Casselman-Wallach property for homological theta lifting

TL;DR

This work proves that archimedean homological theta liftings satisfy the Casselman–Wallach property: for a reductive dual pair $(G,H)$ and a Casselman–Wallach representation $\pi$ of $G$, the Schwartz homology groups $\operatorname{H}^{\mathcal{S}}_i(G, \Omega \widehat{\otimes} \pi)$ are Casselman–Wallach representations of $H$ and vanish for $i>\dim(G/K)$. Consequently, the Euler–Poincaré characteristic $\mathrm{EP}_{G}(\Omega \widehat{\otimes} \pi) := \sum_i (-1)^i \operatorname{H}^{\mathcal{S}}_i(G, \Omega \widehat{\otimes} \pi)$ is well-defined in the Grothendieck group of ${\mathrm{Rep}}_H^{\mathrm{CW}}$. The proof introduces a corank-one parabolic stable filtration on the Weil representation and proceeds by induction by stages, treating Type II and Type I dual pairs separately, with base cases such as $(\mathrm{GL}_1, \mathrm{GL}_m)$ and compact dual pairs. This homological theta lifting framework lays groundwork for a robust archimedean analogue of homological theta correspondence and its Euler–Poincaré invariants. The results enhance understanding of theta lifts in the archimedean setting and enable computable invariants in the Grothendieck group of Casselman–Wallach representations.

Abstract

In this paper, we establish the Casselman-Wallach property for homological theta lifting over archimedean local fields. As a consequence, the Euler-Poincaré characteristic is a well-defined element in the Grothendieck group of Casselman-Wallach representations. Our main tool is a corank-one parabolic stable filtration on the Weil representation.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1: AGS15Derivative
• Lemma 2.2: Casselman's subrepresentation theorem, Wallach1988RRGI
• Lemma 3.1
• proof
• Proposition 3.2: Shapiro's Lemma, CHEN2021108817
• Proposition 3.3: CHEN2021108817
• Proposition 3.4: CHEN2021108817
• Proposition 3.5: CHEN2021108817