Whittaker Normalization Conjecture

• Whittaker Normalization Conjecture is a framework in representation theory that standardizes packet labeling through canonical adjustments when the Whittaker datum changes.
• It connects p-adic, automorphic, and geometric settings by comparing normalized vanishing cycles and microlocal parametrizations with classical Langlands and Arthur correspondences.
• The conjecture implies that while the set of representation packets remains invariant, their labeling twists by a canonical character determined by changes in Whittaker data.

Searching arXiv for the cited works and closely related papers on Whittaker normalization in the ABV, automorphic, and geometric settings. The phrase Whittaker Normalization Conjecture is used for several closely related normalization problems in representation theory and automorphic forms. In the $p$-adic ABV setting, it denotes the statement that ABV-packets are independent of the choice of Whittaker datum, while the parametrization produced by normalized vanishing cycles changes by tensoring with a canonical character determined by Kaletha’s change-of-Whittaker-data pairing; for open parameters this parametrization agrees with the local Langlands correspondence, and for tempered parameters it agrees with Arthur’s parametrization, yielding Vogan’s conjecture on A-packets for tempered representations (Cunningham et al., 2024).

1. Whittaker data and packet normalization

Let $F$ be a non-archimedean local field and $G/F$ a connected quasi-split reductive group. Fix a Borel subgroup $B$ with unipotent radical $U$. A character $\theta:U(F)\to \mathbb C^\times$ is generic if its stabilizer in $T(F)=B(F)/U(F)$ is $Z(F)$, the center of $G(F)$. A representation $\pi$ of $F$0 is $F$1-generic if

$F$2

A Whittaker datum is a $F$3-conjugacy class of pairs $F$4 with $F$5 generic. The set of Whittaker data is a principal homogeneous space for $F$6. Equivalently, after fixing $F$7, the $F$8-orbits of generic characters form a torsor under $F$9, canonically identified with $G/F$0 (Cunningham et al., 2024).

In the LLC normalization, a choice of Whittaker datum $G/F$1 gives a canonical bijection

$G/F$2

for the pure L-packet, including pure inner forms. Writing $G/F$3, one obtains a Whittaker-normalized labeling of the packet by irreducible representations of the component group (Cunningham et al., 2024).

In the ABV construction, the same choice of $G/F$4 enters through the map from the pure L-packet to simple $G/F$5-equivariant perverse sheaves on the Vogan variety,

$G/F$6

and the ABV-packet is defined by the nonvanishing of normalized vanishing cycles:

$G/F$7

The associated parametrization is

$G/F$8

The normalization problem is therefore not whether a Whittaker datum appears, but exactly how its change affects the resulting parametrization (Cunningham et al., 2024).

2. Microlocal structure of the ABV parametrization

For the infinitesimal parameter $G/F$9, the Vogan variety is

$B$0

with $B$1 acting on it. The Langlands parameter $B$2 determines an $B$3-orbit $B$4 (Cunningham et al., 2024).

Above $B$5 sits the microlocal conormal subset

$B$6

together with its open connected part $B$7. For readability, write $B$8 (Editor’s term) for the microlocal equivariant fundamental group

$B$9

The natural $U$0-equivariant projection $U$1 identifies the classical component group with

$U$2

and induces a functor

$U$3

Thus ABV parametrization is microlocal rather than merely component-group-theoretic (Cunningham et al., 2024).

If $U$4, then $U$5 is the simple $U$6-equivariant perverse sheaf $U$7 corresponding to $U$8 under

$U$9

Applying normalized vanishing cycles yields

$\theta:U(F)\to \mathbb C^\times$0

This is the map whose Whittaker dependence is measured exactly in the main theorem (Cunningham et al., 2024).

3. Independence of packets and dependence of parametrizations

The principal theorem states that, assuming the standard LLC desiderata, the ABV-packet attached to a Langlands parameter $\theta:U(F)\to \mathbb C^\times$1 is independent of the Whittaker datum. The dependence on Whittaker normalization is therefore not a dependence of the set of representations, but of the labeling map to microlocal fundamental-group representations (Cunningham et al., 2024).

