Local Non-Tempered Gan–Gross–Prasad Conjecture

• The topic is a branching-law conjecture for GL groups that predicts when a local invariant is nonzero based on the relevance of non-tempered Arthur parameters.
• It focuses on cases where representations carry a nontrivial SL₂(ℂ) factor, replacing classical local factors with an explicit combinatorial relevance condition.
• A concrete framework using Speh modules, derivative analysis, and explicit decision procedures underpins the proofs for both non-Archimedean and Archimedean settings.

Searching arXiv for papers on the local non-tempered Gan–Gross–Prasad conjecture for general linear groups and related settings. The local non-tempered Gan–Gross–Prasad conjecture is a branching-law conjecture for pairs of successive reductive groups, predicting when a local model or restriction functional is nonzero for representations whose local parameters are not tempered because they carry a nontrivial Arthur $SL_2(\mathbf{C})$-factor. In the general linear case, it concerns the restriction problem for $GL_{n+1}(F)$ to $GL_n(F)$, or equivalently the existence of local Rankin–Selberg, Bessel, or Fourier–Jacobi models, and replaces the tempered/generic control by Rankin–Selberg local factors with an explicit combinatorial condition on Arthur parameters, usually called relevance. For non-Archimedean $F$, the Arthur-type case was proved by Chan (Chan, 2020), and later extended to full smooth or unitarizable classes by generalized or extended relevance criteria (Pattanayak, 12 Dec 2025). For Archimedean $F\in\{\mathbf{R},\mathbf{C}\}$, the Arthur-type conjecture for $GL$ is settled by the combination of the “period implies relevance” theorem of Chen–Chen and Boisseau’s converse, as summarized in (Chen et al., 6 Jul 2025). In characteristic zero for the split $GL_n\times GL_{n+1}$ Bessel model, a constructive local invariant functional was obtained from regularized Rankin–Selberg zeta integrals, yielding the “relevance implies distinction” direction for weak Arthur-type representations (Boisseau, 26 Nov 2025).

1. Conjectural framework and basic objects

Let $F$ be a local field and write $G_n=GL_n(F)$, embedded in $G_{n+1}$ by $GL_{n+1}(F)$0. The local problem is to determine when

$GL_{n+1}(F)$1

for irreducible representations $GL_{n+1}(F)$2 of $GL_{n+1}(F)$3 and $GL_{n+1}(F)$4 of $GL_{n+1}(F)$5. In the split Bessel formulation one instead considers $GL_{n+1}(F)$6 with diagonal $GL_{n+1}(F)$7, and the multiplicity

$GL_{n+1}(F)$8

which is equivalent to the same local distinction problem (Boisseau, 26 Nov 2025).

For $GL_{n+1}(F)$9, an Arthur parameter is a morphism

$GL_n(F)$0

that is tempered on $GL_n(F)$1 and algebraic on $GL_n(F)$2. In the non-Archimedean Arthur-type setting this decomposes as

$GL_n(F)$3

with each $GL_n(F)$4 tempered and $GL_n(F)$5, and the associated irreducible representation is denoted $GL_n(F)$6 (Chan, 2020). In the Archimedean $GL_n(F)$7 setting, with $GL_n(F)$8 or $GL_n(F)$9, one similarly writes

$F$0

where $F$1 are tempered $F$2-parameters and the associated Arthur-type representation is a product of generalized Speh representations (Chen et al., 6 Jul 2025).

The adjective non-tempered refers precisely to the presence of a nontrivial Arthur $F$3-factor, equivalently some $F$4 in the Archimedean Speh description (Chen et al., 6 Jul 2025). In the split characteristic-zero refinement, one also allows weak Arthur-type parameters

$F$5

with corresponding Langlands quotients built from twisted Speh blocks (Boisseau, 26 Nov 2025). This suggests a hierarchy: Arthur type is the untwisted case, weak Arthur type allows small complementary exponents, and the full unitary or smooth theories require further extensions of relevance beyond Speh-only data (Pattanayak, 12 Dec 2025).

