---
title: Local Non-Tempered Gan–Gross–Prasad Conjecture
url: https://www.emergentmind.com/topics/local-non-tempered-gan-gross-prasad-conjecture
type: topic
---

# Local Non-Tempered Gan–Gross–Prasad Conjecture

Searching arXiv for recent 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 [2006.02623], and later extended to full smooth or unitarizable classes by generalized or extended relevance criteria [2512.11696]. 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 [2507.04355]. 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 [2511.21798].

## 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 \(g\mapsto \operatorname{diag}(g,1)\). The local problem is to determine when
\[
\operatorname{Hom}_{GL_n(F)}(\pi|_{GL_n(F)},\sigma)\neq 0
\]
for irreducible representations \(\pi\) of \(GL_{n+1}(F)\) and \(\sigma\) of \(GL_n(F)\). In the split Bessel formulation one instead considers \(G(F_v)=GL_n(F_v)\times GL_{n+1}(F_v)\) with diagonal \(H(F_v)\simeq GL_n(F_v)\), and the multiplicity
\[
m(\Pi_v)=\dim \operatorname{Hom}_{H(F_v)}(\Pi_v,\mathbf{C}),
\]
which is equivalent to the same local distinction problem [2511.21798].

For \(GL_N\), an Arthur parameter is a morphism
\[
\psi: W'_F\times SL_2(\mathbf{C})\to GL_N(\mathbf{C})
\]
that is tempered on \(W'_F\) and algebraic on \(SL_2(\mathbf{C})\). In the non-Archimedean Arthur-type setting this decomposes as
\[
\psi=\bigoplus_a M_a\otimes \operatorname{Sym}^{d_a}(\mathbf{C}^2),
\]
with each \(M_a\) tempered and \(d_a\ge 0\), and the associated irreducible representation is denoted \(\pi(\psi)\) [2006.02623]. In the Archimedean \(GL_N(F)\) setting, with \(F=\mathbf{R}\) or \(\mathbf{C}\), one similarly writes
\[
\psi=\bigoplus_i \phi_i\otimes S_{d_i},
\]
where \(\phi_i\) are tempered \(L\)-parameters and the associated Arthur-type representation is a product of generalized Speh representations [2507.04355].

The adjective non-tempered refers precisely to the presence of a nontrivial Arthur \(SL_2(\mathbf{C})\)-factor, equivalently some \(d_i>1\) in the Archimedean Speh description [2507.04355]. In the split characteristic-zero refinement, one also allows weak Arthur-type parameters
\[
\psi_{v,n}=\bigoplus_{i=1}^m (M_i\otimes |\cdot|^{\nu_i})\boxtimes S_{d_i},
\qquad |\nu_i|<1/2,
\]
with corresponding Langlands quotients built from twisted Speh blocks [2511.21798]. 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 [2512.11696].

## 2. Relevance of Arthur parameters

The central concept is relevance, a combinatorial matching condition on the \(SL_2(\mathbf{C})\)-exponents of the two Arthur parameters. In the non-Archimedean \(GL_{n+1}\mid GL_n\) Arthur-type setting, if
\[
\psi=\bigoplus_{d\ge 0}\Big(\bigoplus_i M_{p,i}\otimes \operatorname{Sym}^d\Big)\oplus
\bigoplus_{d\ge 1}\Big(\bigoplus_j M_{q,j}\otimes \operatorname{Sym}^{d-1}\Big),
\]
and
\[
\psi'=\bigoplus_{d\ge 0}\Big(\bigoplus_i M_{p,i}\otimes \operatorname{Sym}^{d-1}\Big)\oplus
\bigoplus_{d\ge 1}\Big(\bigoplus_j M_{q,j}\otimes \operatorname{Sym}^{d}\Big),
\]
with \(\operatorname{Sym}^{-1}=0\), then \((\psi,\psi')\) is called a relevant pair [2006.02623]. The schematic meaning is that the two parameters share the same tempered constituents while their \(SL_2\)-lengths interlace by one-step shifts.

