Venkatesh's strong conjecture: closure of the Arthur-type unitary dual of GL(n) under restriction, induction, and tensor product

The conjecture under study is Conjecture 1 (the "strong conjecture") of A. Venkatesh's paper "The Burger-Sarnak method and operations on the unitary dual of GL(n)" (Representation Theory 9 (2005), 268-286): for a local field k of characteristic 0, the operations of restriction to a standardly embedded GL(m) inside GL(n), induction from Levi subgroups, and tensor product should preserve the Arthur-type part of the unitary dual of GL(n,k) - that is, applying them to irreducible unitary representations of Arthur type never produces complementary series, even weakly (weak containment in the resulting unitary representation). The source paper proves only the weak version, which controls the SL(2)-type (a partition attached to each representation) but cannot exclude complementary series, and it observes that the strong conjecture would follow from the generalized Ramanujan conjecture for GL(n). A large adjacent literature - the non-tempered Gan-Gross-Prasad conjecture and quotient branching laws (Gan-Gross-Prasad 2020; Gurevich 2021; Chan 2022 and 2022/2026; Chen-Chen 2025; Pattanayak 2026) - settles the smooth Hom-space problem for the pair (GL(n+1), GL(n)), but its authors explicitly position it in the smooth category, distinct from Venkatesh's unitary weak-containment question. A second adjacent line (Kobayashi-Speh 2025; Ditlevsen-Frahm 2025) computes explicit direct-integral decompositions of particular unitary restrictions - discrete series of symmetric spaces, and Speh representations and Stein's complementary series of GL(2n,R) restricted to GL(2n-1,R) - but likewise makes no claim about the Arthur-type closure question; no work in the literature proves or refutes the strong conjecture for any of the three operations in the unitary sense. The Ramanujan route remains the only stated implication and is itself open.

Background

Let $k$ be a local field of characteristic 0, i.e. $k$ is $\mathbb{R}$, $\mathbb{C}$, or a finite extension of $\mathbb{Q}_p$. The representations studied here are the smooth complex representations of the general linear groups $\mathrm{GL}(n,k)$ (for $p$-adic $k$: representations in which every vector is fixed by an open subgroup; for $k=\mathbb{R},\mathbb{C}$: smooth Fréchet representations of moderate growth). A representation is unitarizable if it carries a positive-definite invariant Hermitian form; the irreducible unitarizable representations form the unitary dual. The unitary dual carries the Fell topology, in which an irreducible $\sigma$ is weakly contained in a unitary $\pi$ if every matrix coefficient of $\sigma$ is a limit (uniform on compact sets) of sums of matrix coefficients of $\pi$. When $\pi$ is written as a direct integral of irreducibles (always possible for unitary representations of such groups), the weakly contained representations are exactly the irreducibles appearing in the decomposition. Weak containment, rather than literal containment, is the right notion of support because restriction, induction, and tensor product of unitary representations generally decompose continuously.

The irreducible unitary representations of $\mathrm{GL}(n,k)$ are classified combinatorially. In the non-Archimedean case this is the Zelevinsky-Tadić classification: from a discrete-series representation $\delta$ of $\mathrm{GL}(m,k)$ (essentially square-integrable modulo the center; in fact a twist of a Steinberg-type representation $\mathrm{St}(\Delta)$ attached to a Zelevinsky segment $\Delta$, i.e. a string of consecutive twists $\rho\nu^a,\dots,\rho\nu^b$ of a single cuspidal representation $\rho$) one forms a Speh representation $u(\delta,j)$ of $\mathrm{GL}(mj,k)$ - normalized parabolic induction of $j$ copies of $\delta$ with symmetrically shifted exponents - and, for $0<\alpha<1/2$, a complementary series factor $u(\delta,j)[\alpha,-\alpha]$, induced from $u(\delta,j)\nu^{\alpha}\otimes u(\delta,j)\nu^{-\alpha}$ on a two-block Levi (here $\nu(g)=|\det g|_k$). Every irreducible unitary representation is uniquely a product (normalized parabolic induction) of Speh factors and complementary factors. The representations whose factors are all Speh are called of Arthur type (Tadić calls them rigid): they are the local representations predicted by Arthur parameters for $\mathrm{GL}(n)$, and among them the tempered representations (those weakly contained in the regular representation, equivalently those whose matrix coefficients lie in $L^{2+\varepsilon}$ for every $\varepsilon>0$) are exactly those of minimal SL(2)-type (below). The Archimedean case has an analogous classification (due to Vogan) with the same qualitative dichotomy between Arthur-type and complementary-series building blocks.

