Local Converse Problem Overview

• Local Converse Problem is a framework that uses local gamma-factors to determine the isomorphism classes of irreducible generic representations in non-archimedean fields.
• It employs analytic and algebraic methods, including the construction of special Whittaker functions, Bessel functions, and mirabolic comparisons to compare Rankin–Selberg integrals.
• Extensions of the problem encompass classical, unitary, and metaplectic groups, linking its techniques to Langlands-parameter formulations and deepening its impact on representation theory.

Searching arXiv for papers on the local converse problem and related converse theorems. The local converse problem asks how far a family of local factors attached to twists of a representation determines its isomorphism class. In the standard formulation for a non-archimedean local field $F$, one fixes a nontrivial additive character $\psi$, lets $G_n=\mathrm{GL}_n(F)$, and studies whether an irreducible admissible generic representation $\pi$ is determined by the local $\gamma$-factors $\gamma(s,\pi\times\tau,\psi)$ as $\tau$ ranges over generic representations of smaller general linear groups. For $p$-adic $\mathrm{GL}_n$, this was crystallized by Jacquet’s conjecture: twists by $\mathrm{GL}_r(F)$ for $\psi$0 should suffice. The subject now includes proofs for $\psi$1-adic $\psi$2, archimedean analogues, $\psi$3-adic families, classical and metaplectic groups, and parameter-theoretic reformulations in terms of Langlands parameters (Jiang et al., 2015, Jacquet et al., 2016, Matringe, 2024).

1. Formulation for $\psi$4 and the role of local factors

Let $\psi$5 be a non-archimedean local field, $\psi$6, $\psi$7 the standard upper-triangular unipotent subgroup, and $\psi$8 a nontrivial additive character. An irreducible smooth complex representation $\psi$9 of $G_n=\mathrm{GL}_n(F)$0 is called generic if

$G_n=\mathrm{GL}_n(F)$1

and its Whittaker model $G_n=\mathrm{GL}_n(F)$2 realizes $G_n=\mathrm{GL}_n(F)$3 by functions $G_n=\mathrm{GL}_n(F)$4 satisfying $G_n=\mathrm{GL}_n(F)$5 (Jacquet et al., 2016).

For an irreducible generic $G_n=\mathrm{GL}_n(F)$6 of $G_n=\mathrm{GL}_n(F)$7 and an irreducible generic $G_n=\mathrm{GL}_n(F)$8 of $G_n=\mathrm{GL}_n(F)$9, one forms Rankin–Selberg local integrals, for example

$\pi$0

and one obtains a functional equation

$\pi$1

The associated local $\pi$2- and $\pi$3-factors satisfy the usual relation with $\pi$4 (Jacquet et al., 2016, Adrian et al., 2014).

Jacquet’s local converse conjecture may then be stated as follows: if $\pi$5 are irreducible generic representations of $\pi$6 with the same central character and

$\pi$7

for every irreducible generic $\pi$8 of $\pi$9 with $\gamma$0, then $\gamma$1 (Adrian et al., 2014, Jacquet et al., 2016). A standard reduction passes from general generic representations to the unitarizable supercuspidal case by the Zelevinsky classification and multiplicativity of $\gamma$2-factors (Jiang et al., 2015, Chai, 2016).

2. Jacquet’s conjecture for $\gamma$3-adic $\gamma$4

A decisive step was the Jiang–Nien–Stevens approach, which isolates the notion of a special pair of Whittaker functions. If $\gamma$5 is compact-mod-centre open, a Whittaker function $\gamma$6 is called $\gamma$7-special if $\gamma$8 and $\gamma$9 for all $\gamma(s,\pi\times\tau,\psi)$0. For two unitarizable supercuspidals $\gamma(s,\pi\times\tau,\psi)$1 with the same central character, a pair $\gamma(s,\pi\times\tau,\psi)$2 is a special pair if both are $\gamma(s,\pi\times\tau,\psi)$3-special for the same $\gamma(s,\pi\times\tau,\psi)$4 and $\gamma(s,\pi\times\tau,\psi)$5 for all $\gamma(s,\pi\times\tau,\psi)$6 in the mirabolic subgroup $\gamma(s,\pi\times\tau,\psi)$7 (Jiang et al., 2015, Adrian et al., 2014).

