Ginzburg–Kaplan Gamma Factors

• Ginzburg–Kaplan gamma factors are local scalar invariants obtained from zeta integral functional equations in both archimedean and finite-field settings.
• They unify constructions from GL₃’s global integral and doubling-type methods by isolating scalars via equivariant bilinear forms or class-function operators.
• Multiplicity properties and level-zero comparisons are key to their role in bridging analytic and algebraic aspects within the Langlands program.

Searching arXiv for recent and foundational papers on Ginzburg–Kaplan gamma factors, finite-field analogs, and related local theory. Ginzburg–Kaplan gamma factors are local scalar invariants extracted from functional equations of zeta integrals attached to global and local integral representations of automorphic $L$-functions. In the literature represented here, the term encompasses at least two closely related regimes: the archimedean local factors attached to D. Ginzburg’s global integral for the adjoint representation of $\mathrm{GL}_3$, and the tensor-product factors arising from Kaplan’s local doubling-type construction and its finite-field analog for pairs $\pi\in \Irr(\GL_c(\mathbb F_q))$, $\tau\in \Irr(\GL_k(\mathbb F_q))$. The defining mechanism is uniform: one constructs a local zeta object, proves a functional equation, and isolates the scalar by which a dual transform acts. Recent work places finite-field, non-archimedean level-zero, and archimedean theories into a single comparative picture (Carmon et al., 2024, Zelingher, 2024, Tian, 2018).

1. Historical setting and conceptual scope

The global antecedent is Ginzburg’s 1991 construction for the adjoint $L$-function of $\mathrm{GL}_3$, obtained by integrating a cusp form on $\mathrm{GL}_3$ against an Eisenstein series on $G_2$. In that framework, the local theory is extracted place by place, and local gamma factors arise from the corresponding local functional equation. Tian’s archimedean work supplies precisely the missing local analytic package for this construction: convergence, meromorphic continuation, uniqueness of equivariant bilinear forms, and therefore existence of the archimedean gamma factor for the adjoint representation (Tian, 2018).

A distinct but structurally parallel development is the doubling-type tensor-product theory associated with Kaplan locally and Ginzburg globally. Over finite fields, this produces a gamma factor for a pair of irreducible representations of $\GL_c(\mathbb F_q)$ and $\GL_k(\mathbb F_q)$. The construction is explicitly described as a finite-field analog of Kaplan’s local doubling-type construction and Ginzburg’s global construction, and it differs from classical $\mathrm{GL}_3$0 Rankin–Selberg theory in that it only assumes that $\mathrm{GL}_3$1 is generic; $\mathrm{GL}_3$2 may be arbitrary irreducible (Carmon et al., 2024).

The modern usage of “Ginzburg–Kaplan gamma factors” therefore refers less to a single universal formula than to a family of local factors defined from integral representations of Ginzburg–Kaplan type. A recurring theme is that the relevant scalar can be isolated either from a one-dimensional space of equivariant bilinear forms or from a class-function operator that commutes with an irreducible representation and is therefore scalar by Schur’s lemma.

2. Defining mechanism: local integrals, dual transforms, and scalar extraction

In the archimedean adjoint $\mathrm{GL}_3$3 theory, the local setting is $\mathrm{GL}_3$4 or $\mathrm{GL}_3$5. One takes an irreducible admissible generic Casselman–Wallach representation $\mathrm{GL}_3$6 of $\mathrm{GL}_3$7 with trivial central character, Whittaker model $\mathrm{GL}_3$8, and local section $\mathrm{GL}_3$9 for

$\pi\in \Irr(\GL_c(\mathbb F_q))$0

The local zeta integral is

$\pi\in \Irr(\GL_c(\mathbb F_q))$1

After introducing the local intertwining operator attached to $\pi\in \Irr(\GL_c(\mathbb F_q))$2,

$\pi\in \Irr(\GL_c(\mathbb F_q))$3

one defines the dual integral

$\pi\in \Irr(\GL_c(\mathbb F_q))$4

The local functional equation is

$\pi\in \Irr(\GL_c(\mathbb F_q))$5

with Tian’s notation $\pi\in \Irr(\GL_c(\mathbb F_q))$6 for the same scalar. The gamma factor is thus defined abstractly from uniqueness of continuous $\pi\in \Irr(\GL_c(\mathbb F_q))$7-equivariant bilinear forms (Tian, 2018).

