Topic Valued Root Data
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Valued Root Data

A valued root datum is the Bruhat-Tits enrichment of a root datum: it adds a "valuation" \(\varphi_\alpha\) on each root subgroup \(U_\alpha\), encoding the level filtration in the topology of the local field. The valued root datum is the combinatorial input from which the Bruhat-Tits apartment and building are constructed.

Definition

Let \(G\) be a connected reductive group over a non-archimedean local field \(F\) with maximal \(F\)-split torus \(S\). The associated valued root datum is the data:

  • The relative root datum \((X^*(S), \Phi, X_*(S), \Phi^\vee)\);
  • For each root \(\alpha \in \Phi\): a filtration \[ U_\alpha(F) \;=\; \cdots \;\supseteq\; U_{\alpha, r} \;\supseteq\; U_{\alpha, r+1} \;\supseteq\; \cdots, \] indexed by \(r \in \mathbb{R}_{\ge 0}\) (or a discrete subset thereof);
  • A "valuation" \[ \varphi_\alpha : U_\alpha(F) \setminus \{1\} \;\longrightarrow\; \mathbb{R}, \] with \(\varphi_\alpha(u) = r\) iff \(u \in U_{\alpha, r} \setminus U_{\alpha, r^+}\);
  • Compatibility conditions among the filtrations and with the commutator relations \([U_\alpha, U_\beta] \subseteq U_{i\alpha + j\beta}\).

Bruhat-Tits axioms

The valuation \(\varphi = (\varphi_\alpha)_{\alpha \in \Phi}\) must satisfy:

(VR0) Each \(\varphi_\alpha\) is a surjection onto a discrete subgroup \(\Gamma_\alpha \subseteq \mathbb{R}\) (\(\mathbb{Z}\) in the split case).

(VR1) \(\varphi_\alpha(u v) \ge \min(\varphi_\alpha(u), \varphi_\alpha(v))\) (ultrametric inequality).

(VR2) Commutator compatibility: for \(\alpha, \beta\) non-proportional roots, \(\varphi_{i\alpha + j\beta}([u_\alpha, v_\beta]) \ge i \varphi_\alpha(u_\alpha) + j \varphi_\beta(v_\beta)\).

(VR3) Compatibility with the Weyl group: \(W\) permutes \(\varphi\)-data compatibly.

Examples

Split \(\mathrm{SL}_2(F)\)

Single root \(\alpha\); \(U_\alpha(F) = \mathbb{G}_a(F) = F\); \(\varphi_\alpha = v_F\) (the valuation of \(F\)). The valuation defines the standard filtration of \(F\) by powers of the maximal ideal.

Ramified \(\mathrm{SU}_n(F)\)

For a ramified quadratic extension \(K/F\), the unitary group has "twisted" root subgroups, and the valuation lives in a non-integer discrete subgroup of \(\mathbb{R}\): \(\Gamma_\alpha = \frac{1}{2}\mathbb{Z}\) for the affine "short" roots.

Bruhat-Tits theorem

Theorem (Bruhat-Tits). From a valued root datum, one can construct the standard apartment \(\mathbb{A}\) and the entire Bruhat-Tits building \(\mathcal{B}(G, F)\); the construction is functorial and gives the unique CAT(0) building on which \(G(F)\) acts.

Galois descent for non-split groups

For \(G\) non-split over \(F\): construct the valued root datum of \(G_K\) over \(K = \widehat{F^{\mathrm{unr}}}\) (where \(G_K\) is split), then take \(\mathrm{Gal}(K/F)\)-invariants.

The resulting valued root datum has:

  • The relative root system \(\Phi(G, S)\) (not the absolute one);
  • Galois-invariant valuations \(\varphi_\alpha\), which may live in \(\frac{1}{e}\mathbb{Z}\) for ramification index \(e\) (giving non-reduced affine root systems).

Moy-Prasad refinement

For a point \(x \in \mathcal{B}(G, F)\), the Moy-Prasad filtration \(\{G(F)_{x, r}\}_{r \ge 0}\) refines the valued root datum: each \(G(F)_{x, r}\) is a compact open subgroup, generated by:

  • \(T(F)_r\) = the \(r\)-th level subgroup of the torus;
  • \(U_{\alpha, \alpha(x) + r}\) for each affine root \(\psi = \alpha + c\) with \(\psi(x) \ge -r\).

This refinement is essential for the modern theory of Moy-Prasad filtrations.

Role in Langlands

  • Parahoric subgroups: stabilisers in the building come from valuations on root subgroups via the valued root datum.
  • Smooth models: the Bruhat-Tits parahoric group schemes are defined directly from the valued root datum.
  • Reductive quotient: the residue-field reductive group at each parahoric is determined by the valued root datum's combinatorics.

References

  • Bruhat–Tits, Groupes réductifs sur un corps local I, II (1972, 1984).
  • Tits, Reductive groups over local fields (Proc.\ AMS Symp.\ 1979).
  • Landvogt, A compactification of the Bruhat-Tits building (LNM 1996).
  • Yu, Bruhat-Tits theory: a new approach (preprint, 2002).

Prerequisite Topics

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