Local Langlands functoriality for Yu's supercuspidals II: Fargues--Scholze's parametrization

Abstract: This is the second of two papers dedicated to the explicit computation of the Fargues--Scholze correspondence. We compute the Tate cohomology of cuspidal representations arising from Yu's construction. Combining this with modular functoriality in the Local Langlands Correspondence, and the partial characterization of the Local Langlands Correspondence established in the first paper, we compare the Fargues--Scholze and Kaletha parametrizations. Among the consequences (and under an assumption expected to be supplied in forthcoming work), we deduce that the Fargues--Scholze correspondence has finite fibers and is surjective onto inertial L-parameters.

• The paper establishes the equivalence of the Fargues-Scholze inertial parameter and the Kaletha inertial parameter for tame cuspidal representations, extending the comparison to discern singular representations, showing the inertial comparability of both parameters under specific conditions.
• And with the assumption that the hypothesis simplifies to allow parameter comparison on inertia and non-singular parameters into Weyl groups.
• Yu's representations' advanced handling and simplifying will lead us to the main contribution of the finding regarding significant equivalence between parameters.

Scope and mathematical setting

This paper is the second part of a two-paper project computing the Fargues–Scholze parametrization for tame cuspidal representations constructed by Yu. Let $F$ be a non-archimedean local field of residue characteristic $p$, let $G$ be a connected reductive $F$-group splitting over a tamely ramified extension, and let $\ell\neq p$. The paper works primarily under $p\neq 2$ and, in its principal comparison theorem, under the additional assumption that $p$ does not divide the order of the absolute Weyl group.

The central objects are three versions of local Langlands data attached to a Yu datum $\Psi$ and its associated cuspidal representation $\pi(\Psi)$:

• the Fargues–Scholze parameter $\rho^{\mathrm{FS}}(\pi)$;
• Kaletha’s explicit inertial parameter $p$0;
• when $p$1 is non-singular, Kaletha’s full parameter $p$2.

The main objective is to identify the first two on inertia and, under non-singularity, to compare the full Weil-group parameters after a controlled unramified extension. The proof is entirely local and representation-theoretic. Its decisive input is modular functoriality: Tate cohomology in characteristic $p$3 is used to realize descent and base change, after which independence-of-$p$4 arguments recover characteristic-zero statements.

Main comparison theorem

The principal result is conditional on modular functoriality for cyclic base change at every prime $p$5. Under this hypothesis, let $p$6 be a normalized irreducible Yu datum with associated tame cuspidal representation $p$7. Choose a torus-character pair $p$8 attached to $p$9, let $G$0 be the maximal unramified subtorus of $G$1, and let $G$2 be the ramification degree of the splitting field of

$G$3

Assume that $G$4 has degree prime to $G$5.

The paper proves

$G$6

This identifies the Fargues–Scholze inertial parameter with the explicit Kaletha parameter for every tame cuspidal representation in the stated range, not merely for non-singular representations. The extension from non-singular to arbitrary cuspidals is important: singular representations generally do not admit a full explicit Kaletha parameter of the same form, but their inertial data remain accessible.

Let

$G$7

The paper also proves that $G$8 is non-singular if and only if $G$9 is a torus. If $F$0 is non-singular and $F$1 is good for $F$2, then there is an integer $F$3 determined by the relevant cocharacter lattices such that

$F$4

The two parameters can be chosen to agree on inertia and to induce the same conjugation action on $F$5.

The integer $F$6 measures the residual ambiguity in extending the common inertial parameter to $F$7. It is often small; in the type-$F$8 situation, the paper notes that it is at most $F$9. Thus the discrepancy between the two parametrizations is not uncontrolled: it is confined to an explicitly bounded unramified extension and, in the full comparison, to an unramified central cocycle.

Modular functoriality and spectral endoscopy

The paper’s technical architecture is built around modular functoriality. Suppose an automorphism $\ell\neq p$0 of $\ell\neq p$1 has prime order $\ell\neq p$2 and connected fixed-point group

$\ell\neq p$3

The modular functoriality conjecture asserts that if an irreducible representation $\ell\neq p$4 of $\ell\neq p$5 is $\ell\neq p$6-stable and $\ell\neq p$7 occurs in its Tate cohomology, then

$\ell\neq p$8

where $\ell\neq p$9 is the $p\neq 2$0-dual homomorphism.

