Topic L-Groups
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L-Groups

The L-group \({}^L G\) is the Galois-twisted Langlands dual group: the semi-direct product of the complex dual group \(\hat G\) with the Galois (or Weil) group acting through pinned automorphisms. It is the target of L-parameters and the central character of the Langlands correspondence.

Definition

Let \(k\) be a field with separable closure \(\bar k\) and absolute Galois group \(\Gamma = \operatorname{Gal}(\bar k/k)\). Let \(G\) be a connected reductive group over \(k\). Pick a pinning of \(\hat G\) (maximal torus \(\hat T\), Borel \(\hat B\), and a system of root vectors). The Galois group \(\Gamma\) acts on the based root datum of \(G\), hence (via the duality swap) on the based root datum of \(\hat G\), giving a homomorphism \[ \rho_G : \Gamma \;\longrightarrow\; \operatorname{Aut}(\hat G, \text{pinning}). \]

The Galois form of the L-group is the semi-direct product: \[ {}^L G \;:=\; \hat G(\mathbb{C}) \;\rtimes_{\rho_G}\; \Gamma. \]

For local fields \(F\), one usually uses the Weil form: \[ {}^L G \;:=\; \hat G(\mathbb{C}) \;\rtimes\; W_F, \] with \(W_F\) the Weil group of \(F\) acting through its quotient \(W_F \to \Gamma\).

Equivalent formulations

  • Pinned automorphisms of \(\hat G\) are in bijection with diagram automorphisms (\(\mathbb{Z}/2\) for types \(A_n\) (\(n \ge 2\)), \(D_n\) (\(n \ne 4\)), \(E_6\); \(S_3\) for \(D_4\); trivial otherwise).
  • For \(G\) split: \(\rho_G\) is trivial, so \({}^L G = \hat G \times \Gamma\).
  • For \(G\) quasi-split: \(\rho_G\) factors through the diagram automorphism group; the L-group is uniquely determined.

Examples

\(\mathrm{GL}_n\) (split)

\({}^L \mathrm{GL}_n = \mathrm{GL}_n(\mathbb{C}) \times \Gamma\). Continuous L-parameters \(W_F \to {}^L \mathrm{GL}_n\) project to \(n\)-dimensional Galois representations (modulo the Frobenius semisimplification convention).

\(\mathrm{U}(n)\), quasi-split

For the unitary group attached to a quadratic extension \(E/F\), \(\rho_G : \mathrm{Gal}(E/F) = \mathbb{Z}/2 \to \mathrm{Out}(\hat G) = \mathrm{Out}(\mathrm{GL}_n) = \mathbb{Z}/2\) is the non-trivial map (diagram flip). Then \[ {}^L \mathrm{U}(n) \;=\; \mathrm{GL}_n(\mathbb{C}) \rtimes \mathrm{Gal}(E/F). \]

\({}^3 D_4\), the triality form

For the trialitarian form of \(\mathrm{Spin}_8\) over a cubic extension \(K/k\), \(\Gamma\) acts on \(\hat G = \mathrm{Spin}_8\) via the order-\(3\) diagram automorphism. The L-group involves the \(S_3\)-action on the Dynkin diagram of \(D_4\).

Tori

For a torus \(T\), \(\hat T = (X^*(T)) \otimes \mathbb{C}^\times\) with the natural Galois action on \(X^*(T)\). The L-group is \(\hat T \rtimes \Gamma\), and continuous L-parameters into \(\hat T\) are exactly continuous characters \(\chi : W_F^{\mathrm{ab}} \to \hat T\) — these are L-parameters for characters of \(T(F)\) via local class field theory.

L-parameters and L-packets

An L-parameter for \(G/F\) (\(F\) local) is a continuous homomorphism \[ \varphi : W_F \times \mathrm{SL}_2(\mathbb{C}) \;\longrightarrow\; {}^L G \] satisfying:

  • \(\varphi\) commutes with the projection \({}^L G \to \Gamma\);
  • \(\varphi|_{W_F}\) is semisimple (image consists of semisimple elements);
  • \(\varphi|_{\mathrm{SL}_2}\) is algebraic.

L-parameters are taken up to \(\hat G(\mathbb{C})\)-conjugacy. The conjectural map \[ \{\text{irreducible smooth reps of } G(F)\} \;\twoheadrightarrow\; \{\text{L-parameters}\} \] has finite fibres called L-packets.

Functoriality

A homomorphism \({}^L H \to {}^L G\) compatible with the projections to \(\Gamma\) conjecturally induces a transfer of automorphic representations from \(H\) to \(G\). Examples:

  • Base change: \({}^L (G_F) \to {}^L (G_E)\) for \(E/F\) a finite extension.
  • Symmetric power: \({}^L \mathrm{GL}_2 \to {}^L \mathrm{GL}_{n+1}\) via \(\mathrm{Sym}^n\), the \((n+1)\)-th symmetric power.
  • Endoscopic transfer: \({}^L H \to {}^L G\) where \(H\) is the centraliser of a semisimple element in \(\hat G\).

Variants

  • Weil-Deligne form: \({}^L G\) defined with the Weil-Deligne group \(W'_F = W_F \times \mathrm{SL}_2(\mathbb{C})\) in place of just \(W_F\); equivalent for L-parameter purposes.
  • C-group: variant introduced by Buzzard-Gee, more natural for arithmetic Galois representations.

Mathlib status

Not yet formalised; depends on full reductive-group infrastructure plus Galois cohomology.

References

  • Borel, Automorphic L-functions (Corvallis 1979).
  • Kottwitz, Stable trace formula: cuspidal tempered terms (Duke 1984), §1.
  • Gross-Reeder, Arithmetic invariants of discrete Langlands parameters (Duke 2010).
  • Buzzard-Gee, The conjectural connections between automorphic representations and Galois representations (Proc.\ LMS 2014).

Prerequisite Topics

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