Admissible Sets

For a reductive group \(G\) over a local field and a cocharacter \(\mu \in X_*(T)\), the admissible set \(B(G, \mu) \subseteq B(G)\) consists of those \(\sigma\)-conjugacy classes whose Newton point lies in the convex hull of the Weyl orbit of \(\mu\). It indexes the Newton stratification of Shimura varieties of type \(\mu\).

Definition

Let \(G\) be a connected reductive group over a non-archimedean local field \(F\), and let \(\mu \in X_*(T)\) be a cocharacter (typically minuscule for Shimura applications). The set of \(\mu\)-admissible classes is \[ B(G, \mu) \;:=\; \{[b] \in B(G) : \nu([b]) \le \bar\mu \text{ in dominance order}, \;\kappa([b]) = \mu \bmod (\sigma-1)\pi_1(G)\}, \] where \(\bar\mu\) is the dominant representative of the Galois orbit of \(\mu\) modulo the Weyl group, and \(\le\) is the standard dominance order on rational cocharacters.

Equivalent description

\([b] \in B(G, \mu)\) iff there exist:

A representative \(b \in G(\breve F)\) of \([b]\);
An element \(x \in G(\breve F)\) with \(x b \sigma(x)^{-1} \in K \mu(\pi) K\), where \(K = G(\mathcal{O}_F)\) is a hyperspecial maximal compact subgroup and \(\pi\) is a uniformiser.

Equivalently: the \(\sigma\)-conjugacy class \([b]\) is "approachable" from the lattice flag of \(K \mu(\pi) K\) in the affine flag variety.

Examples

Minuscule \(\mu\) for \(\mathrm{GL}_n\)

For \(\mu = (1^d, 0^{n-d})\) (\(d\) ones, \(n-d\) zeros): the admissible set \(B(\mathrm{GL}_n, \mu)\) consists of isocrystals of dimension \(n\) and slope sum \(d\), with Newton polygon below the "Hodge polygon" \((1, \ldots, 1, 0, \ldots, 0)\) (i.e.\ slopes in \([0, 1]\)).

Mazur's inequality ensures this is exactly the condition coming from a \(p\)-divisible group of height \(n\) and dimension \(d\).

Minuscule \(\mu\) for \(\mathrm{Sp}_{2n}\)

For the "minuscule" minuscule \(\mu = (1, 0, \ldots, 0; 0, \ldots, 0, -1)\) (corresponding to symplectic \(p\)-divisible groups of height \(2n\)): admissible Newton polygons have slope multiset \(\{\lambda_i\}\) satisfying \(\lambda_i + \lambda_{2n+1-i} = 0\), \(\lambda_i \in [0, 1]\), and \(\sum \lambda_i = n\).

Rapoport's conjecture (= Mazur's inequality, proven)

Theorem (Rapoport-Richartz, Lucarelli, Gashi). For any reductive \(G\) and dominant \(\mu\): \[ B(G, \mu) \;=\; \{[b] : \nu([b]) \preceq \mu^\diamond, \;\kappa([b]) = \mu^\natural\}, \] where \(\mu^\diamond\) is the Galois average of \(W \mu\), and \(\mu^\natural\) is the image in \(\pi_1(G)_\Gamma\). Originally conjectured by Kottwitz, proven case-by-case by various authors.

Hodge-Newton decomposition

For an admissible class \([b] \in B(G, \mu)\) whose Newton point lies on a wall of the dominant chamber: the corresponding isocrystal admits a canonical Hodge-Newton decomposition into a Levi piece (slope on the wall) and a basic piece.

This gives the base-change stratification of admissible sets, foundational for the Kottwitz conjecture on Shimura varieties.

Shimura variety connection

For a Shimura variety \(\mathrm{Sh}_K(G, X)\) at \(p\) good reduction with minuscule cocharacter \(\mu\):

Newton strata of \(\mathrm{Sh}_K \pmod p\) are indexed by elements of \(B(G, \mu)\).
Each stratum has a description via affine Deligne-Lusztig varieties and Rapoport-Zink spaces.
The basic stratum is the unique class \([b]_0\) with central Newton point; the corresponding RZ space has dimension equal to the dimension of \(\mathrm{Sh}_K\).

Affine Deligne-Lusztig varieties

For \(\tilde w \in \tilde W\) (affine Weyl) and \([b] \in B(G)\): \[ X_w(b) \;:=\; \{g K \in G(\breve F)/K : g^{-1} b \sigma(g) \in K \tilde w K\}. \]

\(X_w(b)\) is non-empty iff certain compatibilities hold; for \(\tilde w = t_\mu\) (translation by \(\mu\)) one recovers the basic geometry of the basic Rapoport-Zink space, and admissibility of \([b]\) in \(B(G, \mu)\) is exactly non-emptiness of \(X_{t_\mu}(b)\).

Role in Langlands

Local Langlands at non-tempered: \(\mu\)-admissibility captures the "discrete series + boundary" decomposition; basic vs.\ non-basic strata distinguish supercuspidal vs.\ non-supercuspidal L-packets.
Shimura variety arithmetic: the Tate conjecture, point counts, and intersection numbers on Shimura varieties are organised by admissible sets and their RZ uniformisation.

References

Kottwitz, Shimura varieties and twisted orbital integrals (Math.\ Ann.\ 1984), §4.
Rapoport–Richartz, On the classification and specialization of F-isocrystals with additional structure (Compositio 1996), §4.
Görtz, Affine Deligne-Lusztig varieties and the geometry of Shimura varieties (notes, 2009).