Bernstein's modified local Langlands correspondence (LLC-prime): extension to all irreducible smooth representations, the role of the two-fold cover $W_2$, and the analytic theory of the attached $R$-functions

In a 2020 note, Joseph Bernstein argued that the $p$-adic local Langlands correspondence (LLC) contains a hidden sign: its standard normalization depends on a non-canonical choice of a square root of the residual characteristic $p$, so the correspondence is not functorial in the coefficient field. He proposed a modified correspondence (LLC-prime) between irreducible smooth representations of a split reductive group $G$ over a $p$-adic field and modified Langlands parameters - morphisms from the Weil group into a canonically enlarged dual group $G_D$ satisfying a modulus condition, equivalently genuine morphisms from a two-fold cover of the Weil group into an enhanced dual group $(G^\vee,\epsilon)$. Bernstein constructed LLC-prime only for representations of principal-series type and explicitly left the rest of the program open: extending the correspondence to all irreducible smooth representations, incorporating the canonical two-fold central extension $W_2$ of the Weil group, formulating functoriality for dual-group morphisms incompatible with the enhancement, defining ramified and Archimedean local factors for the attached automorphic $R$-functions together with proofs of meromorphic continuation and a functional equation, and establishing integral ($\mathbb{Z}$-)forms of his absolute Satake isomorphism. This problem asks for the completion of that program. No resolution of any component of this program is known in the literature to date.

Background

Let $F$ be a $p$-adic field (a finite extension of $\mathbb{Q}_p$) and let $G$ be a connected split reductive algebraic group over $F$, with topological group of $F$-points $\mathbf{G}=G(F)$. The category $\mathcal{M}(\mathbf{G})$ of smooth complex representations (representations in which every vector is stabilized by an open subgroup) is a central object of modern number theory and representation theory, and one of its basic problems is to describe the set $\mathrm{Irr}\,\mathbf{G}$ of irreducible smooth representations.

The Weil group $W_F$ is the subgroup of the absolute Galois group of $F$ consisting of elements acting on the residue field by integer powers of the Frobenius automorphism; it is the Galois-theoretic object on which Langlands parameters live. To $G$, Langlands associated the complex dual group $G^\vee$ (e.g. the dual of $\mathrm{PGL}(2)$ is $\mathrm{SL}(2,\mathbb{C})$), and proposed the local Langlands correspondence (LLC): a natural bijection between $\mathrm{Irr}\,\mathbf{G}$ and the set of Langlands parameters, i.e. continuous homomorphisms $\varphi\colon W_F\to G^\vee$ taken up to conjugation. The LLC is known in important cases ($\mathrm{GL}_n$, classical groups) but remains open in general.

The starting point is the unramified case. Fix a hyperspecial maximal compact subgroup $K\subset\mathbf{G}$ (a maximal compact subgroup coming from a smooth reductive model of $G$ over the ring of integers of $F$) and the spherical Hecke algebra $\mathcal{H}(\mathbf{G})=\mathcal{H}_K(\mathbf{G})$ of compactly supported, $K$-bi-invariant functions on $\mathbf{G}$ under convolution. The Satake isomorphism identifies $\mathcal{H}(\mathbf{G})$ with the $W$-invariant regular functions on the dual torus $T^\vee$, where $W$ is the Weyl group. The classical proof passes through normalized parabolic induction $\mathrm{Ind}_B^{\mathbf{G}}(\chi)$, which twists ordinary induction by a square root $\delta^{1/2}$ of the modulus character; fixing $\delta^{1/2}$ amounts to choosing a square root $\sqrt{p}\in\mathbb{C}$ of the residual characteristic. Bernstein's observation is that this choice is not preserved by field automorphisms of $\mathbb{C}$, so the whole correspondence fails his basic requirement: the construction should be functorial in the coefficient field $\mathbb{C}$, i.e. commute with all automorphisms of $\mathbb{C}$ (equivalently, make sense over any algebraically closed coefficient field of characteristic $0$ without auxiliary choices). Traces of the resulting sign imbalance had appeared earlier in case-by-case analyses of the correspondence (local, global, and geometric).

