# The Generalized Injectivity Conjecture

## TL;DR

This paper proves a conjecture relating to the structure of standard modules in certain reductive groups, establishing that the unique generic subquotient is a subrepresentation, with implications for classical types.

## Contribution

It extends the existence of strategic embeddings for irreducible generic discrete series representations, confirming the conjecture for a broad class of groups.

## Key findings

- Confirmed the conjecture for classical type groups.
- Extended results of Moeglin on embeddings.
- Established the subrepresentation property for generic subquotients.

## Abstract

We prove a conjecture of Casselman and Shahidi stating that the unique irreducible generic subquotient of a standard module is necessarily a subrepresentation for a large class of connected quasi-split reductive groups, in particular for those which have a root system of classical type (or product of such groups). To do so, we prove and use the existence of strategic embeddings for irreducible generic discrete series representations, extending some results of Moeglin.