The paper emphasizes that this phenomenon is intrinsically modular. The prime $p\neq 2$1 simultaneously controls the order of $p\neq 2$2, the characteristic of the coefficient field, and the relevant Tate cohomology. The resulting $p\neq 2$3-dual homomorphism may have no characteristic-zero analogue. A representative example is

$p\neq 2$4

in characteristic $p\neq 2$5, arising from the fixed points

$p\neq 2$6

The existence of this map is a specifically characteristic-$p\neq 2$7 phenomenon and illustrates why modular methods cannot simply be replaced by characteristic-zero endoscopy.

For inner automorphisms, the fixed-point group is a centralizer and the authors call the resulting construction spectral endoscopy. They establish compatibility of $p\neq 2$8-dual homomorphisms with field extensions, passage to Levis, products, central isogenies, and the relevant canonical $p\neq 2$9-embeddings of unramified twisted Levis. In the latter case, the comparison has the form

$p$0

where $p$1 is an unramified cocycle valued in an appropriate central subgroup. On inertia, the cocycle disappears, yielding the precise compatibility required for the main theorem.

A recurring issue is the distinction between $p$2-groups and ordinary pinned $p$3-groups. The map produced naturally by relative Tannakian formalism is untwisted at the level of $p$4-groups, whereas passage to $p$5-groups introduces a cocycle determined by the chosen square root of the cyclotomic character. The appendix gives an explicit formula for this coordinate change and explains the appearance of modulus-character factors in the usual $p$6-group embeddings.

Tate cohomology of tensor products

Yu’s representations are compact inductions of tensor products involving depth-zero cuspidal representations, Heisenberg representations, Weil representations, and sign characters. Consequently, the paper requires a precise analysis of Tate cohomology under tensor products.

A basic obstruction is that ordinary cup products in Tate cohomology can vanish even when the individual Tate cohomology groups are nonzero. The authors classify the relevant behavior using the Jordan decomposition of the order-$p$7 operator $p$8. For indecomposable modules of dimensions $p$9 and $\Psi$0, they obtain exact criteria for the nonvanishing of the three cup products: $\Psi$1 The resulting criteria are expressed by inequalities involving $\Psi$2, $\Psi$3, and $\Psi$4. In particular, the image dimensions are determined combinatorially by the Jordan block sizes.

Two classes of $\Psi$5-modules are especially important:

• minimal modules, of the form $\Psi$6;
• maximal modules, of the form

$\Psi$7

For a minimal first factor, the cup products $\Psi$8 and $\Psi$9 are isomorphisms. For a maximal second factor, $\pi(\Psi)$0 and $\pi(\Psi)$1 are isomorphisms. This yields a tensor-product theorem: if every auxiliary factor is extremal, then Tate cohomology of the full tensor product is the tensor product of the relevant Tate cohomology groups, with a parity shift determined by which factors are maximal.

This result is structurally essential. Without it, Tate cohomology of a Yu representation would not decompose into the individual depth-zero, Heisenberg, and Weil contributions needed to identify the descended representation.

Weil–Heisenberg representations and sign characters

The most delicate local calculation concerns Weil–Heisenberg representations. Let $\pi(\Psi)$2 be a finite-dimensional symplectic space over $\pi(\Psi)$3, let $\pi(\Psi)$4 have prime order $\pi(\Psi)$5, and write

$\pi(\Psi)$6

The authors prove that for either Tate degree,

$\pi(\Psi)$7

as representations of the fixed Heisenberg group. Moreover, the action of $\pi(\Psi)$8 on $\pi(\Psi)$9 is extremal as a $\rho^{\mathrm{FS}}(\pi)$0-module.

The fixed-point symplectic group does not generally act on Tate cohomology solely through the expected Weil–Heisenberg representation. There is an additional quadratic character $\rho^{\mathrm{FS}}(\pi)$1, supported on the non-fixed symplectic quotient. The paper computes this character using the character formula for Weil representations and decomposes it into sign characters indexed by Galois and duality orbits of weights.

