[Paper Review] A Formula for the Geometric Jacquet Functor and its Character Sheaf Analogue

Tsao-Hsien Chen, Alexander Yom Din|arXiv (Cornell University)|Jul 2, 2015

Advanced Algebra and Geometry9 references4 citations

TL;DR

This paper establishes a geometric formula for the nearby cycles functor in the context of symmetric spaces via a 'wonderful degeneration' to the base affine space $MN\backslash G$. It proves that on character sheaves, this functor is isomorphic to a composition of two averaging functors, providing a geometric proof of Casselman's submodule theorem and a second adjointness theorem for Harish-Chandra modules.

ABSTRACT

Let (G,K) be a symmetric pair over the complex numbers, and let X=K\G be the corresponding symmetric space. In this paper we study a nearby cycles functor associated to a degeneration of X to MN\G, which we call the "wonderful degeneration". We show that on the category of character sheaves on X, this functor is isomorphic to a composition of two averaging functors (a parallel result, on the level of functions in the p-adic setting, was obtained in [BK, SV]). As an application, we obtain a formula for the geometric Jacquet functor of [ENV] and use this formula to give a geometric proof of the celebrated Casselman's submodule theorem and establish a second adjointness theorem for Harish-Chandra modules.

Motivation & Objective

• To develop a geometric framework for the Jacquet functor using degeneration techniques in symmetric spaces.
• To establish a formula for the nearby cycles functor associated with the 'wonderful degeneration' of $K\backslash G$ to $MN\backslash G$.
• To prove that this functor is isomorphic to a composition of two averaging functors on the category of character sheaves.
• To apply the formula to give a geometric proof of Casselman's submodule theorem and the second adjointness theorem for Harish-Chandra modules.

Proposed method

• Construct a ${\mathbb{G}}_m$-equivariant degeneration $f: \widetilde{X} = \widetilde{K}\backslash \widetilde{G} \to \mathbb{P}^1$ where $\widetilde{K}$ is a degeneration of $K$ to $MN$ and $M\bar{N}$ at $t=0$ and $t=\infty$, respectively.
• Define the limit functor $L_\gamma := \psi_f \circ a_f^*$ as the composition of the nearby cycles functor $\psi_f$ and the pullback $a_f^*$ from the open stratum $X^\circ \cong \mathbb{G}_m \times X_1$ to $X_1$.
• Use the theory of equivariant derived categories and $!$- and $*$-averaging functors to relate $L_\gamma$ to $\operatorname{av}_{MN}^{M\bar{N}} \circ \operatorname{Av}_{M\bar{N}}^K$ on the category of character sheaves.
• Leverage the $t$-exactness of $L_\gamma$ and the averaging functors under the perverse $t$-structure to establish the isomorphism.
• Construct explicit $G$-varieties $Y$ with $\mathbb{A}^1$-valued morphisms $f: \mathbb{A}^1 \to Y$ such that stabilizers of $f(t)$ contain $K_t(k)$ as finite index subgroups, using representations of $G$ with controlled weights.
• Apply representation-theoretic arguments involving lowest weight representations and cocharacters to control stabilizers and verify the desired geometric properties.

Experimental results

Research questions

• RQ1 How can the nearby cycles functor associated with the wonderful degeneration of a symmetric space be expressed in terms of standard geometric functors?
• RQ2 Is the limit functor $L_\gamma$ on character sheaves isomorphic to a composition of averaging functors?
• RQ3 Can the geometric Jacquet functor be characterized via degeneration and nearby cycles in the context of $D$-modules or $\ell$-adic sheaves?
• RQ4 Does the isomorphism $L_\gamma \simeq \operatorname{av}_{MN}^{M\bar{N}} \circ \operatorname{Av}_{M\bar{N}}^K$ imply $t$-exactness and adjointness properties for Harish-Chandra modules?
• RQ5 Can this geometric approach provide a new proof of Casselman's submodule theorem and the second adjointness theorem?

Key findings

• The limit functor $L_\gamma: D\mathcal{CS}(X_1/K) \to D(X_0/K)$ is naturally isomorphic to the composition $\operatorname{av}_{MN}^{M\bar{N}} \circ \operatorname{Av}_{M\bar{N}}^K$ on the category of character sheaves.
• The composition $\operatorname{av}_{MN}^{M\bar{N}} \circ \operatorname{Av}_{M\bar{N}}^K$ is $t$-exact with respect to the perverse $t$-structure.
• The functor $L_\gamma$ admits both a left and a right adjoint, a consequence of the isomorphism and $t$-exactness.
• The geometric approach provides a new proof of Casselman's submodule theorem for Harish-Chandra modules.
• The second adjointness theorem for Harish-Chandra modules is established geometrically via the derived equivalence induced by the degeneration.
• The construction is valid in the $D$-module, $\ell$-adic, and constructible sheaf settings, generalizing prior results in the $p$-adic function-theoretic setting.