Topic Moy-Prasad Filtrations
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Moy-Prasad Filtrations

The Moy-Prasad filtration is a decreasing sequence of compact open subgroups \(G(F)_{x, r}\) indexed by a point \(x\) of the Bruhat-Tits building and a non-negative real \(r\) (the depth). It is the modern tool for parametrising representations of \(p\)-adic reductive groups by their depth.

Definition

Let \(G\) be a connected reductive group over a non-archimedean local field \(F\) with absolute root system \(\Phi\) and Bruhat-Tits building \(\mathcal{B}(G, F)\). For each point \(x \in \mathcal{B}(G, F)\) and \(r \in \mathbb{R}_{\ge 0}\), define:

\[ G(F)_{x, r} \;:=\; \langle T(F)_r, \;U_{\alpha, \alpha(x) + r} : \alpha \in \Phi \rangle, \]

where:

  • \(T(F)_r\) = the depth-\(r\) part of a maximal torus of \(G\) (typically \(T(F)_r = \{t \in T(\mathcal{O}_F) : t \equiv 1 \pmod {\pi^r}\}\));
  • \(U_{\alpha, \alpha(x) + r}\) = the level-\((\alpha(x) + r)\) part of the root subgroup \(U_\alpha\) (via the valued root datum filtration).

Key features:

  • \(G(F)_{x, 0} = \mathcal{P}_x\), the parahoric subgroup associated to the facet containing \(x\).
  • \(G(F)_{x, 0^+} = \mathcal{P}_x^+\), the pro-unipotent radical of \(\mathcal{P}_x\).
  • For \(r > s\): \(G(F)_{x, r} \subseteq G(F)_{x, s}\) (decreasing filtration).

Successive quotients

The graded pieces \(\mathrm{gr}^r G(F)_x = G(F)_{x, r} / G(F)_{x, r^+}\) have a clean description:

  • For \(r = 0\): \(\mathrm{gr}^0 = \mathcal{P}_x / \mathcal{P}_x^+\) = reductive quotient over \(\mathbb{F}_q\).
  • For \(r > 0\): \(\mathrm{gr}^r\) is a direct sum of root spaces \(\mathfrak{g}_\alpha\) for affine roots \(\psi = \alpha + c\) with \(\psi(x) + r \in \Gamma_\alpha\) (discrete, finite-dimensional \(\mathbb{F}_q\)-vector space).

Lie algebra version

There is a parallel Moy-Prasad Lie algebra filtration: \[ \mathfrak{g}(F)_{x, r} \;:=\; \mathfrak{t}(F)_r \;\oplus\; \bigoplus_\alpha \mathfrak{u}_\alpha(F)_{\alpha(x) + r}, \] with the same properties. The exponential map \(\exp : \mathfrak{g}(F)_{x, r} \to G(F)_{x, r}\) is a homeomorphism for \(r > 0\) (in good characteristic).

Depth of a representation

For a smooth irreducible representation \(\pi\) of \(G(F)\), the depth of \(\pi\) is: \[ \mathrm{depth}(\pi) \;:=\; \inf\{r \ge 0 : \pi^{G(F)_{x, r^+}} \ne 0 \text{ for some } x\}. \]

  • \(\mathrm{depth}(\pi) = 0\) iff \(\pi\) has invariants under the pro-unipotent radical of some parahoric — these are depth-zero representations.
  • General \(\mathrm{depth}(\pi)\) refines the type theory.

Yu's construction of supercuspidals

Theorem (Yu, 2001). For \(G\) a connected reductive group over \(F\) in good residue characteristic, the supercuspidal representations of \(G(F)\) are exhaustively constructed from:

  • A generic cuspidal datum: a tuple \((G_0 \subseteq G_1 \subseteq \cdots \subseteq G_d = G, \pi_0, \chi_1, \ldots, \chi_d)\) of nested twisted Levis with depth-zero supercuspidal data;
  • A point \(x \in \mathcal{B}(G_0, F)\);
  • Compatible characters \(\chi_i\) on intermediate filtration steps.

This is called Yu's construction and depends crucially on the Moy-Prasad filtration.

Refinements

  • Kim's exhaustion theorem (2007): Yu's construction gives all supercuspidals (in residue characteristic \(\ne 2\); refined to more general settings by Fintzen).
  • Fintzen's exhaustion (2021): extends to arbitrary residue characteristic.
  • Kaletha refinement: uses regular supercuspidal parametrisation via embeddings of tori.

Examples

Depth-zero supercuspidals of \(\mathrm{GL}_n(F)\)

Inflated from cuspidal representations of \(\mathrm{GL}_n(\mathbb{F}_q)\) to a maximal compact subgroup of \(\mathrm{GL}_n(F)\), then induced.

Number: equal to the number of cuspidal reps of \(\mathrm{GL}_n(\mathbb{F}_q)\) = number of \(\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\)-orbits of irreducible characters of \(\overline{\mathbb{F}_q}^\times\) of size \(n\).

Positive-depth supercuspidals of \(\mathrm{GL}_2(F)\)

Bushnell-Kutzko's complete classification via "simple types"; in modern language, these are Yu's data with depth \(> 0\).

Role in Langlands

  • Bushnell-Kutzko types: framework for parametrising all representations of \(G(F)\) via the parahoric/Moy-Prasad structure.
  • Local Langlands construction: Kaletha's regular supercuspidal L-packets are constructed via Moy-Prasad data + Yu data.
  • Smooth representations of \(p\)-adic groups: depth is a basic invariant; the category of representations decomposes by depth.

Mathlib

Not formalised; requires extensive \(p\)-adic and reductive-group infrastructure.

References

  • Moy–Prasad, Unrefined minimal K-types for p-adic groups (Inv.\ Math.\ 1994).
  • Moy–Prasad, Jacquet functors and unrefined minimal K-types (Comment.\ Math.\ Helv.\ 1996).
  • Yu, Construction of tame supercuspidal representations (J.\ AMS 2001).
  • Kim, Supercuspidal representations: an exhaustion theorem (J.\ AMS 2007).
  • Fintzen, Types for tame p-adic groups (Annals 2021).

Prerequisite Topics