More precisely, for two Whittaker data $\theta:U(F)\to \mathbb C^\times$2, there is a canonical character $\theta:U(F)\to \mathbb C^\times$3 such that

$\theta:U(F)\to \mathbb C^\times$4

Equivalently,

$\theta:U(F)\to \mathbb C^\times$5

This is the exact Whittaker-dependence formula proved in the paper (Cunningham et al., 2024).

The character $\theta:U(F)\to \mathbb C^\times$6 on $\theta:U(F)\to \mathbb C^\times$7 is canonically determined from Kaletha’s bijection

$\theta:U(F)\to \mathbb C^\times$8

together with the difference $\theta:U(F)\to \mathbb C^\times$9 and the map from $T(F)=B(F)/U(F)$0 into $T(F)=B(F)/U(F)$1. Passing through

$T(F)=B(F)/U(F)$2

produces the microlocal twist. The same construction is compatible with the LLC normalization:

$T(F)=B(F)/U(F)$3

Thus the ABV change-of-normalization law is the microlocal avatar of the familiar LLC change-of-Whittaker-data formula (Cunningham et al., 2024).

The examples recorded in the paper clarify the range of this dependence. For $T(F)=B(F)/U(F)$4, the Whittaker datum is unique up to $T(F)=B(F)/U(F)$5-conjugacy, so the quotient $T(F)=B(F)/U(F)$6 is trivial and there is no dependence. For $T(F)=B(F)/U(F)$7, $T(F)=B(F)/U(F)$8, and $T(F)=B(F)/U(F)$9, the change of datum is controlled by Kaletha’s canonical pairing, producing the twist on $Z(F)$0 and then on $Z(F)$1; the paper uses the general construction rather than spelling out group-specific closed formulas (Cunningham et al., 2024).

4. Open parameters, genericity, and Arthur type

A parameter $Z(F)$2 is open if its orbit $Z(F)$3 is open. It is tempered if its restriction to $Z(F)$4 is bounded. It is of Arthur type if

$Z(F)$5

for some Arthur parameter $Z(F)$6. An ABV-packet is generic relative to a Whittaker datum if it contains a representation admitting a Whittaker model; for a fixed $Z(F)$7, such a generic member is unique inside an L-packet under the LLC desiderata (Cunningham et al., 2024).

The paper isolates two structural equivalences. First,

$Z(F)$8

Second,

$Z(F)$9

The latter reformulates the Gross–Prasad–Rallis criterion for genericity in geometric terms. The conjectural statement denoted “GP2” is that an L-packet is generic if and only if $G(F)$0 is open, while the ABV genericity conjecture asserts that the ABV-packet contains a generic representation if and only if $G(F)$1 is open (Cunningham et al., 2024).

For quasi-split classical groups, and also for $G(F)$2, the paper proves that the packet is generic exactly when $G(F)$3 is open; moreover, the generic member lies in the quasi-split form. Under the LLC desiderata, the ABV genericity conjecture is proved for quasi-split classical groups and their pure inner forms. In this regime, openness is the decisive geometric condition linking genericity, adjoint $G(F)$4-functions, and Arthur-type structure (Cunningham et al., 2024).

5. Open and tempered regimes: agreement with LLC and Arthur

When $G(F)$5 is open, the ABV and LLC parametrizations coincide completely. The theorem called “mainopen” states that the ABV-packet equals the pure L-packet, the microlocal equivariant fundamental group collapses to the classical component group,

$G(F)$6

and the normalized vanishing-cycles map is exactly the LLC map:

$G(F)$7

Equivalently, for every $G(F)$8 in the packet,

$G(F)$9

Open parameters are therefore the range in which the ABV construction recovers the Whittaker-normalized local Langlands parametrization without residue (Cunningham et al., 2024).

For tempered parameters, the result becomes an Arthur-packet statement. Since tempered parameters are precisely the open parameters of Arthur type, if $\pi$0 is a tempered Arthur parameter with corresponding $\pi$1, then for quasi-split classical groups, or more generally for groups for which Arthur’s conjectures are known,

$\pi$2

In formula form,

$\pi$3

This is the announced proof of Vogan’s conjecture on A-packets for tempered representations (Cunningham et al., 2024).