The corresponding reduction theorem states that if $\gamma(s,\pi\times\tau,\psi)$8 are irreducible unitarizable supercuspidal representations of $\gamma(s,\pi\times\tau,\psi)$9, if they satisfy the hypotheses

$\tau$0

for all irreducible supercuspidal $\tau$1 of $\tau$2 and all $\tau$3, and if there exists a special pair of Whittaker functions for $\tau$4, then $\tau$5 (Adrian et al., 2014). This converts the converse problem into a problem of constructing distinguished Whittaker data.

Adrian–Liu–Stevens–Xu proved that any two minimax unitarizable supercuspidals of $\tau$6 that have the same depth and central character admit a special pair of Whittaker functions. As a corollary, Jacquet’s conjecture holds for $\tau$7 when $\tau$8 is prime, because every supercuspidal $\tau$9 of $p$0 is, up to an unramified twist, either depth zero or minimax of positive depth, and the depth-zero case had already been handled in earlier work (Adrian et al., 2014).

The full theorem for generic representations of $p$1-adic $p$2 was then proved in two distinct ways. Jacquet–Liu gave a purely analytic proof: one decomposes $p$3 into double cosets $p$4, compares Whittaker functions height-by-height, and uses only the formal properties of Rankin–Selberg integrals, their analytic continuation, and uniqueness of Whittaker models. Their refined statement is that twists up to $p$5 suffice (Jacquet et al., 2016). Independently, Chai proved Jacquet’s local converse conjecture for $p$6 using Bessel functions: equality of twisted $p$7-factors implies equality of Bessel functions $p$8, and the weak kernel formula then forces $p$9 (Chai, 2016).

3. Principal proof strategies

Three proof patterns recur throughout the subject. The first is the special-Whittaker-function strategy. Its logic is rigid: construct Whittaker functions with controlled support and symmetry, force equality of their Rankin–Selberg zeta integrals by the assumed equality of $\mathrm{GL}_n$0-factors, and then recover equality of the Whittaker functions themselves. In the $\mathrm{GL}_n$1 setting, this is tied to maximal simple types, endo-classes, and the explicit Bessel functions of Paškūnas–Stevens (Jiang et al., 2015, Adrian et al., 2014).

The second is the Bessel-function and Howe-vector strategy. For a generic representation $\mathrm{GL}_n$2, one defines a Bessel function $\mathrm{GL}_n$3 by a stabilized integral over the maximal unipotent subgroup. One then constructs normalized Howe vectors $\mathrm{GL}_n$4, proves strong support and vanishing properties on Bruhat cells, inserts them into local integrals, and derives equalities such as

$\mathrm{GL}_n$5

for diagonal $\mathrm{GL}_n$6. Letting $\mathrm{GL}_n$7 and invoking a kernel formula yields $\mathrm{GL}_n$8, hence $\mathrm{GL}_n$9 (Chai, 2016).

The third is the analytic mirabolic-comparison method. Jacquet–Liu reduce the problem to showing that two Whittaker models coincide on the mirabolic subgroup $\mathrm{GL}_r(F)$0, decompose $\mathrm{GL}_r(F)$1 into double cosets $\mathrm{GL}_r(F)$2, and prove equality layer by layer by averaging the functional equation over suitable translates. A specific simplification in this approach is that no explicit construction of supercuspidals or of special Whittaker pairs is needed (Jacquet et al., 2016).

A common misconception is that the local converse theorem for $\mathrm{GL}_r(F)$3 intrinsically requires twists up to $\mathrm{GL}_r(F)$4. That bound appears in older formulations, but the refined theorem proves that twists up to $\mathrm{GL}_r(F)$5 are sufficient (Jacquet et al., 2016). Another common misconception is that all proofs are representation-theoretically uniform. The literature instead contains substantially different mechanisms: special pairs, partial Bessel functions, and purely analytic Rankin–Selberg arguments (Jiang et al., 2015, Chai, 2016).