Venkatesh attaches to each Arthur-type $\pi$ its SL(2)-type. An Arthur parameter for $\mathrm{GL}(n)$ over $k$ is, roughly, a conjugacy class of homomorphisms from the product of the Weil group of $k$ (the symmetry group governing the local Langlands correspondence) with the complex group $\mathrm{SL}_2(\mathbb{C})$ into the dual group of $\mathrm{GL}(n)$, which is $\mathrm{GL}(n,\mathbb{C})$; the second factor is Arthur's $\mathrm{SL}_2$. Restricting the standard $n$-dimensional representation of $\mathrm{GL}(n,\mathbb{C})$ along the image of this $\mathrm{SL}_2(\mathbb{C})$ and decomposing it into irreducible $\mathrm{SL}_2(\mathbb{C})$-modules gives a partition $\langle n_1,\dots,n_r\rangle$ of $n$ (the parts are the dimensions of the irreducible summands, equivalently the Jordan-block sizes of the image of a regular unipotent element), and this partition is the type of $\pi$; combinatorially a Speh factor $u(\delta,j)$ contributes $\deg(\delta)$ parts equal to $j$. The trivial representation has type $\langle n\rangle$, every tempered representation has type $\langle 1,\dots,1\rangle$, and an Arthur-type representation has type $\langle 1,\dots,1\rangle$ only if it is tempered. The defect of the invariant is that a complementary series representation of $\mathrm{GL}(2)$ also has type $\langle 1,1\rangle$: types cannot see complementary series.

Three natural operations act on the unitary dual. Restriction: $\mathrm{GL}(m,k)$ embeds in $\mathrm{GL}(n,k)$ in the top-left corner (the standard embedding) for $m<n$, and one restricts a unitary representation of $\mathrm{GL}(n,k)$ to this subgroup. Induction: for a Levi subgroup $M\cong\mathrm{GL}(n_1,k)\times\cdots\times\mathrm{GL}(n_r,k)$ (block-diagonal matrices in a parabolic subgroup), one unitarily induces (Mackey's unitary induction, which on smooth vectors is normalized parabolic induction) a tensor product of unitary representations of the blocks. Tensor product: the diagonal action on the Hilbert tensor product of two representations of $\mathrm{GL}(n,k)$. Venkatesh's Proposition 2 (the weak conjecture) computes the effect on types: for $\pi$ of Arthur type with type $\langle n_i\rangle$, the restriction to standardly embedded $\mathrm{GL}(m,k)$ has type $\langle n_i+m-n\rangle$ (dropping non-positive parts and padding with 1s); induction from a Levi with block types $\langle m_{i,k}\rangle$ of $n_i$ gives type $\langle m_{i,k}+n_i-n\rangle$; and the tensor product of types $\langle n_i\rangle$ and $\langle m_j\rangle$ has type $\langle n_i+m_j-n\rangle$. The proof combines Mackey theory with the Burger-Sarnak principle - the closure, in the Fell topology, of the automorphic spectrum (the set of local components of automorphic representations, i.e. of irreducible constituents of the $L^2$-spaces of arithmetic quotients of $\mathrm{GL}(n)$ over number fields) under these operations, established by Burger, Li and Sarnak - via a global argument that requires a complex place. Part (3) had appeared as a conjecture of Clozel, whose IMRN 2004 paper, built on Arthur's conjectures, originated this circle of questions; Venkatesh's paper arose from understanding Clozel's ideas.

The strong conjecture (Conjecture 1 of the source paper) asserts that the three operations preserve the Arthur-type part of the unitary dual: if the inputs are of Arthur type, every irreducible unitary representation weakly contained in the output is again of Arthur type; equivalently, no irreducible unitary representation involving a complementary-series factor is ever weakly contained in the restriction, induction, or tensor product of Arthur-type representations. Venkatesh proves that the Arthur-type dual is contained in the automorphic dual (Lemma 4, using the residual spectrum of Moeglin-Waldspurger and Eisenstein series) and notes that equality is expected - it is equivalent to the generalized Ramanujan conjecture for $\mathrm{GL}(n)$ (temperedness of cuspidal automorphic representations), which remains open. Since the automorphic dual is closed under the three operations by Burger-Sarnak, the strong conjecture would follow from Ramanujan; the paper states that the author does not know how to prove Conjecture 1 in full generality, and that the methods cannot rule out complementary series.

