Stable conjugacy and epipelagic L-packets for Brylinski–Deligne covers of ${\text {Sp}}(2n)$

Abstract

Let F be a local field of characteristic not 2. We propose a definition of stable conjugacy for all the covering groups of $\text {Sp}(2n,F)$ constructed by Brylinski and Deligne, whose degree we denote by m. To support this notion, we follow Kaletha’s approach to construct genuine epipelagic L-packets for such covers in the non-archimedean case with $p \not \mid 2m$, or some weaker variant when $4 \mid m$; we also prove the stability of packets when $F \supset \mathbb {Q}_p$ with p large. When $m=2$, the stable conjugacy reduces to that defined by J. Adams, and the epipelagic L-packets coincide with those obtained by $\Theta$-correspondence. This fits within Weissman’s formalism of L-groups. For $n=1$ and m even, it is also compatible with the transfer factors proposed by K. Hiraga and T. Ikeda.

Index

$(K, K^\sharp , x, c)$

$(\mathcal {E}, S, \theta ^\flat )$

$\Omega (G,T)$

$\Pi (S, \theta ^\flat )$

$\Pi _\phi , \Pi _{[\phi ]}$

$\chi$-data

$\iota _{Q,m}$

$\kappa _-$

$\overline{-1}, \widetilde{-1}$

$\psi _c$

$\theta ^\circ _j, \theta ^\dagger _j$

$\theta ^\flat$

$\{x,y\}_F$

[x, y]

$B_Q$

$\mathbb {B}$

$\mathcal {B}(G, F)$

$\mathbf{C} \text {Ad}$

central character

$\mathsf {CExt}(G,A)$

$\mathbf{C} _m(\nu , \gamma _0)$

contracted product

$\mathcal {D}$

$d^\pm (V,q)$

$\mathcal {D}_{Q,m}, \mathcal {D}_{Q,m}^{\mathrm{sc}}$

epipelagic genuine character

$\epsilon$

$\epsilon (V,q)$

$\epsilon _S$

$F_{\alpha }, F_{\pm \alpha }$

$f_{(G,S)}(\alpha )$

factorization pair

$G(F)_x, G(F)_{x,r}$

$\mathfrak {g}(F)_{x,r}$

$\mathfrak {g}^*(F)_{x,r}$

${}^{{\mathrm{L}}} G$

good element

$\overline{G}_\psi , \overline{G}_\psi ^{(2)}, \overline{G}_\psi ^{(8)}$

$G^T$

$\tilde{G}^\vee$

inertially anisotropic

$\text {inv}(\cdot , \cdot )$

L-parameter

      epipelagic

      toral supercuspidal

Matsumoto’s central extension

moment map

$\mu _m, \varvec{\mu }_m$

$N_F$

$\nabla$

$q\langle Y \rangle$

$R_{B/A}$

representation

      epipelagic supercuspidal

      genuine

$\text {Sgn}_m(T)$

$\sigma _\ell , \sigma _{\text {LP}}$

stable conjugacy

stable distribution

stable system

$S\Theta _\phi , S\Theta _{[\phi ]}$

symmetric root

$T_{Q,m}$

$\tau _{Q,m}$

topological Jordan decomposition

$\tilde{T}_{Q,m}, \tilde{T}_{Q,m}^\sigma$

transfer factor

type (ER)

$\mathcal {W}_F$

Weil restriction

$X_{Q,m}$

$\varvec{x}(\cdot ), \varvec{s}, \varvec{c}$

$Y_0$

$Y_{Q,m}$

$Y_{Q,m}^{\mathrm{sc}}$