For Archimedean \(GL\), relevance is formulated similarly but with a decomposition
\[
\psi_\pi=\sum_{i=1}^r L_i\otimes S_{d_i},
\]
and
\[
\psi_\sigma=\sum_{i\in I} L_i\otimes S_{d_i+1}+\sum_{j\in J} L_j\otimes S_{d_j-1}+\psi_0,
\]
where \(\psi_0\) is generic, meaning a sum of \(S_1\)-terms, and summands \(L\otimes S_0\) are understood to vanish [2507.04355]. The Archimedean formulation explicitly allows a generic remainder on the smaller group.

A useful Archimedean combinatorial reformulation is given by multiplicities \(m(\eta,a;\pi)\) of \(\eta\otimes S_a\) in \(\psi_\pi\), and alternating sums
\[
\Lambda(\eta,a;\pi,\sigma)=m(\eta,a;\pi)-m(\eta,a+1;\sigma)+m(\eta,a+2;\pi)-\cdots.
\]
Then relevance is equivalent to the inequalities
\[
\Lambda(\eta,a;\pi,\sigma)\ge 0,\qquad \Lambda(\eta,a;\sigma,\pi)\ge 0
\]
for all \(\eta\) and \(a\) [2507.04355]. This is the exact combinatorial encoding of the interlacing of \(SL_2\)-sizes.

In the split weak-Arthur setting for \(GL_n\times GL_{n+1}\), relevance is formulated asymmetrically as
\[
\psi_{v,n}=\bigoplus_{i=1}^{m_1} M_{1,i}\boxtimes S_{d(1,i)}
\oplus
\bigoplus_{j=1}^{m_2} M_{2,j}^\vee\boxtimes S_{d(2,j)-1},
\]
\[
\psi_{v,n+1}=\bigoplus_{i=1}^{m_1} M_{1,i}^\vee\boxtimes S_{d(1,i)-1}
\oplus
\bigoplus_{j=1}^{m_2} M_{2,j}\boxtimes S_{d(2,j)},
\]
which is the split-avatar of the Bessel relevance pattern [2511.21798]. In all these versions, the non-temperedness is carried entirely by the \(SL_2\)-factor.

A common misconception is that non-tempered \(GL\)-branching should continue to be governed by local \(L\)- and \(\epsilon\)-factors exactly as in the generic tempered case. In the Arthur-type non-Archimedean \(GL\) theorem, no additional condition involving
\[
L(s,\phi\otimes\phi'),\qquad \epsilon(1/2,\phi\otimes\phi',\psi)
\]
enters the criterion; the condition is purely the relevance of the Arthur \(SL_2\)-parts [2006.02623]. This does not negate the global role of local factors, but it sharply changes the local decision rule in the non-tempered \(GL\) setting.

## 3. Main theorems for general linear groups

The foundational non-Archimedean Arthur-type theorem states that if \(F\) is non-Archimedean, \(\pi=\pi(\psi)\) is an Arthur-type representation of \(GL_{n+1}(F)\), and \(\sigma=\pi(\psi')\) is an Arthur-type representation of \(GL_n(F)\), then
\[
\operatorname{Hom}_{GL_n(F)}(\pi|_{GL_n(F)},\sigma)\neq 0
\quad\Longleftrightarrow\quad
(\psi,\psi') \text{ is relevant}
\]
[2006.02623]. Together with the multiplicity-one theorem of Aizenbud–Gourevitch–Rallis–Schiffmann and Sun–Zhu, the nonzero Hom space is at most one-dimensional [2006.02623].

For Archimedean \(GL\), Chen–Chen prove the necessity direction in full unitary generality:
\[
\operatorname{Hom}_{G_n}(\pi,\sigma)\neq 0 \Longrightarrow (\pi,\sigma)\text{ is relevant},
\]
for irreducible unitary \(\pi\) of \(G_{n+1}\) and \(\sigma\) of \(G_n\), with \(F=\mathbf{R}\) or \(\mathbf{C}\) [2507.04355]. In the Arthur-type case, combined with Boisseau’s converse as stated there, one obtains the full equivalence
\[
\operatorname{Hom}_{G_n}(\pi_{n+1},\pi_n^\vee)\neq 0
\quad\Longleftrightarrow\quad
(\pi_{n+1},\pi_n)\text{ is relevant}
\]
[2507.04355].

