On Irreducible Components of Rapoport–Zink Spaces

Abstract

Under a mild condition, we prove that the action of the group of self-quasi-isogenies on the set of irreducible components of a Rapoport-Zink space has finite orbits. Our method allows both ramified and non-basic cases. As a consequence, we obtain some finiteness results on the representation obtained from the ℓ-adic cohomology of a Rapoport-Zink tower.Theorem 1.3 (Theorem 4.14) Assume that M is the basic Rapoport-Zink space for GSp 2n , in which case G ′ = G = GSp 2n (Q p ) and J is a quaternion unitary similitude group. For every integers i, r ≥ 0 and every irreducible smooth representation ρ of J, the G-representation Ext r J (H i c (M ∞ ), ρ) Dc-sm has finite length. In particular, the G ′ -representation Hom J (H i c (M ∞ ), ρ) sm has finite length.For the precise notation, see Section 4.2. To prove this theorem, we use Theorem 1.2 for two Rapoport-Zink towers: one is the basic Rapoport-Zink tower for GSp 2n , and the other is a Rapoport-Zink tower for a quaternion unitary similitude group. These two towers are isomorphic at the infinite level, thanks to the duality isomorphism. These finiteness results are very useful when we apply representation theory to the study of H i c (M ∞ ). For example, in [31, §2-3], Theorem 1.2 for the Drinfeld tower, which is well-known, enables the author to apply a result in [6] to investigate H i c (M ∞ ) by using the Lefschetz trace formula. Further, by using Theorem 1.3 for the Drinfeld tower ([31, Corollary 4.3]), we obtained a purely local and geometric proof of the local Jacquet-Langlands correspondence in some cases (see [31, Theorem 6.10]). In [30], the author extends these arguments to the basic Rapoport-Zink space for GSp 4 . For this purpose, Theorem 1.3 plays a crucial role. We also remark that Theorem 1.1 and Theorem 1.3 are indispensable in the study of the relation between Now we can give a proof of Theorem 4.14. By a similar method as in the proof of [31, Lemma 3.1], we can show thatOn the other hand, by Lemma 4.15, we know that Ext r J (V, ρ) Dc-sm is generated by Ext r J (V K , ρ). In particular Ext r J (V, ρ) Dc-sm is finitely generated. Therefore, [40, Théorème VI.6.3] tells us that the G-representation Ext r J (V, ρ) Dc-sm has finite length. This completes the proof.

Finiteness Properties of Affine Deligne-Lusztig Varieties

“…In order to be able to apply the usual methods, one needs the cohomology groups to be finitely generated J b (F )-representations, and thus the "infinite level" cohomology groups to be admissible. This follows from the above theorem by a formal argument once the integral model is constructed (see for example [Mie20,Thm. 4.4], [RV14, Prop.…”

“…For the particular case of affine Deligne-Lusztig varieties arising as the underlying reduced subscheme of a Rapoport-Zink moduli space of p-divisible groups with additional structure of PEL type, questions as in Theorem 1.2 have been considered by several people. A recent general theorem along these lines is shown by Mieda [Mie20]. Also, the (rare) cases where an affine Deligne-Lusztig variety is even of finite type have been classified, compare [Gör10, Prop.…”