In the finite-field tensor-product theory, the scalar arises differently but with the same formal logic. One first constructs a class function on $\pi\in \Irr(\GL_c(\mathbb F_q))$8 from a Bessel–Speh special value, then sums it against $\pi\in \Irr(\GL_c(\mathbb F_q))$9. The resulting endomorphism commutes with $\tau\in \Irr(\GL_k(\mathbb F_q))$0, hence is scalar by Schur’s lemma. This scalar is the finite-field Ginzburg–Kaplan gamma factor (Carmon et al., 2024).

Kaplan’s local non-archimedean zeta integral fits the same template. For an irreducible representation $\tau\in \Irr(\GL_k(\mathbb F_q))$1 of $\tau\in \Irr(\GL_k(\mathbb F_q))$2, a generic irreducible representation $\tau\in \Irr(\GL_k(\mathbb F_q))$3 of $\tau\in \Irr(\GL_k(\mathbb F_q))$4, and $\tau\in \Irr(\GL_k(\mathbb F_q))$5, the local zeta integral is

$\tau\in \Irr(\GL_k(\mathbb F_q))$6

together with a dual integral $\tau\in \Irr(\GL_k(\mathbb F_q))$7. Their functional equation defines the local Ginzburg–Kaplan gamma factor $\tau\in \Irr(\GL_k(\mathbb F_q))$8 (Zelingher, 2024).

3. Finite-field construction via Speh representations and Bessel–Speh functions

Let $\tau\in \Irr(\GL_k(\mathbb F_q))$9 be an irreducible generic representation of $L$0, and let $L$1. The finite-field theory begins with the Speh representation $L$2, defined as the unique irreducible subrepresentation of

$L$3

that admits a $L$4-Whittaker vector. Here the $L$5-Whittaker condition is equivariance with respect to the unipotent radical of type $L$6 with character

$L$7

The representation $L$8 has a unique $L$9-Whittaker vector up to scaling (Carmon et al., 2024).

Choosing a $\mathrm{GL}_3$0-invariant inner product and a normalized $\mathrm{GL}_3$1-Whittaker vector $\mathrm{GL}_3$2, one defines the normalized Bessel–Speh function

$\mathrm{GL}_3$3

It satisfies $\mathrm{GL}_3$4, $\mathrm{GL}_3$5, and transforms under the diagonal embedding $\mathrm{GL}_3$6 by the central character $\mathrm{GL}_3$7. The special values

$\mathrm{GL}_3$8

form a class function on $\mathrm{GL}_3$9 (Carmon et al., 2024).

For $\mathrm{GL}_3$0, the central operator is

$\mathrm{GL}_3$1

Since $\mathrm{GL}_3$2 is a class function, $\mathrm{GL}_3$3 commutes with $\mathrm{GL}_3$4, and there exists a scalar $\mathrm{GL}_3$5 such that

$\mathrm{GL}_3$6

This scalar is the finite-field Ginzburg–Kaplan gamma factor. When $\mathrm{GL}_3$7, the construction recovers the twisted Godement–Jacquet/Kondo factor (Carmon et al., 2024).

A common misconception is that this is simply a reformulation of finite-field Rankin–Selberg theory. The construction is instead a finite-field doubling method: $\mathrm{GL}_3$8 must be generic, but $\mathrm{GL}_3$9 can be arbitrary. That asymmetry is one of its defining structural features.

4. Functional equations, multiplicativity, and identification with $G_2$0-factors

The finite-field gamma factor is designed to satisfy a doubling-style functional equation. For $G_2$1 and $G_2$2, one defines

$G_2$3

together with a Fourier-dual operator. If the cuspidal supports of $G_2$4 and $G_2$5 are disjoint, then

$G_2$6

The paper proves a stronger family of identities indexed by $G_2$7, with the basic case $G_2$8 obtained first and the others by descending induction (Carmon et al., 2024).

