On the Local Langlands Correspondence for Split Classical Groups Over Local Function Fields

Abstract

We prove certain depth bounds for Arthur’s endoscopic transfer of representations from classical groups to the corresponding general linear groups over local fields of characteristic 0, with some restrictions on the residue characteristic. We then use these results and the method of Deligne and Kazhdan of studying representation theory over close local fields to obtain, under some restrictions on the characteristic, the local Langlands correspondence for split classical groups over local function fields from the corresponding result of Arthur in characteristic 0.

Rationality and holomorphy of Langlands–Shahidi L-functions over function fields

“…if π ∼ l π ′ , τ ∼ l τ ′ and ψ ∼ l ψ ′ . This is Proposition 13.5.3 of [10] for the split classical groups. For a general split reductive group scheme, we observe that we have:…”

“…We use the notation of [10], to denote π ∼ m π ′ whenever we are in the above situation. A property that is preserved under the transfer is the Moy-Prasad depth [34], depth(π) = depth(π ′ ).…”

“…A property that is preserved under the transfer is the Moy-Prasad depth [34], depth(π) = depth(π ′ ). We refer to [8] for more details, and also § 13 of [10] for a summary of results.…”

Local Langlands and Springer Correspondences

“…In [GaVa], Ganapathy and Varma used Arthur's results to lift the LLC for split symplectic and special orthogonal groups on a non Archimedean field of odd positive characteristic, but using a "Gan-Takeda type" characterization instead of the theory of endoscopy. They proved (see [GaVa,Theorem 13.6.1]) that given a tempered representation π of G, there exists a unique bounded Langlands parameter φ π , defined by suitable compatibility conditions on Langlands-Shahidi L-functions and γ-factors, and on Plancherel measures, together with the requirement that φ π is discrete if π belongs to the discrete series.…”

“…In the case when G is a classical group, the characteristic of F is zero (that is, F is a finite extension of Q p ) and p is odd, Ganapathy and Varma proved in [GaVa,Lemma 8.2.3] that the following inequality holds…”