On the Harris-Viehmann conjecture for Hodge-Newton reducible local Shimura data of abelian type

Technical Summary: On the Harris–Viehmann Conjecture for Hodge–Newton Reducible Local Shimura Data of Abelian Type

Problem Statement
The paper addresses a specific case of the Harris–Viehmann conjecture, which concerns the $\ell$-adic cohomology of local Shimura varieties associated with non-basic local Shimura data. The conjecture posits a parabolic induction formula for these cohomology groups. Specifically, it asserts that for a non-basic local Shimura datum $(G, [b], \{\mu\})$ that is Hodge–Newton reducible with respect to a parabolic subgroup $P$ and Levi factor $L$, the cohomology of the associated Rapoport–Zink (RZ) space $RZ_G$ is induced from the cohomology of the RZ space $RZ_L$ associated with the Levi subgroup. A critical consequence of this conjecture is that all supercuspidal representations appearing in the cohomology of Shimura varieties are concentrated along the basic locus. While the conjecture has been established for PEL-type data (Mantovan), Hodge-type data (Hong), and specific cases like $GL_n$ (Hansen), a general proof for unramified local Shimura data of abelian type under Hodge–Newton reducibility remained open.

Methodology
The authors, Sandra Nair and Xinyu Zhou, extend the proof of the Harris–Viehmann conjecture to the abelian type setting by leveraging Shen's construction of Rapoport–Zink spaces of abelian type. Their methodology involves the following key steps:

1. Reduction to Hodge Type: The authors utilize the definition of abelian type data, which requires an isomorphism between the adjoint local Shimura datum of $G$ and that of an auxiliary Hodge-type datum $(H, [b_H], \{\mu_H\})$. They establish that if $(G, [b_G], \{\mu_G\})$ is Hodge–Newton reducible, there exists an associated Hodge-type datum that is also Hodge–Newton reducible (Lemma 7.1).
2. Geometric Correspondence: The core strategy involves relating the geometry of the abelian type RZ space $RZ_G$ to the Hodge type RZ space $RZ_H$. The authors construct a correspondence diagram involving the RZ spaces of the parabolic $P$ and Levi $L$ for both the abelian and Hodge cases.
3. Weil Descent Data: A significant technical component is the rigorous construction of the Weil descent datum for RZ spaces of abelian type. The authors define a descent datum $\Phi_{Shen}$ relative to a finite extension $E_{opt}$ of the reflex field, ensuring compatibility with the known descent data for Hodge type (via Kim's construction) and the moduli of $p$-adic local shtukas (via Pappas–Rapoport). They prove that these constructions coincide (Theorem 6.3, Corollary 6.5).
4. Functoriality and Quotients: The authors analyze the functoriality of RZ spaces under morphisms of local Shimura data. They carefully handle the passage from the Hodge type datum $H$ to the abelian type datum $G$ via the adjoint quotient $Gad$. This involves managing the kernels of the maps on fundamental groups ($\pi_1$) and establishing isomorphisms between the formal schemes $RZ_G$, $RZ_{Gad}$, and their Hodge counterparts modulo specific group actions (Lemma 5.1, Lemma 10.4).
5. Cohomological Comparison: By establishing isomorphisms between the cohomology of the "plus" components ($RZ^+$) of the abelian and Hodge spaces, and utilizing the known results for the Hodge case (specifically the work of Hong and Mantovan), the authors transfer the parabolic induction property from the Hodge setting to the abelian setting.

Key Contributions and Results

• Main Theorem (Theorem 1.1): The paper proves that for an unramified local Shimura datum $(G, [b], \{\mu\})$ of abelian type that is Hodge–Newton reducible with respect to a parabolic $P$ and Levi $L$, and for any admissible $\ell$-adic representation $\rho$ of $J_b(\mathbb{Q}_p)$, the following equality holds as virtual representations of $G(\mathbb{Q}_p) \times W_{E_{opt}}$:
  $$H^\bullet(RZ^\infty_G)_\rho = \text{Ind}_{P(\mathbb{Q}_p)}^{G(\mathbb{Q}_p)} H^\bullet(RZ^\infty_L)_\rho$$
  Here, $E_{opt}$ is a specific finite extension of the reflex field constructed in Theorem 6.3.
• Construction of Weil Descent: The authors provide a complete and self-contained construction of the Weil descent datum for RZ spaces of abelian type, resolving potential ambiguities regarding the dependence on auxiliary Hodge data (Lemma 6.4).
• Functoriality of RZ Spaces: The paper establishes that the formation of RZ spaces of abelian type respects products and closed embeddings of local Shimura data (Proposition 8.2), generalizing previous results known for Hodge type.
• Reduction to Adjoint Case: The proof effectively reduces the problem for abelian type data to the adjoint case, utilizing the surjectivity of the map $\pi_1(H)^\Gamma \to \pi_1(H^{ad})^\Gamma$ (a condition satisfied by similitude groups like $GL_n$ and $GSp_{2n}$).

Significance
The paper claims to extend the validity of the Harris–Viehmann conjecture to a broader class of local Shimura data, specifically unramified data of abelian type, which strictly contains the class of Hodge-type data. By confirming the parabolic induction formula in this setting, the work reinforces the structural understanding of the cohomology of Shimura varieties. As noted in the introduction, this result supports the broader expectation within the Langlands program that supercuspidal representations are concentrated along the basic locus. The authors do not claim to resolve the conjecture for all local Shimura data (e.g., non-reducible or non-unramified cases) but specifically target the Hodge–Newton reducible case for abelian types, building upon the foundational work of Mantovan, Hong, and Shen.