2. Relevance of Arthur parameters

The central concept is relevance, a combinatorial matching condition on the $F$6-exponents of the two Arthur parameters. In the non-Archimedean $F$7 Arthur-type setting, if

$F$8

and

$F$9

with $F\in\{\mathbf{R},\mathbf{C}\}$0, then $F\in\{\mathbf{R},\mathbf{C}\}$1 is called a relevant pair (Chan, 2020). The schematic meaning is that the two parameters share the same tempered constituents while their $F\in\{\mathbf{R},\mathbf{C}\}$2-lengths interlace by one-step shifts.

For Archimedean $F\in\{\mathbf{R},\mathbf{C}\}$3, relevance is formulated similarly but with a decomposition

$F\in\{\mathbf{R},\mathbf{C}\}$4

and

$F\in\{\mathbf{R},\mathbf{C}\}$5

where $F\in\{\mathbf{R},\mathbf{C}\}$6 is generic, meaning a sum of $F\in\{\mathbf{R},\mathbf{C}\}$7-terms, and summands $F\in\{\mathbf{R},\mathbf{C}\}$8 are understood to vanish (Chen et al., 6 Jul 2025). The Archimedean formulation explicitly allows a generic remainder on the smaller group.

A useful Archimedean combinatorial reformulation is given by multiplicities $F\in\{\mathbf{R},\mathbf{C}\}$9 of $GL$0 in $GL$1, and alternating sums

$GL$2

Then relevance is equivalent to the inequalities

$GL$3

for all $GL$4 and $GL$5 (Chen et al., 6 Jul 2025). This is the exact combinatorial encoding of the interlacing of $GL$6-sizes.

In the split weak-Arthur setting for $GL$7, relevance is formulated asymmetrically as

$GL$8

$GL$9

which is the split-avatar of the Bessel relevance pattern (Boisseau, 26 Nov 2025). In all these versions, the non-temperedness is carried entirely by the $GL_n\times GL_{n+1}$0-factor.

A common misconception is that non-tempered $GL_n\times GL_{n+1}$1-branching should continue to be governed by local $GL_n\times GL_{n+1}$2- and $GL_n\times GL_{n+1}$3-factors exactly as in the generic tempered case. In the Arthur-type non-Archimedean $GL_n\times GL_{n+1}$4 theorem, no additional condition involving

$GL_n\times GL_{n+1}$5

enters the criterion; the condition is purely the relevance of the Arthur $GL_n\times GL_{n+1}$6-parts (Chan, 2020). This does not negate the global role of local factors, but it sharply changes the local decision rule in the non-tempered $GL_n\times GL_{n+1}$7 setting.

3. Main theorems for general linear groups

The foundational non-Archimedean Arthur-type theorem states that if $GL_n\times GL_{n+1}$8 is non-Archimedean, $GL_n\times GL_{n+1}$9 is an Arthur-type representation of $F$0, and $F$1 is an Arthur-type representation of $F$2, then

$F$3

(Chan, 2020). Together with the multiplicity-one theorem of Aizenbud–Gourevitch–Rallis–Schiffmann and Sun–Zhu, the nonzero Hom space is at most one-dimensional (Chan, 2020).

For Archimedean $F$4, Chen–Chen prove the necessity direction in full unitary generality: $F$5 for irreducible unitary $F$6 of $F$7 and $F$8 of $F$9, with $G_n=GL_n(F)$0 or $G_n=GL_n(F)$1 (Chen et al., 6 Jul 2025). In the Arthur-type case, combined with Boisseau’s converse as stated there, one obtains the full equivalence

$G_n=GL_n(F)$2

(Chen et al., 6 Jul 2025).

The split characteristic-zero theorem of Xie–Zhang constructs an explicit local $G_n=GL_n(F)$3-invariant functional for weak Arthur-type representations $G_n=GL_n(F)$4 with relevant parameter $G_n=GL_n(F)$5, showing regularity, factorization through the Langlands quotient, and nonvanishing of a normalized local zeta integral at the relevant point (Boisseau, 26 Nov 2025). Since multiplicity one is classical, this yields

$G_n=GL_n(F)$6

for the split $G_n=GL_n(F)$7 Bessel model over local fields of characteristic zero when combined with the previously established necessity results (Boisseau, 26 Nov 2025).