4. Extensions to classical, unitary, and metaplectic groups

The local converse problem extends beyond $\mathrm{GL}_r(F)$6 by replacing the twisting representations with generic or supercuspidal representations of $\mathrm{GL}_r(F)$7, and by defining the relevant local $\mathrm{GL}_r(F)$8-factors through Rankin–Selberg or Langlands–Shahidi methods. The resulting theorems are group-specific, and the precise twisting range is part of the statement.

Setting	Twists used	Conclusion
$\mathrm{GL}_r(F)$9	characters $\psi$00, $\psi$01	$\psi$02 (Zhang, 2015)
$\psi$03	supercuspidal $\psi$04 of $\psi$05, $\psi$06	$\psi$07 (Zhang, 2015)
$\psi$08	generic $\psi$09 of $\psi$10, $\psi$11	$\psi$12 (Zhang, 2017)
$\psi$13	generic $\psi$14 of $\psi$15, $\psi$16	$\psi$17 (Zhang, 2017)
$\psi$18, $\psi$19, char$\psi$20	supercuspidal $\psi$21 of $\psi$22, $\psi$23	$\psi$24 (Jo, 2022)
$\psi$25	supercuspidal generic $\psi$26 of $\psi$27, $\psi$28	$\psi$29 (Haan, 2022)
split $\psi$30	generic $\psi$31 of $\psi$32, $\psi$33	$\psi$34 or $\psi$35 (Hazeltine et al., 2023)
quasi-split non-split $\psi$36	generic $\psi$37 of $\psi$38, $\psi$39	$\psi$40 or $\psi$41 (Hazeltine, 13 Jan 2025)
$\psi$42, refined form	$\psi$43-, $\psi$44-twists and $\psi$45	$\psi$46 (Yan et al., 2023)

The principal technical devices in these results are Howe vectors, partial Bessel functions, local zeta integrals, and vanishing lemmas of Jacquet–Shalika type. This suggests that the $\psi$47 methods based on Whittaker models and Bessel distributions were not isolated phenomena, but part of a broader local-analytic pattern (Zhang, 2017, Zhang, 2017, Jo, 2022).

Even special orthogonal groups exhibit a specific obstruction: the outer automorphism $\psi$48. For split $\psi$49 and quasi-split non-split $\psi$50, the standard twisted $\psi$51-factors determine an irreducible generic representation only up to outer conjugacy (Hazeltine et al., 2023, Hazeltine, 13 Jan 2025). In rank $\psi$52, Yan–Zhang showed that an additional twisted exterior square local $\psi$53-factor resolves this ambiguity for $\psi$54 (Yan et al., 2023).

The metaplectic case is conceptually different. Haan establishes the local converse theorem for $\psi$55 by transferring the problem through the precise local theta correspondence to $\psi$56, using preservation of genericity and preservation of $\psi$57-factors under theta lift (Haan, 2022).

5. Families and archimedean analogues

The converse problem also admits a deformation-theoretic version. Moss extended Rankin–Selberg integrals to $\psi$58-adic families of smooth representations of $\psi$59, with coefficient rings $\psi$60 and $\psi$61, and constructed $\psi$62-factors

$\psi$63

characterized by a family functional equation. In this setting, equality of $\psi$64-factors against all absolutely irreducible generic integral twists of rank $\psi$65 determines the same supercuspidal support (Moss, 2014).

Liu–Moss then formulated a Jacquet-type theorem for co-Whittaker $\psi$66-families. If $\psi$67 is reduced, $\psi$68-torsion-free, and finite-type over $\psi$69, and if two co-Whittaker $\psi$70-modules with the same central character have matching $\psi$71-factors against every irreducible generic integral representation $\psi$72 of $\psi$73 for all $\psi$74, then they have the same Whittaker model, equivalently the same supercuspidal support. The same paper proves a descent theorem characterizing the smallest subring of definition by the coefficients of finitely many $\psi$75-factors (Liu et al., 2017).

Over archimedean fields, the problem takes a different form because the local factors are explicit products of classical $\psi$76-functions. Adrian–Takeda first proved that over $\psi$77, $\psi$78-twists of local $\psi$79-factors determine an irreducible admissible representation of $\psi$80, while over $\psi$81 one needs twists by $\psi$82 and $\psi$83 (Adrian et al., 2017). They later proved a local converse theorem for archimedean $\psi$84 in terms of twisted local $\psi$85-factors: if $\psi$86 are generic irreducible admissible representations of $\psi$87, $\psi$88 or $\psi$89, with the same central character, and

$\psi$90

for every unitary character $\psi$91, then $\psi$92 (Adrian et al., 2023). The proof proceeds on the Weil-group side by a pole comparison for explicit $\psi$93-factors.

