Proposition 1 supplies Lee’s defining vector bundle over the full non-split rank-two source and its schematically dense unramified twisting family. The contraction Lemmas 3 and 4 show that global functions on ${\mathcal{C}}_{\sigma }$ are detected by its split zero section. Proposition 3 extends this section to the reduced rank-two component and yields an injection

$$\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\hookrightarrow \Gamma ({\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }\times {\mathbf{G}}_m,\mathcal{O}).$$

By Lemma 7 and Proposition 4, the target embeds into

$$\Gamma ({\mathcal{C}}_{{\sigma }^{\prime }}^{\prime },\mathcal{O}){\otimes }_{k}k\left[q^{\pm 1}\right]\hookrightarrow k[D^{\pm 1},q^{\pm 1}].$$

This proves (2). Since a subalgebra of a Laurent polynomial algebra in two variables has transcendence degree at most two, the first inequality in (3) follows. The spherical Hecke algebra has the form (13) by Herzig’s Satake theorem and Lemma 2, so it has transcendence degree three. Transcendence degree is invariant under k-algebra isomorphism, proving the abstract non-isomorphism.    □

8.1. The Initial Prime-83 Example

For $p=83$, $d=25$, and $c=0$, the full values are

$$(a,b;c)=(106,81;0),(e_1,e_2,e_3,e_4)=(190,108,82,0).$$

Reduction modulo 82 gives $(26,26,0,0)$. The auxiliary rank-two weight is

$${\sigma }^{\prime }=F_{{\mathrm{GL}}_2}(189,108),189-108=81=p-2.$$

The three Siegel radical graded inertial characters have exponent $d+1=26(mod82)$, different from the cyclotomic exponent 1. There are two noncentral antidominant directions and a central invertible direction in the Hecke algebra. These observations establish the input conditions of Theorem 1 in the original numerical case without any special computation in characteristic 83 beyond modular arithmetic.

8.2. Further Instances and the Endpoint

The table records several points in the family. The entries are exact modular calculations, not independent proofs of the stack geometry.

p	d	c	$(a,b)$	$(e_1,e_2,e_3,e_4)mod(p-1)$	${\omega }^{d+1}$
5	1	0	(4,3)	$(2,2,0,0)$	${\omega }^2$
7	5	0	(10,5)	$(0,0,0,0)$	1
11	9	2	(18,9)	$(2,2,2,2)$	1
83	25	0	(106,81)	$(26,26,0,0)$	${\omega }^{26}$
83	81	0	(162,81)	$(0,0,0,0)$	1
101	99	7	(198,99)	$(7,7,7,7)$	1

The endpoint examples include some with all four inertial characters identical. This does not invalidate the argument: Lee’s chosen Siegel component and the role of its complete defining extension family are what select the relevant geometry, not the distinctness of the four inertial characters. At $d=p-2$, the Siegel obstruction is trivial on inertia, but the cyclotomic character remains nontrivial and the cohomological obstruction vanishes on the specified family.

Corollary 2 

(An infinite collection of counterexamples). There are infinitely many regular non-Steinberg symplectic Serre-weight components over varying residual characteristics for which the global-function ring is not abstractly isomorphic to the associated spherical mod-p Hecke algebra.

Proof. 

For every prime $p>3$ choose $d=1$ and $c=0$. Theorem 1 applies, and there are infinitely many such primes.    □

8.3. Interpretation of the Dimension Drop

The dimension of Lee’s symplectic component itself is four for $K={\mathbf{Q}}_p$, as stated in ([1], Proposition 4.1.13). The number four is a dimension of an algebraic stack, not the transcendence degree of its global regular-function ring. The comparison in this paper is between a global-function algebra of transcendence degree at most two and a spherical Hecke algebra of transcendence degree three. There is no contradiction in a four-dimensional stack having far fewer globally detectable parameters.

Geometrically, the extension coordinates are invisible to global functions because the unframed group action contracts them to zero, while repeated-inertia symmetry together with the one-sided Satake support constraint leaves only the rank-two determinant direction. This gives a conceptual two-stage loss of possible global parameters. We stress that the argument is an upper bound: it does not show that both D and q survive as independent symplectic global functions.

9. Boundaries, Alternative Mechanisms, and Open Questions

9.1. The Exact Symplectic Ring

The first open problem is to determine the image of (2). It is tempting to guess that some elementary Laurent combinations of D and q descend, but the split Levi source may identify points or meet additional boundary strata imposing further equations. The injection alone proves neither finite generation nor a particular normality property for the image. An exact answer would require an independent extension theorem or a sufficiently complete geometric description of the component’s boundary.

Nevertheless, the similitude character supplies a genuine invertible global function. The similitude morphism from the reduced irreducible component factors through the rank-one reduced component with its prescribed tame character. The presentation $[{\mathbf{G}}_m/{\mathbf{G}}_m]$ with trivial action ([2], §5.2) makes the remaining unramified Frobenius coordinate u and its inverse global regular functions. Along the dense twisting family (16), its pullback is $u=\alpha q^2$ for a fixed $\alpha \in k^{\times }$, so the map $k\left[u^{\pm 1}\right]\to \Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})$ is injective. Consequently

$$1\le {trdeg}_k\Gamma ({\mathcal{C}}_{\sigma },\mathcal{O})\le 2.$$

This improves only the lower bound, not the unknown exact presentation of the symplectic ring.

9.2. A Possible Stronger Bound and a Missing Wall

A historical exploratory calculation suggested an inclusion into a ring of the form $k[W,z^{\pm 1}]$, with only one freely invertible direction, via a middle Klingen wall. Its W is a specified torus monomial in a different coordinate normalization; it should not be identified with the rank-two determinant D without checking the normalizations. This proposed improvement is not a consequence of the Siegel proof. Establishing it would require a genuine morphism from the entire reduced wall source to the selected component, with the correct relative family, ordered labels and similitude line, and a compatible pullback to a second Laurent cone. The corresponding geometric factorization is not certified here. Consequently no stronger theorem is stated.

