L-Morphisms

An L-morphism is a continuous group homomorphism between L-groups that is compatible with the Galois action. L-morphisms are the "morphism layer" of Langlands functoriality: a single L-morphism conjecturally encodes a transfer of automorphic representations.

Definition

Let \(G\) and \(H\) be connected reductive groups over a field \(k\) (local or global), with L-groups \[ {}^L G \;=\; \hat G \rtimes W_k, \qquad {}^L H \;=\; \hat H \rtimes W_k \] (using the Weil-group form). An L-morphism \(\eta : {}^L H \to {}^L G\) is a continuous group homomorphism such that:

\(\eta\) commutes with the projections to \(W_k\): \(\mathrm{pr}_W \circ \eta = \mathrm{pr}_W\);
the restriction \(\eta|_{\hat H} : \hat H \to \hat G\) is a homomorphism of complex algebraic groups.

L-morphisms are taken up to inner conjugation by \(\hat G\).

Equivalent description via 1-cocycles

Equivalent data: a pair \((f, c)\) where:

\(f : \hat H \to \hat G\) is a complex algebraic group homomorphism;
\(c : W_k \to \hat G\) is a continuous 1-cocycle (with respect to the \(W_k\)-action on \(\hat G\)) "intertwining" the Galois actions on \(\hat H\) and \(\hat G\) via \(f\): \[ f(\rho_H(w)(h)) \;=\; c(w) \cdot \rho_G(w)(f(h)) \cdot c(w)^{-1}. \]

Two pairs are equivalent if they differ by an inner automorphism of \(\hat G\).

Examples

Inclusion of a Levi

For a parabolic subgroup \(P \subseteq G\) with Levi \(L\): \[ \eta : {}^L L \;\hookrightarrow\; {}^L G. \] Conjecturally induces parabolic induction at the automorphic level: irreducible reps of \(L\) lift to subquotients of induced reps of \(G\).

Base change

For a finite Galois extension \(K/k\): \[ \eta : {}^L G_k \;\to\; {}^L G_K \] encodes base change. Concretely: the restriction \(\eta|_{\hat G}\) is the diagonal, and the Galois cocycle records the restriction \(W_k \to W_K\).

Symmetric power

For \(G = \mathrm{GL}_2\) and \(G' = \mathrm{GL}_{n+1}\): \[ \mathrm{Sym}^n : \mathrm{GL}_2(\mathbb{C}) \;\to\; \mathrm{GL}_{n+1}(\mathbb{C}) \] is an L-morphism (trivial cocycle since both groups split). Conjecturally induces the symmetric power lift (Kim-Shahidi for \(n \le 4\), Newton-Thorne 2021 for general \(n\)).

Tensor product

For \(G_1 = \mathrm{GL}_m\), \(G_2 = \mathrm{GL}_n\), \(G_3 = \mathrm{GL}_{mn}\): \[ \mathrm{GL}_m(\mathbb{C}) \times \mathrm{GL}_n(\mathbb{C}) \;\to\; \mathrm{GL}_{mn}(\mathbb{C}) \] is the tensor product L-morphism. Conjecturally induces the automorphic tensor product (= Rankin-Selberg L-function in pieces).

Endoscopic embedding

For an endoscopic group \(H\) of \(G\): there is a canonical L-morphism \[ \eta : {}^L H \;\to\; {}^L G \] arising from \(\hat H \subseteq \hat G\) (as the connected component of the centraliser of a semisimple element). Endoscopic functoriality conjecturally transfers representations from \(G\) to \(H\) (via trace identities and Shelstad-Kottwitz-Ngô).

Composition

L-morphisms compose: if \(\eta_1 : {}^L A \to {}^L B\) and \(\eta_2 : {}^L B \to {}^L C\), then \(\eta_2 \circ \eta_1 : {}^L A \to {}^L C\) is an L-morphism. The composition is well-defined up to inner conjugation.

Functoriality conjecture

Langlands functoriality conjecture. For every L-morphism \(\eta : {}^L H \to {}^L G\) (\(H, G\) reductive over \(k\)), there is a transfer of automorphic representations: \[ \mathcal{A}(H) \;\to\; \mathcal{A}(G), \qquad \pi \mapsto \eta_*(\pi), \] sending an automorphic representation \(\pi\) of \(H(\mathbb{A}_k)\) to one of \(G(\mathbb{A}_k)\), such that the L-parameter of \(\eta_*(\pi)\) is \(\eta \circ \varphi_\pi\).

This is the "fundamental conjecture" of the Langlands programme, unifying base change, symmetric power, tensor product, etc.

L-morphisms and characters

An L-morphism \(\eta\) restricted to the centre induces a homomorphism \(Z(\hat G) \to Z(\hat H)\), which dualises to: \[ \pi_1(H) \;\to\; \pi_1(G). \]

This is the "central character compatibility" of L-morphisms.

Verification: how functoriality has been proved

Symmetric square (\(\mathrm{Sym}^2\)): Gelbart-Jacquet (1978).
Symmetric cube and 4th power: Kim-Shahidi (2000s).
Symmetric \(n\)-th power for \(\mathrm{GL}_2\): Newton-Thorne (2021); first complete proof.
Base change for \(\mathrm{GL}_n\) in cyclic case: Arthur-Clozel (1989).
Endoscopic transfer (fundamental lemma): Ngô (2008); Fields medal 2010.
Tensor product: largely conjectural for general \(\mathrm{GL}_m \times \mathrm{GL}_n\) (gives the Rankin-Selberg integral, not yet a direct functoriality).

Mathlib

Not formalised; requires the full Langlands programme infrastructure.

References

Borel, Automorphic L-functions (Corvallis 1979).
Gelbart, An elementary introduction to the Langlands program (Bull.\ AMS 1984).
Ngô, Le lemme fondamental pour les algèbres de Lie (Publ.\ IHES 2010).
Newton–Thorne, Symmetric power functoriality for holomorphic modular forms (Publ.\ IHES 2021).