The resulting formula is

$\rho^{\mathrm{FS}}(\pi)$2

This calculation explains one component of the Fintzen–Kaletha–Spice sign character. In particular, the sign $\rho^{\mathrm{FS}}(\pi)$3, which appears in the explicit local Langlands parametrization, is shown to arise from Tate cohomology of Weil–Heisenberg representations and ultimately from the sign computations of Gérardin.

The restriction-of-scalars compatibility of $\rho^{\mathrm{FS}}(\pi)$4 is also established. This is needed when passing between symplectic spaces over $\rho^{\mathrm{FS}}(\pi)$5 and their underlying spaces over $\rho^{\mathrm{FS}}(\pi)$6, as occurs in the Moy–Prasad quotients underlying Yu’s construction.

Tate cohomology of Yu representations

The preceding calculations are assembled into explicit modular base-change and descent results for Yu representations.

For a sufficiently large banal prime $\rho^{\mathrm{FS}}(\pi)$7, specifically under the condition

$\rho^{\mathrm{FS}}(\pi)$8

the paper constructs a Yu datum $\rho^{\mathrm{FS}}(\pi)$9 over the unramified extension $p$00 such that Tate cohomology of $p$01 contains the Frobenius twist of $p$02 in both Tate degrees. The resulting parameter identity is

$p$03

For toral data and a cyclic extension $p$04 of prime degree $p$05 such that $p$06 remains elliptic, an analogous result holds at small primes, subject to modular functoriality at $p$07 when $p$08 is not good for $p$09. This is the crucial small-degree input in the comparison theorem.

The paper also treats descent to unramified twisted Levis. If $p$10 is an unramified twisted Levi and the relevant torus is elliptic in the depth-zero subgroup, the authors construct a cuspidal Yu representation $p$11 such that

$p$12

The corresponding identity holds for Kaletha’s inertial parameters. If the original datum is non-singular, the descended datum is non-singular and the full Kaletha parameter satisfies

$p$13

The proof requires a substantial compatibility between Yu’s construction and parabolic induction. Because a modular representation extracted by Tate cohomology need not lift directly to a cuspidal characteristic-zero representation, the authors introduce intermediate compact-mod-center subgroups and use Kim–Yu and Ohara-type covers. The resulting proposition shows that a tame cuspidal representation for a Levi can be chosen so that its parabolic induction contains the desired representation of the ambient group. This resolves the lifting obstruction without assuming that every cuspidal modular representation has a cuspidal characteristic-zero lift.

Characteristic-zero recovery via independence of $p$14

The passage from modular computations to characteristic-zero statements uses two ingredients.

First, the paper proves compatibility of Fargues–Scholze parameters with reduction modulo $p$15: every irreducible constituent of the reduction of a characteristic-zero representation has parameter equal to the semisimplified reduction of the original parameter.

Second, Scholze’s independence-of-$p$16 theorem provides a common excursion-algebra character over a number field. Consequently, congruences established at several auxiliary primes can be compared inside a single characteristic-zero parameter space.

The proof of unramified-Levi functoriality uses this mechanism in a particularly elaborate way. An auxiliary large prime is selected so that:

1. an unramified extension supplies enough torsion in a chosen simple torus;
2. a second prime becomes the coefficient characteristic for Tate cohomology;
3. the relevant centralizers are connected;
4. all inertia images have order prime to the coefficient prime;
5. the modular functoriality hypotheses are available.

After applying modular descent in this auxiliary setting, independence of $p$17 transfers the result back to the original parameter. The argument is technically demanding but conceptually coherent: positive-characteristic Tate cohomology supplies the representation-theoretic comparison, while independence of $p$18 removes the auxiliary characteristic.

Consequences for depth, fibers, and inertial surjectivity

Under the global modular-functoriality hypothesis, if $p$19 splits after a tamely ramified extension and

$p$20

then the Fargues–Scholze correspondence preserves depth and has finite fibers.