A later non-Archimedean extension proves a quotient branching law for all irreducible smooth representations: $G_n=GL_n(F)$8 (Pattanayak, 12 Dec 2025). For irreducible unitary pairs, the same paper shows equivalence with an extended GGP relevance condition in terms of Tadić blocks, Speh data, and complementary series parameters $G_n=GL_n(F)$9–$G_{n+1}$0 (Pattanayak, 12 Dec 2025). This suggests that the Arthur-type non-tempered conjecture for $G_{n+1}$1 is the first layer of a broader branching theory controlled by derivative-compatible multisegment combinatorics.

4. Speh representations, derivatives, and multisegments

The representation-theoretic realization of Arthur-type $G_{n+1}$2-representations is through Speh modules. In the non-Archimedean case, if $G_{n+1}$3 is an irreducible unitarizable cuspidal representation of $G_{n+1}$4, one defines generalized Steinberg representations $G_{n+1}$5 for segments $G_{n+1}$6, and the Speh representation

$G_{n+1}$7

as the unique irreducible quotient of a rectangular product of translated Steinberg segments (Chan, 2020). Arthur-type representations for $G_{n+1}$8 are exactly finite products of Speh representations, irreducible up to permutation of factors (Chan, 2020).

In the Archimedean setting, the generalized Speh module attached to a discrete series $G_{n+1}$9 of $GL_{n+1}(F)$00 and $GL_{n+1}(F)$01 is

$GL_{n+1}(F)$02

and an Arthur-type representation is a finite parabolic product of such Speh representations (Chen et al., 6 Jul 2025).

The derivative formalism explains why relevance is a one-step shift condition. A key non-Archimedean identity states that the highest shifted derivative of a Speh lowers the $GL_{n+1}(F)$03-length by one: $GL_{n+1}(F)$04 (Chan, 2020). Accordingly, if

$GL_{n+1}(F)$05

$GL_{n+1}(F)$06

then $GL_{n+1}(F)$07 is the Speh reformulation of relevance (Chan, 2020).

This picture admits a precise multisegment translation. For general irreducible pairs, Chan introduces generalized relevance via right and left derivatives $GL_{n+1}(F)$08, $GL_{n+1}(F)$09, right and left integrals, and strong RdLi-commutativity of triples of segments and multisegments (Pattanayak, 12 Dec 2025). The core criterion is the existence of multisegments $GL_{n+1}(F)$10 such that

$GL_{n+1}(F)$11

and $GL_{n+1}(F)$12 is strongly RdLi-commutative (Pattanayak, 12 Dec 2025). In the unitary case this reduces to explicit block relations $GL_{n+1}(F)$13–$GL_{n+1}(F)$14 between Speh parameters and complementary exponents (Pattanayak, 12 Dec 2025).

An earlier stage of this development is Gurevich’s result that for p-adic $GL_{n+1}(F)$15, nonvanishing in the Arthur-type case forces the GGP position of Speh blocks, and in the generic case the converse holds (Gurevich, 2018). That work already exhibits the central role of derivatives, quasi-Speh representations, and the reduction of restriction problems to derivative Hom spaces (Gurevich, 2018).

5. Methods of proof and explicit decision procedures

The non-Archimedean Arthur-type proof relies on Bernstein–Zelevinsky derivatives and the mirabolic filtration of $GL_{n+1}(F)$16, together with an exact transfer between Hom spaces on successive layers (Chan, 2020). The reduction isolates the unique Bernstein–Zelevinsky layer that can contribute, and the highest-derivative behavior of Speh blocks turns the branching question into a recursive block-matching problem (Chan, 2020).

A concrete decision procedure is available in the Arthur-type non-Archimedean case. Given Speh factorizations

$GL_{n+1}(F)$17

one orders factors by Tadić order, computes the highest shifted derivative of the maximal factor, compares with a matching factor in $GL_{n+1}(F)$18, strips matched factors, transfers to a reduced pair using a cuspidal character $GL_{n+1}(F)$19, and iterates (Chan, 2020). If every step matches the prescribed one-step shift, then Hom is nonzero; otherwise it vanishes (Chan, 2020).

