Topic Splittings over Integral and Parahoric Subgroups
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Splittings over Integral and Parahoric Subgroups

A metaplectic cover \(\overline G \to G(F)\) of a reductive group over a local field often splits when restricted to integral / parahoric subgroups. These splittings are the key technical input for organising the representation theory of metaplectic groups via parahoric representations.

Setup

Let \(F\) be a non-archimedean local field, \(G/F\) a connected reductive group, and \[ 1 \;\to\; \mu_n \;\to\; \overline G \;\to\; G(F) \;\to\; 1 \] a metaplectic cover specialised from a Brylinski-Deligne extension.

A splitting of the cover over a subgroup \(H \subseteq G(F)\) is a group homomorphism \[ s : H \;\hookrightarrow\; \overline G \] such that \(\overline G \to G(F)\) composed with \(s\) is the identity on \(H\).

Splittings over hyperspecial parahorics

Theorem (Brylinski-Deligne; Moore). Let \(K = G(\mathcal{O}_F)\) be a hyperspecial maximal compact subgroup (exists iff \(G\) is unramified at \(F\)). The metaplectic cover \(\overline G \to G(F)\) splits over \(K\) canonically (up to inner automorphism by \(\mu_n\)).

In particular: the genuine representations of \(\overline G\) with non-trivial central character can have \(K\)-invariant vectors, and the \(K\)-spherical representations of \(\overline G\) are organised via Satake-style transforms compatible with the splitting.

Splittings over general parahorics

For a general parahoric \(P \subseteq G(F)\):

  • The metaplectic cover may or may not split over \(P\), depending on the BD class and the parahoric type.
  • Hyperspecial \(\subseteq\) special vertices typically have splittings.
  • Iwahori subgroups often have partial splittings with cohomological obstructions.

The set of splittings over \(P\) (when non-empty) is a torsor under characters \(\mathrm{Hom}(P/[P, P], \mu_n)\).

Pro-unipotent radical

The pro-unipotent radical \(P^+ \subseteq P\) (= kernel of reduction) is a pro-\(p\) group. For \(p \nmid n\) (e.g.\ \(p \ne 2\) for the \(\mu_2\)-cover): the cover splits canonically over \(P^+\) (since \(H^2(P^+, \mu_n) = 0\)).

So the cover descends to a cover of the reductive quotient \(\bar{P}^{\mathrm{red}}(\mathbb{F}_q)\), which is a finite reductive group. Studying covers of finite reductive groups (much simpler!) gives the "depth-zero" piece of metaplectic representation theory.

Global splittings

For a global metaplectic cover \(\overline{G(\mathbb{A}_F)}\): the product formula ensures the cover splits over the diagonal \(G(F)\) (see local and global metaplectic covers).

In addition, the cover splits over the global integral subgroup \(K_S := \prod_{v \notin S} G(\mathcal{O}_v)\) for a finite "bad" set \(S\) of places, with splittings at each \(v \notin S\) determined by the local hyperspecial splitting.

Examples

\(\overline{\mathrm{SL}_2(\mathbb{Q}_p)}^{(2)}\) for \(p\) odd

The double cover splits over \(\mathrm{SL}_2(\mathbb{Z}_p)\) canonically. Genuine \(\mathrm{SL}_2(\mathbb{Z}_p)\)-spherical representations of \(\overline{\mathrm{SL}_2(\mathbb{Q}_p)}^{(2)}\) are parametrised by characters of the diagonal torus (after passage through the splitting).

Iwahori splitting for \(\overline{\mathrm{SL}_2}\)

The Iwahori \(I \subseteq \mathrm{SL}_2(\mathbb{Q}_p)\) admits a canonical splitting iff \(p\) is odd. For \(p = 2\), \(\mathrm{Sp}_{2n}(\mathbb{Q}_2)\) doesn't split over the Iwahori — the cohomological obstruction is the Hilbert symbol \((2, \cdot)_2\).

Brylinski-Deligne global splitting

For a global BD extension, the local-global splitting structure is governed by:

  • Local Hilbert symbols at each place;
  • The global reciprocity (= product formula).

The compatibility of local splittings (when they exist) is what makes global automorphic forms on \(\overline G\) well-defined.

Role in Langlands

  • Local Langlands at depth zero: parahoric splittings reduce metaplectic depth-zero theory to finite-reductive-group covers.
  • Spherical Hecke algebras: defined for the genuine \(K\)-spherical representations via the canonical splitting at hyperspecial vertices.
  • Eisenstein series on covers: require local splittings at almost all places + compatible global section.
  • \(p\)-integral structure: splittings are essential for any arithmetic / integral version of automorphic theory on covers.

References

  • Moore, Group extensions of p-adic and adelic linear groups (Publ.\ IHES 1968).
  • Matsumoto, Sur les sous-groupes arithmétiques des groupes semi-simples déployés (Ann.\ Sci.\ ENS 1969).
  • Brylinski–Deligne, Central extensions of reductive groups by \(\mathbf{K}_2\) (Publ.\ IHES 2001).
  • Weissman, Metaplectic tori over local fields (Pac.\ J.\ Math.\ 2009).

Prerequisite Topics