Dualities for root systems with automorphisms and applications to non-split groups

Abstract

This article establishes some elementary dualities for root systems with automorphisms. We give several applications to reductive groups over nonarchimedean local fields: (1) the proof of a conjecture of Pappas-Rapoport-Smithling characterizing the extremal elements of the {µ}-admissible sets attached to general non-split groups; (2) for quasi-split groups, a simple uniform description of the Bruhat-Titséchelonnage root system Σ0, the Knop root system Σ0, and the Macdonald root system Σ1, in terms of Galois actions on the absolute roots Φ; and (3) for quasi-split groups, the construction of the geometric basis of the center of a parahoric Hecke algebra, and the expression of certain important elements of the stable Bernstein center in terms of this basis. The latter gives an explicit form of the test function conjecture for general Shimura varieties with parahoric level structure. ) is an Iwahori subgroup, and if M µ is the associated Rapoport-Zink local model [RZ], then Theorem C ensures that we have a good understanding of the set of irreducible components in the special fiber of M µ . See §4 for more discussion.We now assume again that G/F is quasi-split. With Theorems B and C in hand, we can construct and study a geometric basis for the center Z(G(F ), J) of the parahoric Hecke algebra associated to G(F ) and an parahoric subgroup J ⊂ G(F ). Recall that W F acts on the dual group G, preserving a splitting ( B, T , X). A dual-group consequence of Theorem C is that there is an isomorphism of G I -modulesfor aλ ,µ ∈ Z ≥0 . Hereλ ∈ X * ( T I ) is the image of λ ∈ X * ( T ), V µ and Vλ are irreducible highest weight representations of the reductive groups G and G I , and Wt(μ) is the set of T I -weights in Vμ. These extend to representations V I µ and Vλ ,1 of G I ⋊ τ (see §5.2 and §7.1). The following theorem summarizes Lemma 7.4 and Theorem 7.5.Theorem D. There is a basis {Cλ ,J }λ for Z(G(F ), J) indexed byλ ∈ X * ( T I ) +,τ , characterized by:Cλ ,J acts on π J by the scalar tr(s(π) ⋊ τ | Vλ ,1 ) whenever π is an irreducible smooth representation of G(F ) with π J = 0 and Satake parameter s(π) (see [H15]). Furthermore, in terms of Bernstein functions zν ,J ∈ Z(G(F ), J) and certain Kazhdan-Lusztig polynomials P wν ,wλ (q 1/2 ) associated to the affine Hecke algebra with parameters H( W τ , S τ aff , L) there is an equalityWe call the Cλ ,J the geometric basis elements for the following reason: when F = F q ((t)) and) is a very special maximal parahoric subgroup of G(F q ((t))), Cλ ,J is the element of H(G(F q ((t))), J) arising, via the function-sheaf dictionary, from the equivariant perverse sheaf on the affine flag variety LG/L + P f which corresponds to Vλ ∈ Rep( G I ) under the geometric Satake isomorphism (see [Ric2], [Zhu1]). Moreover, in the

On the structure of some p-adic period domains

“…This is not the case anymore in general, even if G is quasi-split. One can compare He's result with ours using theorem 6.1 of [29] to obtain that the (ω α ) α∈∆0 are exactly the ω O of [34] when O goes through the set of σ 0 -orbits of simple roots in the Bruhat-Tits échelonnage root system attached to G F . An analysis of the construction of this root systems shows that we can take ξ = σ(0) with the notations of [34].…”

Smoothness of Schubert varieties in twisted affine Grassmannians

“…Let M denote the set of minimal elements of X * (T ) + I \{0} with respect to the partial ordering ≤ defined by the échelonnage coroots Σ∨ ⊂ X * (T ) I , cf. [Hai18]. Recall that μ ∈ M is • minuscule if α, μ ∈ {0, ±1} for all roots α ∈ Σ • quasi-minuscule, otherwise.…”

Smooth locus of twisted affine Schubert varieties and twisted affine Demazure modules

“…We also recall that αi is a simple coroot of g for each i ∈ I, and γ j is the image of αi in X * (T ) σ . The following lemma already appears in [Ha,Lemma 3.2] in a slighly different setting.…”