The scope of these theorems is explicit. The ambient assumption is that $\pi$4 is a connected reductive group over a $\pi$5-adic field $\pi$6; for quasi-split classical groups and their pure inner forms, the requisite LLC and A-packets are supplied by Arthur, Mok, and KMSW, while for $\pi$7 they are supplied by the known LLC. The paper assumes the LLC desiderata and concentrates on microlocal geometry and compatibility with Langlands and Arthur parametrizations; endoscopic transfer factors and stability are not developed there (Cunningham et al., 2024).

6. Related formulations in automorphic, geometric, categorical, and finite-ring settings

The expression “Whittaker normalization conjecture” also appears in several adjacent settings, all centered on the problem of turning a Whittaker model, Whittaker coefficient, or Whittaker datum into a canonical normalization.

Setting	Normalized object	Representative statement
$\pi$8-adic ABV/LLC	Packet parametrization under change of Whittaker datum	Packet independent of $\pi$9; labeling twisted canonically (Cunningham et al., 2024)
Global automorphic periods	Local/global Whittaker functionals and measures	Conjectural constant $F$00; metaplectic local constants identified with $F$01 at finite places (Lapid et al., 2013, Lapid et al., 2014)
Geometric Langlands	Whittaker coefficient functors and Hecke eigensheaf normalization	Existence of Whittaker-normalized eigensheaves with $F$02 (Faergeman et al., 2022)
Loop group categories	$F$03-twisted invariants versus coinvariants	Conjectural equivalence $F$04; proved for $F$05 (Beraldo, 2013)
Finite local rings	Degenerate Whittaker space as a canonically normalized $F$06-module	For strongly cuspidal $F$07 of $F$08, $F$09 (Parashar et al., 2024)

In the global automorphic setting, Lapid–Mao formulate a Whittaker analogue of the Ichino–Ikeda conjecture. The normalization consists of canonical local Whittaker relative characters and measure conventions, characterized at unramified places by

$F$10

together with a conjectural global constant

$F$11

For the metaplectic double cover, the $F$12-adic local normalization is sharpened to

$F$13

which makes the global Whittaker period formula canonical and unramified local factors equal to $F$14 (Lapid et al., 2013, Lapid et al., 2014).

In geometric Langlands, the Whittaker coefficient functor is interpreted microlocally, and the enhanced Whittaker functor $F$15 is conservative on the tempered category. The paper proves that Whittaker-normalized Hecke eigensheaves exist for irreducible $F$16-local systems, are perverse, and are irreducible on each connected component of $F$17 when the local system is very irreducible. Here normalization plays the role of fixing the automorphic eigensheaf by requiring its first Whittaker coefficient to equal the spectral skyscraper, up to the intrinsic shift by $F$18 (Faergeman et al., 2022).

In categorical local geometric representation theory, Beraldo defines Whittaker invariant and coinvariant categories for an $F$19-action on a DG category and constructs normalization functors

$F$20

The conjecture is that $F$21 is an equivalence whenever the $F$22-action extends to a $F$23-action; this is proved for $F$24 by combining Fourier–Deligne transform on Tate vector spaces with explicit pro-unipotent stabilizers $F$25 (Beraldo, 2013).

Further representation-theoretic evidence for canonical Whittaker normalization appears in characteristic $F$26. For irreducible $F$27-compact representations, one has

$F$28

and for $F$29-generic irreducibles,

$F$30

Together with multiplicity-at-most-one, these statements isolate the uniqueness and contragredient compatibility that a Whittaker normalization is expected to encode (Matringe et al., 2022).

A more distant but structurally analogous usage occurs for $F$31-Whittaker functions, where normalization is fixed by the rank-one condition

$F$32

by the Baxter-operator recursion, and by the Sklyanin measure that makes the $F$33-Whittaker transform unitary. In that context the normalized Whittaker functions satisfy completeness and orthogonality relations analogous to a Plancherel theorem (Schrader et al., 2018).

Taken together, these formulations suggest a common pattern: Whittaker normalization is the process of converting a generic character or Whittaker coefficient into a canonical parametrization, scalar normalization, or categorical equivalence. What varies from one setting to another is the object being rigidified—packets, periods, eigensheaves, twisted invariants, or degenerate Whittaker spaces—whereas the organizing principle is consistently the same.