On a geometric comparison of representations of complex and $p$-adic $\mathbf{GL}_n$
Taiwang Deng, Chang Huang, Bin Xu, Qixian Zhao
TL;DR
The paper builds a geometric framework linking representations of $ ext{GL}_n(C)$ and unramified representations of $ ext{GL}_m(Q_p)$ via Langlands parameter spaces, establishing an open immersion from ABV spaces into Vogan-type parameter spaces and proving a compatibility with Chan–Wong’s algebraic functor. It shows that the geometric pullback on parameter spaces corresponds to Chan–Wong’s functor on Grothendieck groups, up to an explicit sign, and demonstrates that this geometric picture intertwines translation functors with partial Bernstein–Zelevinsky derivatives. A detailed comparison between Lusztig induction and push–pull functors is developed through the extension space formalism, culminating in a comparison theorem that identifies Lusztig induction with a fiberwise push–forward construction. The methods set up a robust geometric–categorical dictionary that the authors intend to extend to real and $p$-adic classical groups and Arthur packets in forthcoming work, potentially illuminating the structure of Arthur packets via parameter-space microlocal data.
Abstract
In this paper, we use geometric methods to study the relations between admissible representations of $\mathbf{GL}_n(\mathbb{C})$ and unramified representations of $\mathbf{GL}_m(\mathbb{Q}_p)$. We show that the geometric relationship between Langlands parameter spaces of $\mathbf{GL}_n(\mathbb{C})$ and $\mathbf{GL}_m(\mathbb{Q}_p)$ constructed by the first named author is compatible with the functor recently defined algebraically by Chan-Wong. We then show that the said relationship intertwines translation functors on representations of $\mathbf{GL}_n(\mathbb{C})$ and partial Bernstein-Zelevinskii derivatives on representations of $\mathbf{GL}_m(\mathbb{Q}_p)$, providing purely geometric counterparts to some results of Chan-Wong. In the sequels, the techniques of this work will be extended to real and $p$-adic classical groups and used to study their Arthur packets.