---
title: Whittaker Normalization Conjecture
url: https://www.emergentmind.com/topics/whittaker-normalization-conjecture
type: topic
---

# Whittaker Normalization Conjecture

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 [2412.06824].

## 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 $G(F)$ is $\theta$-generic if
$$
\operatorname{Hom}_{U(F)}(\pi,\theta)\neq 0.
$$
A Whittaker datum is a $G(F)$-conjugacy class of pairs $(B,\theta)$ with $\theta$ generic. The set of Whittaker data is a principal homogeneous space for $G_{\mathrm{ad}}(F)/\operatorname{Im}(G(F))$. Equivalently, after fixing $B$, the $T(F)$-orbits of generic characters form a torsor under $T_{\mathrm{ad}}(F)/\operatorname{Im}(T(F))$, canonically identified with $G_{\mathrm{ad}}(F)/\operatorname{Im}(G(F))$ [2412.06824].

In the LLC normalization, a choice of Whittaker datum $\mathfrak w$ gives a canonical bijection
$$
J(\mathfrak w): \Pi_\phi(G)\to \operatorname{Irr}(A_\phi),
\qquad
A_\phi=\pi_0(Z_{\widehat G}(\phi)),
$$
for the pure L-packet, including pure inner forms. Writing $\iota_{\mathfrak w}=J(\mathfrak w)^{-1}$, one obtains a Whittaker-normalized labeling of the packet by irreducible representations of the component group [2412.06824].

In the ABV construction, the same choice of $\mathfrak w$ enters through the map from the pure L-packet to simple $H_\lambda$-equivariant perverse sheaves on the Vogan variety,
$$
\mathcal P_{\mathfrak w}:\Pi_\lambda(G)\to \operatorname{Per}_{H_\lambda}(V_\lambda)^{\mathrm{simple/iso}},
$$
and the ABV-packet is defined by the nonvanishing of normalized vanishing cycles:
$$
\Pi_{\phi,\mathfrak w}(G)=\{\pi\in \Pi_\lambda(G): \operatorname{NEvs}_\phi(\mathcal P_{\mathfrak w}(\pi))\neq 0\}.
$$
The associated parametrization is
$$
f^{\mathrm{nvc}}_{\mathfrak w}=\operatorname{NEvs}_\phi\circ \mathcal P_{\mathfrak w}.
$$
The normalization problem is therefore not whether a Whittaker datum appears, but exactly how its change affects the resulting parametrization [2412.06824].

## 2. Microlocal structure of the ABV parametrization

For the infinitesimal parameter $\lambda=\lambda_\phi:W_F\to {}^LG$, the Vogan variety is
$$
V_\lambda=\{x\in \widehat{\mathfrak g} : \operatorname{Ad}(\lambda(w))(x)=|w|_F x \text{ for all } w\in W_F\},
$$
with $H_\lambda=Z_{\widehat G}(\lambda)$ acting on it. The Langlands parameter $\phi$ determines an $H_\lambda$-orbit $C_\phi\subset V_\lambda$ [2412.06824].

Above $C_\phi$ sits the microlocal conormal subset
$$
\Lambda_{C_\phi}=\{(x,y)\in V_\lambda\times V_\lambda^\ast:[x,y]=0\},
$$
together with its open connected part $\Lambda^{\mathrm{gen}}_{C_\phi}$. For readability, write $A_\phi^{\mathrm{mic}}$ (*Editor’s term*) for the microlocal equivariant fundamental group
$$
A_\phi^{\mathrm{mic}}=\pi_1^{H_\lambda}(\Lambda^{\mathrm{gen}}_{C_\phi},(x_\phi,y)).
$$
The natural $H_\lambda$-equivariant projection $\Lambda^{\mathrm{gen}}_{C_\phi}\to C_\phi$ identifies the classical component group with
$$
A_\phi=\pi_0(Z_{\widehat G}(\phi))\simeq \pi_1^{H_\lambda}(C_\phi,x_\phi),
$$
and induces a functor
$$
\operatorname{Rep}(A_\phi)\to \operatorname{Rep}(A_\phi^{\mathrm{mic}}).
$$
Thus ABV parametrization is microlocal rather than merely component-group-theoretic [2412.06824].

If $J(\mathfrak w)(\pi)=(\phi,\rho)$, then $\mathcal P_{\mathfrak w}(\pi)$ is the simple $H_\lambda$-equivariant perverse sheaf $IC(\mathcal L_{C_\phi})$ corresponding to $\rho$ under
$$
\operatorname{Rep}(A_\phi)\simeq \operatorname{Loc}_{H_\lambda}(C_\phi).
$$
Applying normalized vanishing cycles yields
$$
f^{\mathrm{nvc}}_{\mathfrak w}: \Pi_\phi(G)\to \operatorname{Rep}(A_\phi^{\mathrm{mic}}).
$$
This is the map whose Whittaker dependence is measured exactly in the main theorem [2412.06824].

## 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 $\phi$ 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 [2412.06824].

