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Multiplicities of Representations in Algebraic Families

Li Cai, Yangyu Fan

Abstract

In this short notes, we consider multiplicities of representations in general algebraic families, especially the upper semi-continuity of homological multiplicities and the locally constancy of Euler-Poincare numbers. This generalizes the main result of Aizenbud-Sayag for unramified twisting families.

Multiplicities of Representations in Algebraic Families

Abstract

In this short notes, we consider multiplicities of representations in general algebraic families, especially the upper semi-continuity of homological multiplicities and the locally constancy of Euler-Poincare numbers. This generalizes the main result of Aizenbud-Sayag for unramified twisting families.
Paper Structure (3 sections, 13 theorems, 37 equations)

This paper contains 3 sections, 13 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Homological algebra
  3. Homological multiplicities

Key Result

Theorem 1.3

Let $\pi$ be a finitely generated torsion-free smooth admissible $R[G(F)]$-module whose fiber rank is locally constant on $\Sigma$. Assume moreover there exists a finitely generated torsion-free admissible R[G(F)]-module $\tilde{\pi}$ such that for any $x\in\Sigma$, $\tilde{\pi}|_x\cong (\pi|_x)^\ve

Theorems & Definitions (29)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Proposition 1.6: Upper semi-continuous theorem
  • proof
  • Remark 1.7
  • Corollary 1.8
  • Lemma 2.1
  • ...and 19 more