The split characteristic-zero theorem of Xie–Zhang constructs an explicit local \(H(F_v)\)-invariant functional for weak Arthur-type representations \(\Pi_v\) with relevant parameter \(\psi_v\), showing regularity, factorization through the Langlands quotient, and nonvanishing of a normalized local zeta integral at the relevant point [2511.21798]. Since multiplicity one is classical, this yields
\[
m(\Pi_v)=1 \quad\Longleftrightarrow\quad \psi_v \text{ is relevant}
\]
for the split \(GL_n\times GL_{n+1}\) Bessel model over local fields of characteristic zero when combined with the previously established necessity results [2511.21798].

A later non-Archimedean extension proves a quotient branching law for all irreducible smooth representations:
\[
\operatorname{Hom}_{GL_n(F)}(\pi,\pi')\neq 0
\quad\Longleftrightarrow\quad
(\pi,\pi') \text{ is a generalized GGP relevant pair}
\]
[2512.11696]. 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 \((R1)\)–\((R4)\) [2512.11696]. This suggests that the Arthur-type non-tempered conjecture for \(GL\) 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 \(GL\)-representations is through Speh modules. In the non-Archimedean case, if \(\rho\) is an irreducible unitarizable cuspidal representation of \(GL_m(F)\), one defines generalized Steinberg representations \(St([\nu^a\rho,\nu^b\rho])\) for segments \([\nu^a\rho,\nu^b\rho]\), and the Speh representation
\[
U_\rho(m,d)
\]
as the unique irreducible quotient of a rectangular product of translated Steinberg segments [2006.02623]. Arthur-type representations for \(GL\) are exactly finite products of Speh representations, irreducible up to permutation of factors [2006.02623].

In the Archimedean setting, the generalized Speh module attached to a discrete series \(\delta\) of \(GL_k(F)\) and \(d\ge 1\) is
\[
(\delta,d):=LQ\big(\delta|\det|^{(d-1)/2}\times \delta|\det|^{(d-3)/2}\times\cdots\times \delta|\det|^{-(d-1)/2}\big),
\]
and an Arthur-type representation is a finite parabolic product of such Speh representations [2507.04355].

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 \(SL_2\)-length by one:
\[
\big(U_\rho(m,d+1)\big)^{[\mathrm{level}]}=U_\rho(m,d)
\]
[2006.02623]. Accordingly, if
\[
\pi=\prod_{p,i} U_{\rho_i}(m_i,d_i+1)\times \prod_{q,j} U_{\rho'_j}(m'_j,e_j),
\]
\[
\sigma=\prod_{p,i} U_{\rho_i}(m_i,d_i)\times \prod_{q,j} U_{\rho'_j}(m'_j,e_j+1),
\]
then \((\pi,\sigma)\) is the Speh reformulation of relevance [2006.02623].

This picture admits a precise multisegment translation. For general irreducible pairs, Chan introduces generalized relevance via right and left derivatives \(D^R_{\mathfrak m}\), \(D^L_{\mathfrak n}\), right and left integrals, and strong RdLi-commutativity of triples of segments and multisegments [2512.11696]. The core criterion is the existence of multisegments \(\mathfrak m,\mathfrak n\) such that
\[
D^R_{\mathfrak m}(\nu^{1/2}\pi)\cong D^L_{\mathfrak n}(\pi')
\]
and \((\mathfrak m,\mathfrak n,\nu^{1/2}\pi)\) is strongly RdLi-commutative [2512.11696]. In the unitary case this reduces to explicit block relations \((R1)\)–\((R4)\) between Speh parameters and complementary exponents [2512.11696].

An earlier stage of this development is Gurevich’s result that for p-adic \(GL_n\to GL_{n-1}\), nonvanishing in the Arthur-type case forces the GGP position of Speh blocks, and in the generic case the converse holds [1808.02640]. That work already exhibits the central role of derivatives, quasi-Speh representations, and the reduction of restriction problems to derivative Hom spaces [1808.02640].

