Arithmetic invariants of discrete Langlands parameters. (English) Zbl 1207.11111

Gross, Benedict H.; Reeder, Mark

Duke Math. J. 154, No. 3, 431-508 (2010).

Let \(G\) be a connected reductive group over a nonarchimedean local field \(k\) of characteristic 0. Let \({\mathcal W}\) be the Weil group of \(k\). It is assumed that the connected centre of \(G\) is a \(k\)-anisotropic torus. Under this assumption a Langlands parameter \(\phi:{\mathcal W}\times \text{SL}_2({\mathbb C})\rightarrow {}^L G\) is discrete if and only if the centralizer in \(\hat G\) of the image of \(\phi\) is finite. The latter condition is equivalent to the condition that the adjoint \(L\)-function \(L(\phi, Ad, s)\) is regular at \(s = 0\). Thus for any discrete parameter \(\phi\) the \(\gamma\)-factor \(\gamma (\phi, Ad, s)\) is regular and nonzero at \(s = 0\). Put \(\gamma (\phi)=\gamma(\phi, Ad, 0)\). The authors show that, for any discrete parameter \(\phi\), there is a rational function \(\Gamma_{\phi}(X)\) with coefficients in \(\mathbb Q\) such that \(\gamma (\phi)/\gamma(\phi_0) = \Gamma_{\phi}(q)\), when the residue field of \(k\) has cardinality \(q\). Here \(\phi_0\) is the Steinberg parameter.
There is a conjecture, by Hiraga, Ichino and Ikeda, for the formal degree of a discrete series representation of \(G(k)\). This conjecture depends on the conjectural local Langlands correspondence. When reformulated using the Euler-Poincaré measure on \(G(k)\), the expression for the formal degree takes the form of a product of \(\Gamma_{\phi}(q)\) by a factor independent of \(q\).
The order of \(\Gamma_{\phi}(X)\) at \(X = 0\) is computed. It is conjectured that this order is always \(\geq0\), which amounts to an inequality for the Swan conductor of \(Ad\, \phi\). The conjecture is true for tamely ramified \(\phi\) and it is verified for some wildly ramified \(\phi\). Special attention is paid to the case where the inequality for the Swan conductor is an equality. In connection with the formal degree conjecture this leads to the definition and construction of ”simple” wild parameters and ”simple” supercuspidal representations of \(G(k)\). Assuming the formal degree conjecture (hence also the local Langlands conjecture), one can prove that the parameters of the simple supercuspidal representations are simple wild parameters.

Reviewer: J. G. M. Mars (Utrecht)

MSC:

11S37	Langlands-Weil conjectures, nonabelian class field theory
11S15	Ramification and extension theory

Keywords:

Swan conductor; formal degree

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