A major theorem is multiplicativity in both variables. If $G_2$9 and $\GL_c(\mathbb F_q)$0 is the unique irreducible generic subrepresentation of $\GL_c(\mathbb F_q)$1, then

$\GL_c(\mathbb F_q)$2

If $\GL_c(\mathbb F_q)$3 and $\GL_c(\mathbb F_q)$4 is a subrepresentation of $\GL_c(\mathbb F_q)$5, then

$\GL_c(\mathbb F_q)$6

These identities are proved from convolution identities for Bessel–Speh functions, including a convolution formula in the $\GL_c(\mathbb F_q)$7-variable and an averaging identity over a unipotent radical (Carmon et al., 2024).

The same work identifies the finite-field gamma factor with the tensor-product $\GL_c(\mathbb F_q)$8-factor: $\GL_c(\mathbb F_q)$9 for every irreducible $\GL_k(\mathbb F_q)$0 and generic irreducible $\GL_k(\mathbb F_q)$1. For cuspidal $\GL_k(\mathbb F_q)$2, one also obtains the absolute value formula

$\GL_k(\mathbb F_q)$3

where $\GL_k(\mathbb F_q)$4 is the multiplicity of $\GL_k(\mathbb F_q)$5 in the cuspidal support of $\GL_k(\mathbb F_q)$6 (Carmon et al., 2024).

The Bessel–Speh special values admit an explicit interpretation in terms of exotic matrix Kloosterman sums. For generic principal series, repeated use of the convolution identity yields a twisted matrix Kloosterman sum, and for general generic $\GL_k(\mathbb F_q)$7 one has

$\GL_k(\mathbb F_q)$8

This leads to multiplicativity identities for exotic matrix Kloosterman sums and to a converse theorem characterizing generic representations from sufficiently many Bessel–Speh special values (Carmon et al., 2024).

5. Level-zero supercuspidals and the bridge between local and finite-field theories

The paper on level-zero supercuspidals establishes a direct comparison between finite-field Ginzburg–Kaplan factors and local Kaplan factors. Starting from an irreducible cuspidal representation $\GL_k(\mathbb F_q)$9 of $\mathrm{GL}_3$00 and a compatible central character $\mathrm{GL}_3$01 on $\mathrm{GL}_3$02, one forms the compact induction

$\mathrm{GL}_3$03

which is an irreducible level-zero supercuspidal representation. A lift map carries vectors, tensors, and sections from finite-field objects to their local counterparts (Zelingher, 2024).

This lift is compatible with the formation of Speh representations. On the finite-field side, $\mathrm{GL}_3$04 is the irreducible subrepresentation of minimal dimension in $\mathrm{GL}_3$05; on the local side, $\mathrm{GL}_3$06 is defined as the image of the standard intertwining operator

$\mathrm{GL}_3$07

on the normalized induced representation $\mathrm{GL}_3$08. The main commutative diagram shows that lifting commutes with passing to Speh quotients or images. Equivalently, if $\mathrm{GL}_3$09, then its lift $\mathrm{GL}_3$10 lies in $\mathrm{GL}_3$11 (Zelingher, 2024).

The same paper compares the finite-field and local $\mathrm{GL}_3$12-Whittaker models. In both settings, the unipotent subgroup $\mathrm{GL}_3$13 carries the character

$\mathrm{GL}_3$14

The lift carries finite-field $\mathrm{GL}_3$15-Whittaker functions to local ones and matches values on maximal compact elements: $\mathrm{GL}_3$16 Support restrictions for the lifted Whittaker functions are also proved; in particular, when $\mathrm{GL}_3$17, only the trivial diagonal survives in a specified family of diagonal evaluations (Zelingher, 2024).

The comparison of zeta integrals is the key consequence. If $\mathrm{GL}_3$18 and $\mathrm{GL}_3$19 are irreducible cuspidal representations of $\mathrm{GL}_3$20 and $\mathrm{GL}_3$21, and $\mathrm{GL}_3$22 are the associated level-zero supercuspidals, then in the non-exceptional case $\mathrm{GL}_3$23,

$\mathrm{GL}_3$24

An analogous equality holds for the dual integral. When $\mathrm{GL}_3$25 and hence $\mathrm{GL}_3$26, an explicit correction term appears in both comparisons. From these identities one obtains the gamma-factor comparison theorem

$\mathrm{GL}_3$27

in the non-exceptional case, and in the exceptional case the finite-field factor is explicitly

$\mathrm{GL}_3$28

Combining the non-exceptional comparison, the exceptional computation, and known identifications with local Rankin–Selberg gamma factors yields

$\mathrm{GL}_3$29

for all irreducible cuspidal pairs over finite fields (Zelingher, 2024).