More precisely, for two Whittaker data $\mathfrak w,\mathfrak w'$, there is a canonical character $\varrho^{\mathrm{mic},\phi}_{\mathfrak w,\mathfrak w'}\in \operatorname{Irr}(A_\phi^{\mathrm{mic}})$ such that
$$
f^{\mathrm{nvc}}_{\mathfrak w'}(\pi)
=
f^{\mathrm{nvc}}_{\mathfrak w}(\pi)\otimes \varrho^{\mathrm{mic},\phi}_{\mathfrak w,\mathfrak w'}
\qquad
(\pi\in \Pi_\phi(G)).
$$
Equivalently,
$$
f^{\mathrm{nvc}}_{\mathfrak w'}=f^{\mathrm{nvc}}_{\mathfrak w}\otimes \varrho^{\mathrm{mic},\phi}_{\mathfrak w,\mathfrak w'}.
$$
This is the exact Whittaker-dependence formula proved in the paper [2412.06824].

The character $\varrho^\phi_{\mathfrak w,\mathfrak w'}$ on $A_\phi$ is canonically determined from Kaletha’s bijection
$$
G_{\mathrm{ad}}(F)/G(F)\simeq
\bigl(\ker(H^1(W_F,Z_{\mathrm{sc}})\to H^1(W_F,Z))\bigr)^D,
$$
together with the difference $(\mathfrak w,\mathfrak w')\in G_{\mathrm{ad}}(F)/G(F)$ and the map from $Z_{\widehat G}(\phi)$ into $\ker(H^1(W_F,Z_{\mathrm{sc}})\to H^1(W_F,Z))$. Passing through
$$
\operatorname{Rep}(A_\phi)\to \operatorname{Rep}(A_\phi^{\mathrm{mic}})
$$
produces the microlocal twist. The same construction is compatible with the LLC normalization:
$$
J(\mathfrak w')=J(\mathfrak w)\otimes \varrho^\phi_{\mathfrak w,\mathfrak w'}.
$$
Thus the ABV change-of-normalization law is the microlocal avatar of the familiar LLC change-of-Whittaker-data formula [2412.06824].

The examples recorded in the paper clarify the range of this dependence. For $GL_n(F)$, the Whittaker datum is unique up to $G(F)$-conjugacy, so the quotient $G_{\mathrm{ad}}(F)/G(F)$ is trivial and there is no dependence. For $Sp(2n)$, $SO(2n+1)$, and $U(n)$, the change of datum is controlled by Kaletha’s canonical pairing, producing the twist on $A_\phi$ and then on $A_\phi^{\mathrm{mic}}$; the paper uses the general construction rather than spelling out group-specific closed formulas [2412.06824].

## 4. Open parameters, genericity, and Arthur type

A parameter $\phi$ is **open** if its orbit $C_\phi\subset V_\lambda$ is open. It is **tempered** if its restriction to $W_F$ is bounded. It is **of Arthur type** if
$$
\phi(w,x)=\psi(w,x,d_w)
$$
for some Arthur parameter $\psi:W_F\times SL_2\times SL_2\to {}^LG$. An ABV-packet is **generic** relative to a Whittaker datum if it contains a representation admitting a Whittaker model; for a fixed $\mathfrak w$, such a generic member is unique inside an L-packet under the LLC desiderata [2412.06824].

The paper isolates two structural equivalences. First,
$$
\phi \text{ tempered }
\Longleftrightarrow
\phi \text{ open and of Arthur type}.
$$
Second,
$$
\phi \text{ open }
\Longleftrightarrow
L(s,\phi,\operatorname{Ad}) \text{ is regular at } s=1.
$$
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 $\phi$ is open, while the ABV genericity conjecture asserts that the ABV-packet contains a generic representation if and only if $\phi$ is open [2412.06824].

For quasi-split classical groups, and also for $GL_n$, the paper proves that the packet is generic exactly when $\phi$ 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 $L$-functions, and Arthur-type structure [2412.06824].

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

When $\phi$ 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,
$$
A_\phi^{\mathrm{mic}}=A_\phi,
$$
and the normalized vanishing-cycles map is exactly the LLC map:
$$
f^{\mathrm{nvc}}_{\mathfrak w}=J(\mathfrak w).
$$
Equivalently, for every $\pi$ in the packet,
$$
\operatorname{NEvs}_\phi(\mathcal P_{\mathfrak w}(\pi))=J(\mathfrak w)(\pi).
$$
Open parameters are therefore the range in which the ABV construction recovers the Whittaker-normalized local Langlands parametrization without residue [2412.06824].

