[Paper Review] Zelevinsky involution and l-adic cohomology of the Rapoport-Zink tower

Yoichi Mieda|arXiv (Cornell University)|Jan 21, 2014

Advanced Algebra and Geometry25 references3 citations

TL;DR

This paper generalizes Fargues' result on the Zelevinsky involution in $
ell$-adic cohomology from the Drinfeld tower to Rapoport-Zink towers for $\mathrm{GSp}(2n)$, introducing a new cohomology $H^i_{\mathcal{C}_{\mathcal{M}}}(M_\infty)$ that depends on the formal model $\mathcal{M}$, and establishes an isomorphism between the Zelevinsky dual of compactly supported cohomology and a shifted, dual cohomology group with respect to $\mathcal{C}_{\mathcal{M}}$, extending geometric realizations of the local Langlands correspondence.

ABSTRACT

In this paper, we investigate how the Zelevinsky involution appears in the l-adic cohomology of the Rapoport-Zink tower. We generalize the result of Fargues on the Drinfeld tower to the Rapoport-Zink towers for symplectic similitude groups.

Motivation & Objective

• To extend Fargues' result on the Zelevinsky involution in $\ell$-adic cohomology from the Drinfeld tower to Rapoport-Zink towers for $\mathrm{GSp}(2n)$.
• To introduce a new cohomology theory $H^i_{\mathcal{C}_{\mathcal{M}}}(M_\infty)$ that depends on the formal model $\mathcal{M}$ of the Rapoport-Zink space, addressing the non-schematic nature of $\mathcal{M}$ in this setting.
• To establish a duality isomorphism between the Zelevinsky dual of compactly supported cohomology and a shifted, dual cohomology group with respect to $\mathcal{C}_{\mathcal{M}}$, generalizing Poincaré duality in this context.
• To provide a framework for understanding the contribution of non-supercuspidal representations of $J$ to the cohomology of the Rapoport-Zink tower, particularly in relation to Bernstein components and $\ell$-adic representations of the Weil group.

Proposed method

• Introduce a new cohomology group $H^i_{\mathcal{C}_{\mathcal{M}}}(M_\infty)$, defined as cohomology with compact support in the direction of the formal model $\mathcal{M}$, which reduces to the standard compactly supported cohomology when $\mathcal{M}$ is a $p$-adic formal scheme.
• Use the action of $G \times J \times W_{\mathbb{Q}_p}$ on $H^i_c(M_\infty)$ and $H^i_{\mathcal{C}_{\mathcal{M}}}(M_\infty)$ to relate representation-theoretic structures to geometric cohomology.
• Apply the Zelevinsky involution $\operatorname{Zel}$ on smooth representations of $J$, defined via Bernstein-Zelevinsky theory, to relate dual cohomology groups.
• Utilize the Bernstein decomposition and the notion of $\mathfrak{s}$-components of cohomology, indexed by Bernstein components $\mathfrak{s}$ of $J$-representations with fixed central character $\chi$, to localize the cohomological isomorphism.
• Employ the Hecke algebra $\mathcal{H}(Z_G)$ of the center $Z_G$ of $G$ to define the $\tau$-isotypic component $V_\tau$ of a $G$-representation $V$, enabling the study of cohomology in terms of smooth $\widetilde{K}$-representations.
• Use results from [IM10] to compare $H^i_c(M_\infty/p^{\mathbb{Z}})$ and $H^i_{\mathcal{C}_{\mathcal{M}}}(M_\infty/p^{\mathbb{Z}})$, crucial for proving the main theorem and Corollary 5.12.

Experimental results

Research questions

• RQ1 How does the Zelevinsky involution manifest in the $\ell$-adic cohomology of Rapoport-Zink towers for $\mathrm{GSp}(2n)$, beyond the Drinfeld and Lubin-Tate cases?
• RQ2 What cohomology theory is needed to generalize Fargues' duality result when the Rapoport-Zink space $\mathcal{M}$ is not a $p$-adic formal scheme?
• RQ3 How do the $\mathfrak{s}$-components of cohomology, indexed by Bernstein components of $J$, transform under the Zelevinsky involution?
• RQ4 What is the precise relationship between the cohomology of the Rapoport-Zink tower modulo $p^{\mathbb{Z}}$ and the dual cohomology with respect to the $\mathcal{C}_{\mathcal{M}}$-cohomology?
• RQ5 Can the cohomological duality involving the Zelevinsky involution be used to detect the presence of non-supercuspidal representations in the cohomology of the tower?

Key findings

• The main result establishes an isomorphism of $J \times W_{\mathbb{Q}_p}$-representations: $H^{2d + \iota(\mathfrak{s}) - i}_{\mathcal{C}_{\mathcal{M}}}(M_\infty)_{\tau^\vee, \mathfrak{s}^\vee}(d) \cong \operatorname{Zel}\bigl{(}H^i_c(M_\infty)_{\tau, \mathfrak{s}}^\vee\bigr{)}$, where $d = n(n+1)/2$ and $\iota(\mathfrak{s})$ is an invariant measuring the depth of the Bernstein component $\mathfrak{s}$.
• For $n=2$, the paper proves that $\pi \otimes \rho \otimes \sigma$ appears as a subquotient of $H^3_c(M_\infty / p^{\mathbb{Z}})$ if and only if $\pi^\vee \otimes \operatorname{Zel}(\rho^\vee) \otimes \sigma^\vee(-3)$ appears as a subquotient of $H^4_c(M_\infty / p^{\mathbb{Z}})$, under technical Assumption 5.8.
• The cohomology $H^i_{\mathcal{C}_{\mathcal{M}}}(M_\infty)$ is shown to coincide with the standard compactly supported cohomology $H^i_c(M_\infty)$ when $\mathcal{M}$ is a $p$-adic formal scheme, validating the new cohomology as a natural generalization.
• The paper shows that the $G$-cuspidal part of $H^2_c(M_\infty / p^{\mathbb{Z}})$ vanishes for $n=2$, as $\dim \mathcal{M}^{\mathrm{red}} = 1$, extending a result from [Ito-Mieda].
• The authors speculate that the cohomology of the Rapoport-Zink tower may realize local $A$-packets, with $\pi \otimes \rho^\vee$ and $\pi \otimes \rho^{\prime\vee}$ appearing in $H^3_c(M_\infty / p^{\mathbb{Z}})$ and $\pi \otimes \operatorname{Zel}(\rho)^\vee$ in $H^4_c(M_\infty / p^{\mathbb{Z}})$, suggesting a link between $A$-packets and cohomological duality.
• The method is shown to be extendable to other Rapoport-Zink towers, as evidenced by the applicability of the geometric properties in [Mie13], and the author plans to extend the result to $\mathrm{GSp}(4)$ in future work.