## 5. Methods of proof and explicit decision procedures

The non-Archimedean Arthur-type proof relies on Bernstein–Zelevinsky derivatives and the mirabolic filtration of \(\pi|_{GL_n}\), together with an exact transfer between Hom spaces on successive layers [2006.02623]. 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 [2006.02623].

A concrete decision procedure is available in the Arthur-type non-Archimedean case. Given Speh factorizations
\[
\pi=\times_i U_{\rho_i}(m_i,d_i^\pi),\qquad
\sigma=\times_j U_{\rho_j}(m_j,d_j^\sigma),
\]
one orders factors by Tadić order, computes the highest shifted derivative of the maximal factor, compares with a matching factor in \(\sigma\), strips matched factors, transfers to a reduced pair using a cuspidal character \(o\), and iterates [2006.02623]. If every step matches the prescribed one-step shift, then Hom is nonzero; otherwise it vanishes [2006.02623].

The proof also uses an Ext-theoretic dual restriction. For suitable cuspidal \(o\),
\[
\operatorname{Ext}^i_{GL_n(F)}(\pi|_{GL_n},\sigma)
\simeq
\operatorname{Ext}^i_{GL_{n+1}(F)}((\sigma\times o)|_{GL_{n+1}},\pi)
\]
[2006.02623]. This transfer permits control of cuspidal supports and vanishing of many layers, and motivates the conjectural higher-Ext branching law formulated there [2006.02623].

In the Archimedean case, the necessity proof uses annihilator varieties. Gourevitch–Sahi identify the \(SL_2\)-type \(TP(\pi)\) with the transpose of the annihilator partition \(AP(\pi)\), while Gourevitch–Sayag–Karshon show that higher-corank Rankin–Selberg distinction imposes
\[
|TP(\pi)_i-TP(\sigma)_i|\le 1
\]
for all \(i\) [2507.04355]. Chen–Chen convert this microlocal inequality into the alternating-sum relevance inequalities via deformation, Mackey-theoretic reductions, and vanishing of closed-orbit contributions [2507.04355].

The split constructive theorem proceeds differently. It introduces a local Jacquet–Whittaker functional
\[
W_{P,\delta}^{\psi_0}(\varphi,\lambda)
\]
and the local Rankin–Selberg zeta integral
\[
Z_\delta(\varphi,\lambda)
=
\int_{N_{0,H}(F_v)\backslash H(F_v)} W_{P,\delta}^{\psi_0}(h,\varphi,\lambda)\,dh,
\]
with normalized version
\[
Z_\delta^\natural(\varphi,\lambda)
=
b(\lambda,\delta)\cdot
\frac{Z_\delta(\varphi,\lambda)}{L(\lambda+1/2,\delta_n\times \delta_{n+1})}
\]
[2511.21798]. Restricting to the relevant affine subspace and evaluating at the inducing point \(\lambda_0=-\nu_\pi+\nu_\Pi\) produces the local invariant functional
\[
L_{\mathrm{loc}}(\varphi)=RZ_\delta^\natural(\varphi,-\nu_\pi+\nu_\Pi)
\]
[2511.21798]. The normalized functional equation
\[
Z_\delta^\natural(\varphi,\lambda)=
Z_{w\delta}^\natural(N_\delta(w,\lambda)\varphi,w\lambda)
\]
is the key analytic input ensuring regularity and factorization through the Langlands quotient [2511.21798].

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

The local non-tempered \(GL\) 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 \(GL\)-restriction, obtaining
\[
\operatorname{Hom}_H(TM\otimes \beta,TN)\neq 0
\quad\Longleftrightarrow\quad
\text{the Arthur parameters are relevant}
\]
for the considered Bessel, Fourier–Jacobi, and mixed models [2006.02623]. The same work proves multiplicity one and finite-dimensionality of the corresponding Ext spaces [2006.02623].