Since 2020 a major program has settled a formally different question: which irreducible smooth representations occur as quotients of the restriction of an Arthur-type representation from $\mathrm{GL}(n+1,k)$ to $\mathrm{GL}(n,k)$ (equivalently, for which pairs is the Hom space $\mathrm{Hom}_{\mathrm{GL}(n,k)}(\pi,\sigma)$ nonzero). This is the non-tempered Gan-Gross-Prasad conjecture of Gan, Gross and Prasad, with subsequent work by Gurevich, Chan, Chen-Chen and Pattanayak. Those theorems live in the smooth category (algebraic Hom spaces); the strong conjecture is about weak containment in the unitary category, and, as Gurevich states, there is no immediate relation between the two problems. The source paper also raises (Sections 1 and 3.7) extensions beyond $\mathrm{GL}(n)$ - other reductive groups, base change and twisted forms - which remain largely open even on the type level and lie outside the scope of the conjecture stated here.

Problem Statement

Let $k$ be a local field of characteristic 0, and let $\mathrm{GL}(n,k)_{\mathrm{Ar}}$ denote the Arthur-type part of the unitary dual of $\mathrm{GL}(n,k)$: the set of irreducible unitary representations that are products of Speh representations $u(\delta,j)$ in the Zelevinsky-Tadić classification (non-Archimedean case) or their Vogan-classification analogues (Archimedean case) - equivalently, whose classification involves no complementary-series factor $u(\delta,j)[\alpha,-\alpha]$ with $0<\alpha<1/2$.

Venkatesh's strong conjecture (Conjecture 1 of "The Burger-Sarnak method and operations on the unitary dual of GL(n)", Representation Theory 9 (2005), 268-286) asserts that $\mathrm{GL}(n,k)_{\mathrm{Ar}}$ is preserved, in the sense of weak containment (Fell topology), under the three operations of his Proposition 2:

(1) Restriction. For $m<n$, let $\mathrm{GL}(m,k)$ be embedded in $\mathrm{GL}(n,k)$ in the top-left corner (the standard embedding). For every $\pi\in\mathrm{GL}(n,k)_{\mathrm{Ar}}$, every irreducible unitary representation weakly contained in the restriction of $\pi$ to $\mathrm{GL}(m,k)$ again lies in $\mathrm{GL}(m,k)_{\mathrm{Ar}}$.

(2) Induction. For a Levi subgroup $M\cong\mathrm{GL}(n_1,k)\times\cdots\times\mathrm{GL}(n_r,k)$ of $\mathrm{GL}(n,k)$ and Arthur-type representations $\pi_i\in\mathrm{GL}(n_i,k)_{\mathrm{Ar}}$, every irreducible unitary representation weakly contained in the (unitary, Mackey) induction of $\pi_1\otimes\cdots\otimes\pi_r$ from $M$ to $\mathrm{GL}(n,k)$ lies in $\mathrm{GL}(n,k)_{\mathrm{Ar}}$.

(3) Tensor product. For $\pi_1,\pi_2\in\mathrm{GL}(n,k)_{\mathrm{Ar}}$, every irreducible unitary representation weakly contained in $\pi_1\otimes\pi_2$ (diagonal action on the Hilbert tensor product) lies in $\mathrm{GL}(n,k)_{\mathrm{Ar}}$.

Equivalently: prove that no irreducible unitary representation that is not of Arthur type - that is, whose classification data involve at least one complementary-series factor $u(\delta,j)[\alpha,-\alpha]$ with $0<\alpha<1/2$ - is ever weakly contained in the restriction, induction, or tensor product of Arthur-type representations, for any $n$, any $m<n$, and any local field $k$ of characteristic 0; or refute the conjecture by exhibiting one explicit counterexample, with all representations specified by their classification data (Zelevinsky multisegments/Tadić parameters, or Archimedean Vogan parameters). The source paper proves the weak version, which controls only SL(2)-types and is compatible with complementary series, and shows the strong conjecture would follow from the generalized Ramanujan conjecture for GL(n) (temperedness of cuspidal automorphic representations), which remains open; a proof must therefore be either unconditional-local or rest on new global input. Partial resolutions (fixed corank, fixed field type, tempered or generic subclasses) do not by themselves settle the conjecture as stated. The extensions to other reductive groups and twisted forms raised in the source paper are context, not part of this core conjecture.

References

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