Induced Representations on Character Varieties of Surfaces

TL;DR

This work identifies a Poisson-geometric realization of Frobenius reciprocity on character varieties of surfaces. By decomposing the direct image and pullback processes into restriction $\mathsf{Res}$ and induction $\mathsf{Ind}$ maps and proving their injectivity and Poisson properties, the authors show that the Frobenius reciprocity map $\mathsf{Frob}$ is a Poisson embedding. Central to the analysis are explicit constructions of the direct image $\phi_*V$ and its monodromy as induced representations, together with Poisson-compatible maps between character varieties and their tangent/cotangent descriptions via group cohomology. The paper further establishes compatibility under isomorphisms of pulled-back and pushed-forward connections, and extends the Poisson/isomonodromy framework to families of surfaces, highlighting a robust interplay between geometric representation theory and integrable structures in the setting of surface groups.

Abstract

We show that a Frobenius reciprocity map on character varieties of surfaces is a Poisson embedding.

Theorems & Definitions (13)

• Theorem 1.1
• Proposition 2.1
• proof
• Theorem 3.1
• proof
• Proposition 4.1
• proof
• Proposition 4.2
• proof
• Lemma 5.1