For tempered parameters, the result becomes an Arthur-packet statement. Since tempered parameters are precisely the open parameters of Arthur type, if $\psi$ is a tempered Arthur parameter with corresponding $\phi=\phi_\psi$, then for quasi-split classical groups, or more generally for groups for which Arthur’s conjectures are known,
$$
\Pi_\psi(G)=\Pi_\phi(G),
\qquad
A_\psi=A_\phi^{\mathrm{mic}},
\qquad
f^{\mathrm{nvc}}_{\mathfrak w}=f^{\mathrm{Arthur}}_{\mathfrak w}.
$$
In formula form,
$$
\operatorname{NEvs}_\phi(\mathcal P_{\mathfrak w}(\pi))=A(\mathfrak w)(\pi)
\qquad
(\pi\in \Pi_\psi(G)).
$$
This is the announced proof of Vogan’s conjecture on A-packets for tempered representations [2412.06824].

The scope of these theorems is explicit. The ambient assumption is that $G$ is a connected reductive group over a $p$-adic field $F$; 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 $GL_n$ 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 [2412.06824].

## 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 |
|---|---|---|
| $p$-adic ABV/LLC | Packet parametrization under change of Whittaker datum | Packet independent of $\mathfrak w$; labeling twisted canonically [2412.06824] |
| Global automorphic periods | Local/global Whittaker functionals and measures | Conjectural constant $c_{\pi_{\psi_N}(\phi),\psi_N}=|S_\phi|$; metaplectic local constants identified with $\epsilon(1/2,\pi_v,\psi_v)$ at finite places [1309.3190], [1404.2905] |
| Geometric Langlands | Whittaker coefficient functors and Hecke eigensheaf normalization | Existence of Whittaker-normalized eigensheaves with $\operatorname{coeff}^{\mathrm{enh}}(\operatorname{Aut}_\sigma)\simeq k_\sigma[-\dim \operatorname{Bun}_G]$ [2207.02955] |
| Loop group categories | $\chi$-twisted invariants versus coinvariants | Conjectural equivalence $\Theta_k:\mathcal C_{N((t)),\chi}\to \mathcal C^{N((t)),\chi}$; proved for $GL_n$ [1310.5127] |
| Finite local rings | Degenerate Whittaker space as a canonically normalized $GL_n(\mathfrak o_l)$-module | For strongly cuspidal $\pi$ of $GL_4(\mathfrak o_2)$, $\pi_{N,\psi}\simeq \operatorname{Ind}_{O_2^\times}^{GL_2(\mathfrak o_2)}(\theta|_{O_2^\times})$ [2407.21165] |

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
$$
\mathcal J_v(\mathrm{sph})=\Delta_{G,v}(1)\cdot L(1,\pi_v,\operatorname{Ad})^{-1},
$$
together with a conjectural global constant
$$
c_{\pi_{\psi_N}(\phi),\psi_N}=|S_\phi|.
$$
For the metaplectic double cover, the $p$-adic local normalization is sharpened to
$$
c_{\pi_v}=\epsilon(1/2,\pi_v,\psi_v),
$$
which makes the global Whittaker period formula canonical and unramified local factors equal to $1$ [1309.3190], [1404.2905].

In geometric Langlands, the Whittaker coefficient functor is interpreted microlocally, and the enhanced Whittaker functor $\operatorname{coeff}^{\mathrm{enh}}$ is conservative on the tempered category. The paper proves that Whittaker-normalized Hecke eigensheaves exist for irreducible $\check G$-local systems, are perverse, and are irreducible on each connected component of $\operatorname{Bun}_G$ 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 $\dim \operatorname{Bun}_G$ [2207.02955].

In categorical local geometric representation theory, Beraldo defines Whittaker invariant and coinvariant categories for an $N((t))$-action on a DG category and constructs normalization functors
$$
\Theta_k:\mathcal C_{N((t)),\chi}\to \mathcal C^{N((t)),\chi}.
$$
The conjecture is that $\Theta_k$ is an equivalence whenever the $N((t))$-action extends to a $G((t))$-action; this is proved for $GL_n$ by combining Fourier–Deligne transform on Tate vector spaces with explicit pro-unipotent stabilizers $H_k$ [1310.5127].

Further representation-theoretic evidence for canonical Whittaker normalization appears in characteristic $\ell\neq p$. For irreducible $Z$-compact representations, one has
$$
\dim \operatorname{Hom}_N(\pi,\theta)=\dim \operatorname{Hom}_N(\pi^\vee,\theta^{-1}),
$$
and for $\theta$-generic irreducibles,
$$
\pi^\vee\simeq \pi\circ I_\theta.
$$
Together with multiplicity-at-most-one, these statements isolate the uniqueness and contragredient compatibility that a Whittaker normalization is expected to encode [2209.15353].

A more distant but structurally analogous usage occurs for $b$-Whittaker functions, where normalization is fixed by the rank-one condition
$$
\Psi_b^{(1)}(x_1;\lambda_1)=e^{2\pi i \lambda_1 x_1},
$$
by the Baxter-operator recursion, and by the Sklyanin measure that makes the $b$-Whittaker transform unitary. In that context the normalized Whittaker functions satisfy completeness and orthogonality relations analogous to a Plancherel theorem [1806.00747].

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.

Source: https://www.emergentmind.com/topics/whittaker-normalization-conjecture