### Abstract

Let G be any reductive p-adic group. We discuss several conjectures, some of them new, that involve the representation theory and the geometry of G. At the heart of these conjectures are statements about the geometric structure of Bernstein components for G, both at the level of the space of irreducible representations and at the level of the associated Hecke algebras. We relate this to two well-known conjectures: the local Langlands correspondence and the Baum-Connes conjecture for G. In particular, we present a strategy to reduce the local Langlands correspondence for irreducible G-representations to the local Langlands correspondence for supercuspidal representations of Levi subgroups.

Original language | English (US) |
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Title of host publication | Contemporary Mathematics |

Publisher | American Mathematical Society |

Pages | 15-51 |

Number of pages | 37 |

DOIs | |

State | Published - Jan 1 2017 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 691 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Contemporary Mathematics*(pp. 15-51). (Contemporary Mathematics; Vol. 691). American Mathematical Society. https://doi.org/10.1090/conm/691/13892