The split characteristic-zero paper is explicitly formulated for the split Bessel model \(G=GL_n\times GL_{n+1}\) with diagonal \(H\simeq GL_n\), and it identifies the local normalized zeta integral as the local factor in the refined global Ichino–Ikeda-type formula [2511.21798]. 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 \(\pi_\rho(u,v)\) and complementary series \(\pi_\rho(u,v)(\alpha)\), with explicit relations
\[
(R1)\text{--}(R4)
\]
linking the parameters of \(\pi\) and \(\pi'\) [2512.11696]. The equivalence between this extended relevance and generalized derivative-based relevance gives a complete classification of unitary branching for \(GL_{n+1}\to GL_n\) [2512.11696]. 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 \(GL\), 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 [1708.06705]. For \(U(3)\times U(2)\), 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 [1501.00885]. For certain non-tempered \(A\)-packets of \(SO(3,2)\) and \(SO(4,1)\), packet-level uniqueness can fail altogether, so the generic local GGP principle does not extend naively [1310.4187]. These contrasts underscore that the \(GL\) 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 \(A\)-parameters is still conjectural in general; available results prove only one global implication and local vanishing statements for specific residual representations [2201.03270]. This marks an important boundary of present knowledge: the \(GL\) 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 \(GL\) Arthur-type theory is the Speh-to-Speh drop:
\[
\pi=U_\rho(m,d),\qquad \sigma=U_\rho(m,d-1).
\]
Then
\[
\operatorname{Hom}_{GL_{m(d-1)}(F)}(\pi|,\sigma)\neq 0,
\]
and the corresponding Arthur parameters are \(M_\rho\otimes \operatorname{Sym}^d\) and \(M_\rho\otimes \operatorname{Sym}^{d-1}\), so relevance is immediate [2006.02623].

A mixed-product example from the same work takes on \(GL_5(F)\)
\[
\pi=([\nu^{-1},\nu])\times 1\times 1,
\]
and on \(GL_4(F)\)
\[
\sigma=([\nu^{-1/2},\nu^{1/2}])\times St([\nu^{-1/2},\nu^{1/2}]).
\]
A filtration argument shows that a specific Bernstein–Zelevinsky layer of \(\pi|_{GL_4}\) maps onto \(\sigma\), hence
\[
\operatorname{Hom}_{GL_4}(\pi|,\sigma)\neq 0,
\]
again matching relevance at the Arthur-parameter level [2006.02623].

In the Archimedean Arthur-type setting, an illustrative case is
\[
\pi_4=(\delta,2)
\]
for a unitary discrete series \(\delta\) of \(GL_2(\mathbf{R})\), with parameter \(L(\delta)\otimes S_2\). Relevance predicts that a \(GL_3(\mathbf{R})\)-representation \(\sigma_3\) must contain \(L(\delta)\otimes S_1\) plus a generic remainder; then the local non-tempered GGP implies
\[
\operatorname{Hom}_{G_3}(\pi_4,\sigma_3^\vee)\neq 0,
\]
whereas \(S_3\) in place of \(S_1\) breaks relevance and forces vanishing [2507.04355].

The present status may be summarized succinctly.

| Setting | Result | Reference |
|---|---|---|
| Non-Archimedean \(GL\), Arthur type | \( \operatorname{Hom}\neq 0 \iff \) relevance | [2006.02623] |
| Archimedean \(GL\), Arthur type | full equivalence via Chen–Chen and Boisseau | [2507.04355] |
| Split \(GL_n\times GL_{n+1}\), weak Arthur type, char. \(0\) | constructive local functional and full local conjecture in split case | [2511.21798] |
| Non-Archimedean \(GL\), full smooth/unitary classes | generalized or extended relevance criterion | [2512.11696] |

The significance of these results is twofold. First, they identify the exact local branching rule for non-tempered \(GL\)-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 [2006.02623] [2511.21798].

The remaining open directions are not within basic \(GL\) 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 [2006.02623], full non-tempered analogues for classical or covering groups where uniqueness may fail or the packet selection rule is still incomplete [1310.4187] [2201.03270], and the systematic integration of complementary-series blocks into a unified local GGP formalism beyond the already established non-Archimedean \(GL\) unitary theory [2512.11696].

Source: https://www.emergentmind.com/topics/local-non-tempered-gan-gross-prasad-conjecture