Depth preservation follows from the explicit inertial calculation. The depth of the parameter is detected by the depth of the torus character $p$21 attached to a Yu datum, and the equality

$p$22

forces the parameter depth to equal the final Yu depth $p$23.

Finite fibers are obtained by reducing arbitrary irreducible representations to cuspidal representations of Levi subgroups using compatibility with parabolic induction. For cuspidal representations, the explicit inertia formula bounds:

• the possible tamely ramified twisted-Levi sequences;
• the relevant building vertices;
• the depth sequence;
• the orders of the generic characters;
• the depth-zero cuspidal representations with fixed central character.

These are all finite choices. Hence only finitely many Yu data can yield a fixed Fargues–Scholze parameter.

The paper also proves inertial surjectivity. Under the same hypotheses, every irreducible semisimple parameter

$p$24

is the inertial restriction of the Fargues–Scholze parameter of some irreducible smooth $p$25-representation: $p$26 For quasi-split $p$27, irreducibility of $p$28 is unnecessary, because parabolic induction realizes parameters normalizing proper parabolics.

This is a deliberately weaker assertion than full surjectivity onto Weil-group parameters. The proof generally identifies the parameter only after restriction to a finite unramified extension. When the inertial parameter factors through a maximal torus, however, the paper obtains the stronger conclusion

$p$29

Type A and full parameter equality

For groups whose unramified base change is $p$30, the paper strengthens the comparison substantially. If $p$31, $p$32 does not divide $p$33 for $p$34, and $p$35 is a non-singular normalized irreducible Yu datum, then

$p$36

over the full Weil group $p$37.

The proof embeds $p$38 into a reductive group whose unramified base change is $p$39. For $p$40-type groups, centralizers of subsets are connected, allowing the residual ambiguity between two parameters with the same inertia and the same abelian quotient to be reduced to a central cocycle. The type-A structure then eliminates the remaining ambiguity after composing with the embedding.

This argument is purely local and applies beyond the cases where the result follows from previously established comparisons with classical local Langlands correspondences. In particular, it covers inner forms of $p$41 and unramified special unitary groups in the stated range.

Limitations and open questions

The principal results remain conditional on modular functoriality for cyclic base change at every prime $p$42. The paper notes that the hypothesis is known for sufficiently large primes and for good primes in the currently available literature, while an all-prime proof is expected from a forthcoming revision of the underlying modular-functoriality work. Thus the main comparison and its finiteness and surjectivity consequences are not unconditional in the full stated generality.

The assumptions $p$43, tame splitting, and $p$44 are also substantive. The authors expect parts of the method to extend to more general constructions of cuspidal representations, including settings with small residue characteristic, but the especially difficult missing case is cyclic descent of degree $p$45, where the present modular Tate-cohomological arguments do not directly apply.

For unramified twisted-Levi functoriality, the full Weil-group comparison is stated with an unramified central cocycle. The authors expect this cocycle to be removable, but their current calculation of the $p$46-dual homomorphism does not establish that it is always a coboundary. This leaves open whether the displayed cocycle is an artifact of the chosen $p$47-group normalization or an intrinsic residual ambiguity.

Finally, inertial surjectivity does not generally imply surjectivity onto full L-parameters. The paper identifies a sufficient condition for full equality—toral inertial image—but does not resolve the general problem of choosing a representation whose entire Weil-group parameter equals a prescribed irreducible parameter.

Conclusion

The paper establishes an explicit comparison between the Fargues–Scholze and Kaletha parametrizations for tame cuspidal representations arising from Yu’s construction. Its main technical contribution is the calculation of Tate cohomology for the tensor factors occurring in Yu representations, especially Weil–Heisenberg representations and their sign characters. Modular functoriality, spectral endoscopy, parabolic-induction compatibility, and independence of $p$48 then combine to produce characteristic-zero functoriality.

Under the stated hypotheses, the Fargues–Scholze parameter agrees with Kaletha’s parameter on inertia, agrees with it on the full Weil group after a controlled unramified extension for non-singular data, preserves depth, has finite fibers, and is surjective onto irreducible inertial L-parameters. For groups of unramified type $p$49, the paper obtains full conjugacy without passing to an unramified extension.