Bernstein's remedy (arXiv:2004.10487) replaces both sides of the LLC. On the dual side, he constructs from the flag variety of $G$ a canonical central extension $1\to\mathbb{G}_m\to\widetilde{G}\to G\to 1$ - built from a square root of the anti-canonical line bundle on the flag variety, the variety of Borel subgroups of $G$ - carrying a canonical weight $r$ of type $\rho$ (a weight behaving like the half-sum of positive roots: $\sigma_\alpha(r)=r/\alpha$ for every simple root $\alpha$, written multiplicatively). The extension also carries a canonical morphism $j\colon\widetilde{G}\to\mathbb{G}_m$ whose composition with the embedding $\mathbb{G}_m\to\widetilde{G}$ is the squaring map; its kernel $H=\ker j$ is a central two-fold cover of $G$ itself, which may be connected or disconnected - Bernstein remarks that $H$ is 'what we really are interested in' and that for disconnected $H$ it is not clear how to define its dual group, his construction being a way to define that dual group via the resolution $1\to H\to\widetilde{G}\to\mathbb{G}_m\to 1$. Dually this produces an exact sequence $1\to G^\vee\to G_D\to\mathbb{G}_m\to 1$ together with a central two-fold cover morphism $j\colon\mathbb{G}_m\to G_D$, and the Langlands dual group acquires the structure of an enhanced group $(G^\vee,\epsilon)$: a distinguished central element $\epsilon$ of order dividing $2$, equivalently a morphism $\epsilon\colon\mu_2\to Z(G^\vee)$ from the group of square roots of unity into the center. For $G=\mathrm{PGL}(2)$ one gets $\widetilde{G}=\mathrm{GL}(2)$, $G_D=\mathrm{GL}(2,\mathbb{C})$, and $G^\vee=\mathrm{SL}(2,\mathbb{C})$. Let $\omega\colon W_F\to\mathbb{C}^\times$ be the modulus character of the Weil group (the unramified character sending a geometric Frobenius to the cardinality $q$ of the residue field). A modified Langlands parameter is a morphism $\psi\colon W_F\to G_D$ whose composition with $G_D\to\mathbb{G}_m$ equals $\omega$, taken up to $G^\vee$-conjugation; equivalently, a genuine morphism (one sending the generator of the covering kernel to $\epsilon$) from the two-fold cover $\widetilde{W}_F$ of $W_F$ induced by the squaring map on $\mathbb{C}^\times$ via $\omega$, into $G^\vee$. The modified correspondence (LLC-prime) posits a canonical, coefficient-field-functorial correspondence between $\mathrm{Irr}\,\mathbf{G}$ and these modified parameters. Bernstein constructs it for representations of principal-series type - those with nonzero Jacquet module along a Borel, a class containing all unramified representations - via the non-normalized Jacquet functor $J(\pi)=V_U$ (coinvariants under the unipotent radical) and the dot action $\sigma_\alpha\cdot\chi=\sigma_\alpha(\chi)/|\alpha|$ of the Weyl group; for supercuspidal representations (those with vanishing Jacquet modules along every proper parabolic) the construction gives nothing, and the extension is conjectural.

Along the way Bernstein proves an absolute Satake isomorphism: the Hecke algebra $\mathcal{H}(\mathbf{G})$ is canonically the algebra of $G^\vee$-invariant regular functions on the fiber $X_q$ of $G_D\to\mathbb{G}_m$ over $q$, a statement meaningful over any coefficient field; for the universal Hecke algebra over $\mathbb{C}[q^{\pm 1}]$ he notes the isomorphism is in fact definable over $\mathbb{Q}$ and 'probably, after taking care of some normalizations, ... over $\mathbb{Z}$ as well'. He also sketches how the geometric Satake theorem of Ginzburg and Mirkovic-Vilonen (the categorical upgrade of the Satake isomorphism to perverse sheaves on the affine Grassmannian of $G$) already secretly produces the enhanced group: its natural fiber functor lands in super vector spaces, and the parity operator is exactly $\sigma(\epsilon)$.

Two further loose ends complete the picture. First, local class field theory provides a unique non-trivial two-fold central extension $W_2\to W_F$ of the Weil group itself (induced from the squaring cover of $\overline{F}^{\,\times}$); Bernstein believes $W_2$ should play an important role in the correspondence but was 'not able to bring it to the picture' - his cover $\widetilde{W}_F$ splits non-canonically, whereas $W_2$ does not. Second, the functoriality principle (a morphism $\nu\colon H^\vee\to G^\vee$ of dual groups should induce a transfer of representations) becomes problematic when $\nu$ does not respect the enhancements $\epsilon$ - already for the standard embedding $\mathbb{C}^\times\to\mathrm{SL}(2,\mathbb{C})$ - and Bernstein proposes a generalized principle for his almost algebraic groups (reductive groups equipped with a central enhancement $\epsilon\colon\mu_2\to G^\vee$, represented on the side of $F$-points by genuine representations of an associated two-fold topological cover).

Finally, for a split reductive group over a global field $k$ and an automorphic representation $\Pi$ (an irreducible representation of the adelic group $G(\mathbb{A}_k)$, the restricted product of the groups $G(k_v)$ over all places $v$ of $k$, realized in the space of functions of moderate growth on the automorphic quotient), Bernstein attaches to every complex-analytic finite-dimensional representation $\tau$ of $G_D$ a partial $R$-function: at an unramified place $v$ the local factor is $R(\pi_v,\tau;s)=\det(1-q_v^{-s}\,\tau\circ\psi_v(\mathrm{Frob}_v))^{-1}$, where $q_v$ is the residue-field cardinality and the twist by $\omega^{-s}$ is the source's regularization, and the partial $R$-function $R_S(\Pi,\tau;s)=\prod_{v\notin S}R(\pi_v,\tau;s)$ is absolutely convergent for $\mathrm{Re}(s)\gg 0$ - a direct analogue of Langlands's automorphic $L$-functions but in the modified normalization, where the special points $s\in\mathbb{Z}$ carry arithmetic significance for algebraic $\Pi$ and $\tau$. Here $S$ is a finite set of places containing the ramified and Archimedean ones (the real and complex places), since Bernstein can define local factors only at unramified places; he conjectures meromorphic continuation to all $s\in\mathbb{C}$, and can only guess, for a cuspidal automorphic representation $\Pi$, the shape of the functional equation relating $R(\Pi,\tau;s)$ and $R(\widetilde{\Pi},\tau^\vee;-s)$ for the contragredient representation $\widetilde{\Pi}$ and the dual $\tau^\vee=p\otimes\tau^*$, where $p$ denotes the one-dimensional representation of $G_D$ given by the projection $p\colon G_D\to\mathbb{G}_m$, 'if all this is done'.