The proof also uses an Ext-theoretic dual restriction. For suitable cuspidal $GL_{n+1}(F)$20,

$GL_{n+1}(F)$21

(Chan, 2020). This transfer permits control of cuspidal supports and vanishing of many layers, and motivates the conjectural higher-Ext branching law formulated there (Chan, 2020).

In the Archimedean case, the necessity proof uses annihilator varieties. Gourevitch–Sahi identify the $GL_{n+1}(F)$22-type $GL_{n+1}(F)$23 with the transpose of the annihilator partition $GL_{n+1}(F)$24, while Gourevitch–Sayag–Karshon show that higher-corank Rankin–Selberg distinction imposes

$GL_{n+1}(F)$25

for all $GL_{n+1}(F)$26 (Chen et al., 6 Jul 2025). Chen–Chen convert this microlocal inequality into the alternating-sum relevance inequalities via deformation, Mackey-theoretic reductions, and vanishing of closed-orbit contributions (Chen et al., 6 Jul 2025).

The split constructive theorem proceeds differently. It introduces a local Jacquet–Whittaker functional

$GL_{n+1}(F)$27

and the local Rankin–Selberg zeta integral

$GL_{n+1}(F)$28

with normalized version

$GL_{n+1}(F)$29

(Boisseau, 26 Nov 2025). Restricting to the relevant affine subspace and evaluating at the inducing point $GL_{n+1}(F)$30 produces the local invariant functional

$GL_{n+1}(F)$31

(Boisseau, 26 Nov 2025). The normalized functional equation

$GL_{n+1}(F)$32

is the key analytic input ensuring regularity and factorization through the Langlands quotient (Boisseau, 26 Nov 2025).

6. Models, extensions, and broader non-tempered GGP landscape

The local non-tempered $GL_{n+1}(F)$33 theory extends beyond the basic restriction Hom space. In the non-Archimedean Arthur-type setting, Chan proves Bessel and Fourier–Jacobi analogues via functorial reductions to basic $GL_{n+1}(F)$34-restriction, obtaining

$GL_{n+1}(F)$35

for the considered Bessel, Fourier–Jacobi, and mixed models (Chan, 2020). The same work proves multiplicity one and finite-dimensionality of the corresponding Ext spaces (Chan, 2020).

The split characteristic-zero paper is explicitly formulated for the split Bessel model $GL_{n+1}(F)$36 with diagonal $GL_{n+1}(F)$37, and it identifies the local normalized zeta integral as the local factor in the refined global Ichino–Ikeda-type formula (Boisseau, 26 Nov 2025). This makes the local non-tempered theorem constructive and places it directly inside the global period formalism.

The unitary and smooth extensions go further. For non-Archimedean unitary pairs, extended GGP relevance is defined through Tadić’s classification by unitary Speh blocks $GL_{n+1}(F)$38 and complementary series $GL_{n+1}(F)$39, with explicit relations

$GL_{n+1}(F)$40

linking the parameters of $GL_{n+1}(F)$41 and $GL_{n+1}(F)$42 (Pattanayak, 12 Dec 2025). The equivalence between this extended relevance and generalized derivative-based relevance gives a complete classification of unitary branching for $GL_{n+1}(F)$43 (Pattanayak, 12 Dec 2025). A plausible implication is that the Arthur-type conjecture should be viewed as a special case of a more general derivative/integral calculus on multisegments rather than as an isolated non-generic phenomenon.

Beyond $GL_{n+1}(F)$44, non-tempered local GGP behaves less rigidly. For unitary groups, a non-generic parameter on the larger side can still yield explicit multiplicity formulas depending on whether a distinguished character occurs in the smaller parameter, with packet labels determined by epsilon factors (Haan, 2017). For $GL_{n+1}(F)$45, Haan shows that the naive extension of the generic recipe fails for some non-generic pairs, even though one still gets precise theta-theoretic criteria and a regularized local Ichino–Ikeda-type statement (Haan, 2015). For certain non-tempered $GL_{n+1}(F)$46-packets of $GL_{n+1}(F)$47 and $GL_{n+1}(F)$48, packet-level uniqueness can fail altogether, so the generic local GGP principle does not extend naively (Gurevich et al., 2013). These contrasts underscore that the $GL_{n+1}(F)$49 non-tempered theorem is unusually clean: the relevance criterion remains exact and multiplicity one survives.