6. Archimedean local theory for the adjoint representation of $\mathrm{GL}_3$30

The archimedean side is represented by Tian’s study of the local theory attached to Ginzburg’s global Rankin–Selberg integral for the adjoint $\mathrm{GL}_3$31-function of $\mathrm{GL}_3$32. The representation $\mathrm{GL}_3$33 is an irreducible admissible generic Casselman–Wallach representation of $\mathrm{GL}_3$34 with trivial central character, $\mathrm{GL}_3$35 or $\mathrm{GL}_3$36, and the paper establishes four foundational results: absolute convergence for $\mathrm{GL}_3$37 large, meromorphic continuation of the local zeta integrals, a local functional equation relating the two local zeta integrals, and hence the existence of the local gamma factor $\mathrm{GL}_3$38, denoted $\mathrm{GL}_3$39 in the paper (Tian, 2018).

The main analytic input is refined asymptotic expansion of Whittaker functions along the torus of $\mathrm{GL}_3$40, followed by an analysis of singular behavior near the origin of the relevant integration variables. Theorem 1.6 states that $\mathrm{GL}_3$41 extends meromorphically to all $\mathrm{GL}_3$42, and if $\mathrm{GL}_3$43 is a principal series with complex parameter $\mathrm{GL}_3$44, then $\mathrm{GL}_3$45 is also meromorphic in $\mathrm{GL}_3$46 (Tian, 2018).

Several normalizations are explicit. For the standard maximal unipotent subgroup $\mathrm{GL}_3$47 of $\mathrm{GL}_3$48, the generic character is

$\mathrm{GL}_3$49

If $\mathrm{GL}_3$50 and $\mathrm{GL}_3$51, then the associated Whittaker functions satisfy

$\mathrm{GL}_3$52

and the local integrals obey the scaling relation

$\mathrm{GL}_3$53

This scaling behavior is part of the normalization data for the gamma factor (Tian, 2018).

A recurrent misunderstanding is that the archimedean theory already provides an explicit closed formula. It does not. Tian’s paper proves existence and the functional equation, while the explicit computation of the archimedean gamma factor is postponed to a forthcoming paper. The result is therefore foundational rather than computational.

7. Related gamma-factor formalisms and broader significance

Two adjacent bodies of work clarify the broader landscape. First, the paper on level-zero supercuspidal representations gives an explicit formula for the twisted Rankin–Selberg gamma factor for a pair of irreducible supercuspidal representations of level zero and shows that level-zero $\mathrm{GL}_3$54-adic gamma factors reduce to finite-field gamma factors via the Nien–Zhang comparison formula. In this reduction, finite-field gamma factors are then computed explicitly in terms of Gauss sums of Green parameters. The paper does not directly develop a Ginzburg–Kaplan formalism, but it is explicitly described as being in the same circle of ideas and as providing a precise bridge from local Rankin–Selberg gamma factors on $\mathrm{GL}_3$55-adic $\mathrm{GL}_3$56 to explicit finite-field Gauss sums (Yang, 2019).

Second, the Braverman–Kazhdan/Ngo multiplicativity paper proves, under a commutativity assumption between the relevant Fourier transform and a generalized Harish-Chandra transform, that

$\mathrm{GL}_3$57

This work does not discuss Ginzburg–Kaplan gamma factors by name, but it addresses the same multiplicativity principle that any gamma-factor theory of Langlands type is expected to satisfy (Shahidi et al., 2021).

Taken together, these results suggest a stratified picture. In the finite-field setting, Ginzburg–Kaplan gamma factors admit an explicit construction from Speh and Bessel–Speh theory, satisfy functional equations and multiplicativity, and coincide with $\mathrm{GL}_3$58-factors. In the level-zero non-archimedean setting, local Kaplan integrals can be compared directly with their finite-field analogs, including an explicit exceptional correction when the cuspidal data coincide. In the archimedean adjoint $\mathrm{GL}_3$59 setting, the essential analytic machinery for defining the local factor is in place, while explicit formulas remain a separate problem. The resulting theory is therefore both local and comparative: it links global integral representations, local zeta integrals, finite-field harmonic analysis, and the structural properties expected of gamma factors in the broader Langlands program.