6. Langlands-parameter formulations and sharpness

A recent reformulation treats the converse problem directly at the level of Langlands parameters. Let $\psi$94 be quasi-split, $\psi$95, and let $\psi$96 be an algebraic representation. For an admissible homomorphism $\psi$97 and a representation $\psi$98 of $\psi$99, one forms

$G_n=\mathrm{GL}_n(F)$00

Matringe proves a parameter-level local converse theorem when $G_n=\mathrm{GL}_n(F)$01 is acceptable: equality of these twisted $G_n=\mathrm{GL}_n(F)$02-factors for every irreducible algebraic representation $G_n=\mathrm{GL}_n(F)$03 and every irreducible $G_n=\mathrm{GL}_n(F)$04 forces the semisimple parts of the parameters to be $G_n=\mathrm{GL}_n(F)$05-conjugate. When $G_n=\mathrm{GL}_n(F)$06 is $G_n=\mathrm{GL}_n(F)$07-split, this yields full equality of generic parameters; for split semisimple simply-connected $G_n=\mathrm{GL}_n(F)$08, it suffices to use the fundamental representations $G_n=\mathrm{GL}_n(F)$09 and twists of dimension at most

$G_n=\mathrm{GL}_n(F)$10

The same note gives variants for $G_n=\mathrm{GL}_n(F)$11 and quasi-split classical groups (Matringe, 2024).

Sharpness questions then ask whether the standard twisting range can be reduced. Adrian–Stevens show that when $G_n=\mathrm{GL}_n(F)$12 is a symplectic or special orthogonal group, or the exceptional group $G_n=\mathrm{GL}_n(F)$13, and the residue characteristic $G_n=\mathrm{GL}_n(F)$14 is large enough, the optimal standard local converse theorem requires twisting by representations of $G_n=\mathrm{GL}_n(F)$15 with $G_n=\mathrm{GL}_n(F)$16 up to half the dimension of the standard representation of the dual group (Adrian et al., 26 Sep 2025). For generic supercuspidal representations of $G_n=\mathrm{GL}_n(F)$17, there is an improvement when $G_n=\mathrm{GL}_n(F)$18 is odd: twisting up to $G_n=\mathrm{GL}_n(F)$19 suffices, and this bound is itself optimal (Adrian et al., 26 Sep 2025).

The same paper also gives counterexamples to possible improvements based on non-standard representations of the dual group. For $G_n=\mathrm{GL}_n(F)$20, there exist inequivalent irreducible supercuspidal parameters $G_n=\mathrm{GL}_n(F)$21 such that all twisted $G_n=\mathrm{GL}_n(F)$22-factors

$G_n=\mathrm{GL}_n(F)$23

agree for every character $G_n=\mathrm{GL}_n(F)$24 and every $G_n=\mathrm{GL}_n(F)$25. Analogous counterexamples are given for $G_n=\mathrm{GL}_n(F)$26 and $G_n=\mathrm{GL}_n(F)$27 (Adrian et al., 26 Sep 2025). A plausible implication is that the optimality problem is not merely a matter of enlarging the repertoire of dual-side algebraic representations; the specific twisting range is itself a structural invariant of the local converse theorem.

The modern picture is therefore stratified. For $G_n=\mathrm{GL}_n(F)$28-adic $G_n=\mathrm{GL}_n(F)$29, Jacquet’s conjectural bound $G_n=\mathrm{GL}_n(F)$30 is exact in the standard formulation (Jacquet et al., 2016, Chai, 2016). For many classical groups, one has direct local converse theorems with twists by $G_n=\mathrm{GL}_n(F)$31 up to the natural half-rank bound, but even orthogonal groups retain an outer-automorphism phenomenon unless extra twisted factors are introduced (Hazeltine et al., 2023, Yan et al., 2023). At the same time, parameter-level results show that the subject fits naturally into the structure of the local Langlands correspondence (Matringe, 2024).