9.3. Cyclotomic Resonance Is a Different Phenomenon

Other non-generic weights can have cyclotomic resonance in adjacent extension characters. Prescribed successive extensions may then face a nontrivial Yoneda cup-product obstruction, with local existence equations such as $ts=0$ in appropriate models. Even when such equations are rigorously derived for fixed-character cohomology functors, they do not automatically describe the global functions of a real Emerton–Gee component. Our mechanism is instead a repeated-inertia permutation of ordered Frobenius data, coupled with a one-sided Satake cone. Both themes deserve study, but they should not be merged without a theorem connecting their moduli problems.

9.4. What This Work Does Not Construct

There is no categorical action of ${\mathcal{H}}_{{\mathrm{GSp}}_4}\left(\sigma \right)$ on an automorphic representation category in this paper. The formal action of global functions on quasi-coherent sheaves on a stack is a different and much weaker statement. Nor is there an assertion about all non-generic weights, about a canonical comparison morphism in the symplectic setting, or about patched local spectral coordinates. The abstract inequality of transcendence degrees is already sufficient for the main result.

9.5. A Criterion for Other Groups

The proof suggests a reusable sufficient mechanism. Suppose an irreducible component admits a schematically dense parabolic-extension family; suppose the unframed Levi scales the extension coordinates so that global functions restrict injectively to the Levi base; and suppose a repeated-inertia permutation preserves the semisimplified objects while the Levi Satake image is confined to a proper polyhedral cone. Then the algebra of global functions may be confined to the monoid algebra of the intersection of that cone with its permutation images. When this invariant intersection has smaller rank than the spherical Hecke monoid, an abstract Satake mismatch follows. The geometric hypotheses must be verified anew for each group and parabolic; this observation is a strategy, not an additional blanket theorem.

Appendix C.1. A Divisor Is Not a Dense Source

Set $A=k[u^{\pm 1},v^{\pm 1}]$. The restriction to the “fixed parameter” hypersurface $v=1$ is the quotient map

$$A⟶A/(v-1)\simeq k\left[u^{\pm 1}\right].$$

It has nonzero kernel $(v-1)$. In particular, a function may vanish on the whole fixed slice without vanishing on the two-parameter torus. This gives an elementary model of the danger of replacing Lee’s full non-split $U_{{\sigma }^{\prime }}$ by his codimension-one $t_2=1$ locus in the otherwise branch. The extra unramified twisting parameter of the Siegel family does not repair the missing injectivity by itself: one must know the actual scheme-theoretic image of the chosen source.

Appendix C.2. Set-theoretic Density Is Too Weak on Nonreduced Targets

Consider $X=Speck\left[\epsilon \right]/\left({\epsilon }^2\right)$ and the unique reduced closed subscheme $E=Speck$. The map $E\to X$ is surjective on underlying topological spaces, but its pullback on functions kills the nonzero nilpotent $\epsilon$. Hence it is not schematically dense. In contrast, the component used in this paper is reduced, and more importantly the chosen defining Siegel family is known to have the component itself as its scheme-theoretic image. This guarantees injectivity even before one uses contraction. For general nonreduced deformation stacks the same argument needs the stronger schematic-image hypothesis.

Appendix C.3. Regularity on an Open Need Not Extend to Its Closure

The regular function $x^{-1}$ on ${\mathbf{G}}_m={\mathbf{A}}^1\setminus \left\{0\right\}$ does not extend to ${\mathbf{A}}^1$. Thus an injection into $\Gamma (U_{{\sigma }^{\prime }}\times {\mathbf{G}}_m,\mathcal{O})$ would not imply an injection into $\Gamma ({\mathcal{C}}_{{\sigma }^{\prime }}^{\prime }\times {\mathbf{G}}_m,\mathcal{O})$ merely because $U_{{\sigma }^{\prime }}$ is dense. The split Levi map (19), combined with the closed-target principle, is the additional geometric ingredient that justifies replacing functions on the open by functions on the complete reduced rank-two component.

Appendix C.4. An Upper Bound Never Gives Descent of Its Generators

The inclusion

$$k[D^2,z^{\pm 1}]⊊k[D,z^{\pm 1}]$$

shows that a two-variable polynomial–Laurent upper bound can have a proper subalgebra with the same transcendence degree. Likewise $k\left[z^{\pm 1}\right]\subset k[D,z^{\pm 1}]$ has strictly smaller transcendence degree. Neither ring-theoretic possibility is excluded by an injection alone. More delicate normality and fraction-field arguments can sometimes distinguish such subrings, but they require input about the actual global-function image. Consequently one cannot conclude either the exact image of (2) or the existence of named global symplectic generators from the present theorem.

Appendix C.5. The Cyclotomic and Tr Ès-Ramifi Ée Endpoints Are Distinct

Two ways to leave our family should not be conflated. At $d=0$, the Siegel graded inertial character becomes $\omega$, so the easy $H^2$ exclusion fails, even though the weight label is still in the same broad p-restricted chamber. At $d=p-1$ the graded character becomes $\omega$ again, but an independent geometric change also occurs: Lee’s source switches from the full $U_{{\sigma }^{\prime }}$ to a fixed-parameter locus. Both endpoints are excluded from our uniform theorem. At the included endpoint $d=p-2$, by contrast, ${\omega }^{d+1}=1$ and the full source remains available. This triage explains why the precise interval $1\le d\le p-2$ is the natural one for the argument.