Problem Statement

Complete Bernstein's modified local Langlands program from arXiv:2004.10487. The overarching objective is to turn the modified correspondence LLC-prime - the conjectured canonical, coefficient-field-functorial correspondence between irreducible smooth representations of $\mathbf{G}=G(F)$ (for $G$ connected split reductive over a $p$-adic field $F$) and modified Langlands parameters, i.e. $G^\vee$-conjugacy classes of morphisms $\psi\colon W_F\to G_D$ with $p\circ\psi=\omega$, or equivalently genuine morphisms $\widetilde{W}_F\to G^\vee$ for the enhanced dual group $(G^\vee,\epsilon)$ - from a construction on principal-series-type representations into an established theory. Concretely, the source's jointly linked open questions are: (1) construct, or prove the existence of, the correspondence LLC-prime on all of $\mathrm{Irr}\,\mathbf{G}$ - the case of supercuspidal representations is the crux - extending Bernstein's assignment on principal-series type and satisfying the basic requirement of functoriality under all automorphisms of the coefficient field; (2) clarify and incorporate the canonical non-trivial two-fold central extension $W_2\to W_F$ of local class field theory into this picture, in particular its relation to the cover $\widetilde{W}_F$ and to the modified parameters, which the author was 'not able to bring to the picture'; (3) establish the generalized functoriality principle: formulate and justify canonical transfers for morphisms of dual groups that do not respect the enhancements (Bernstein's almost-algebraic-group framework), resolving the fate of the naive principle for enhancement-incompatible morphisms such as the standard embedding $\mathbb{C}^\times\to\mathrm{SL}(2,\mathbb{C})$; (4) for a split reductive group over a global field, define the local $R$-factors at ramified and Archimedean places, prove meromorphic continuation of the full $R$-functions $R(\Pi,\tau;s)$, and establish, for a cuspidal automorphic representation $\Pi$, the functional equation relating $R(\Pi,\tau;s)$ and $R(\widetilde{\Pi},\tau^\vee;-s)$ for $\tau^\vee=p\otimes\tau^*$ ($p$ the one-dimensional representation of $G_D$ given by the projection $G_D\to\mathbb{G}_m$); and (5) establish the integral refinement of the absolute Satake isomorphism for the universal Hecke algebra over $\mathbb{Z}$ (exhibiting the required normalizations). These components jointly constitute one theory: (1) is the correspondence itself, (2)-(3) are its structural coherence on the Galois side, and (4)-(5) are its arithmetic payoff, in the same sense in which the classical local Langlands program spans the correspondence, its functoriality, and its $L$-functions. Completion of the program requires all five components (and, within (4), all three of its parts); the supercuspidal case identified in (1) is the central difficulty of the program, not a self-contained alternative whose resolution completes it.

References

1. Joseph Bernstein, 'Hidden sign in Langlands' correspondence', arXiv:2004.10487 (2020), https://arxiv.org/abs/2004.10487 (the primary source for the open program)
2. Laurent Fargues and Peter Scholze, 'Geometrization of the local Langlands correspondence', arXiv:2102.13459 (2021), https://arxiv.org/abs/2102.13459
3. Peter Scholze, 'Geometrization of the local Langlands correspondence, motivically', arXiv:2501.07944 (2025), https://arxiv.org/abs/2501.07944
4. Kevin Buzzard and Toby Gee, 'The conjectural connections between automorphic representations and Galois representations', arXiv:1009.0785 (2010), https://arxiv.org/abs/1009.0785
5. Joseph Bernstein, 'Stacks in Representation Theory. What is a continuous representation of an algebraic group?', arXiv:1410.0435 (2014), https://arxiv.org/abs/1410.0435
6. Ivan Mirkovic and Kari Vilonen, 'Geometric Langlands duality and representations of algebraic groups over commutative rings', Annals of Mathematics 166 (2007), 95-143, doi:10.4007/annals.2007.166.95, https://arxiv.org/abs/math/0401222
7. D. Gaitsgory and S. Lysenko, 'Parameters and duality for the metaplectic geometric Langlands theory', arXiv:1608.00284 (2016), https://arxiv.org/abs/1608.00284