A related but distinct line appears in the metaplectic–symplectic Fourier–Jacobi setting. There the non-tempered local selection rule for $GL_{n+1}(F)$50-parameters is still conjectural in general; available results prove only one global implication and local vanishing statements for specific residual representations (Haan, 2022). This marks an important boundary of present knowledge: the $GL_{n+1}(F)$51 case is effectively complete in characteristic zero, while analogous non-tempered local GGP problems for classical and covering groups remain more delicate.

7. Examples, significance, and current status

A basic example in the non-Archimedean $GL_{n+1}(F)$52 Arthur-type theory is the Speh-to-Speh drop: $GL_{n+1}(F)$53 Then

$GL_{n+1}(F)$54

and the corresponding Arthur parameters are $GL_{n+1}(F)$55 and $GL_{n+1}(F)$56, so relevance is immediate (Chan, 2020).

A mixed-product example from the same work takes on $GL_{n+1}(F)$57

$GL_{n+1}(F)$58

and on $GL_{n+1}(F)$59

$GL_{n+1}(F)$60

A filtration argument shows that a specific Bernstein–Zelevinsky layer of $GL_{n+1}(F)$61 maps onto $GL_{n+1}(F)$62, hence

$GL_{n+1}(F)$63

again matching relevance at the Arthur-parameter level (Chan, 2020).

In the Archimedean Arthur-type setting, an illustrative case is

$GL_{n+1}(F)$64

for a unitary discrete series $GL_{n+1}(F)$65 of $GL_{n+1}(F)$66, with parameter $GL_{n+1}(F)$67. Relevance predicts that a $GL_{n+1}(F)$68-representation $GL_{n+1}(F)$69 must contain $GL_{n+1}(F)$70 plus a generic remainder; then the local non-tempered GGP implies

$GL_{n+1}(F)$71

whereas $GL_{n+1}(F)$72 in place of $GL_{n+1}(F)$73 breaks relevance and forces vanishing (Chen et al., 6 Jul 2025).

The present status may be summarized succinctly.

Setting	Result	Reference
Non-Archimedean $GL_{n+1}(F)$74, Arthur type	$GL_{n+1}(F)$75 relevance	(Chan, 2020)
Archimedean $GL_{n+1}(F)$76, Arthur type	full equivalence via Chen–Chen and Boisseau	(Chen et al., 6 Jul 2025)
Split $GL_{n+1}(F)$77, weak Arthur type, char. $GL_{n+1}(F)$78	constructive local functional and full local conjecture in split case	(Boisseau, 26 Nov 2025)
Non-Archimedean $GL_{n+1}(F)$79, full smooth/unitary classes	generalized or extended relevance criterion	(Pattanayak, 12 Dec 2025)

The significance of these results is twofold. First, they identify the exact local branching rule for non-tempered $GL_{n+1}(F)$80-representations in terms of Arthur-parameter interlacing, thereby extending the local Gan–Gross–Prasad paradigm beyond the tempered/generic range. Second, they supply the local input for global period formulas at places where automorphic representations carry non-tempered local components, including Bessel and Fourier–Jacobi models and refined Ichino–Ikeda-type identities (Chan, 2020, Boisseau, 26 Nov 2025).

The remaining open directions are not within basic $GL_{n+1}(F)$81 Arthur type over characteristic-zero local fields, where the conjecture is effectively settled, but rather in broader extensions: higher Ext-branching laws conjectured in derivative form (Chan, 2020), full non-tempered analogues for classical or covering groups where uniqueness may fail or the packet selection rule is still incomplete (Gurevich et al., 2013, Haan, 2022), and the systematic integration of complementary-series blocks into a unified local GGP formalism beyond the already established non-Archimedean $GL_{n+1}(F)$82 unitary theory (Pattanayak, 12 Dec 2025).