# Characteristic cycles, micro local packets and packets with cohomology

## TL;DR

This paper investigates how characteristic cycles behave under geometric induction, linking representation theory and microlocal geometry, and shows that certain cohomology packets are micro-packets, unifying different sheaf-theoretic approaches.

## Contribution

It describes the characteristic cycle of induced representations in terms of the original cycle, extending the understanding of geometric induction in representation theory.

## Key findings

- Characteristic cycle splits into two terms under regular infinitesimal character.
- The first term of the cycle is described explicitly.
- Cohomology packets by Adams-Johnson are shown to be micro-packets.

## Abstract

Relying on work of Kashiwara-Schapira and Schmid-Vilonen, we describe the behaviour of characteristic cycles with respect to the operation of geometric induction, the geometric counterpart of taking parabolic or cohomological induction in representation theory. By doing this, we are able to describe to some extent the characteristic cycle associated to an induced representation, in terms of the characteristic cycle of the representation being induced. More precisely, under the hypothesis that the infinitesimal character is regular (and dominant), we show that the characteristic cycle of an induced representation splits in two terms. We describe the first term precisely, but we are not able to do the same for the second one. What we are able to say, is that this second term is supported on the boundary of the space generated by the inclusion in the flag variety of $G$, of the flag variety of the Levi subgroup. As a consequence, we prove that the cohomology packets defined by Adams and Johnson are micro-packets, that is to say that the cohomological constructions of Adams-Johnson are particular cases of the sheaf-theoretic ones of Adams-Barbasch-Vogan.