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Introduction to Automorphic Forms for $GL(n,\BZ)\ltimes \BZ^{(m,n)}$

Jae-Hyun Yang

Abstract

In this paper, we introduce the notion of automorphic forms for $GL(n,\BZ)\ltimes \BZ^{(m,n)}$ and discuss invariant differential operators on the Minkowski-Euclid space. The group $GL{n,\BR}\ltimes \BR^{(m,n)}$ is the semidirect product of $GL(n,\BR)$ and the additive group $\BR^{(m,n)}$ and is {\it not} a reductive group. The Minkowski-Euclid space is the quotient space of $GL(n,\BR)\ltimes \BR^{(m,n)}$ by $O(n,\BR)$. The Minkowski-Euclid space is an important non-symmetric homogeneous space geometrically and number theoretically. We present some open problems to be solved in the future.

Introduction to Automorphic Forms for $GL(n,\BZ)\ltimes \BZ^{(m,n)}$

Abstract

In this paper, we introduce the notion of automorphic forms for and discuss invariant differential operators on the Minkowski-Euclid space. The group is the semidirect product of and the additive group and is {\it not} a reductive group. The Minkowski-Euclid space is the quotient space of by . The Minkowski-Euclid space is an important non-symmetric homogeneous space geometrically and number theoretically. We present some open problems to be solved in the future.
Paper Structure (5 sections, 11 theorems, 174 equations)

This paper contains 5 sections, 11 theorems, 174 equations.

Table of Contents

  1. Introduction
  2. Automorphic Forms for $GL(n,\mathbb Z)$
  3. Automorphic Forms for $SL(n,\mathbb Z)$
  4. Automorphic Forms for $GL(n,\mathbb Z)\ltimes \mathbb Z^{(m,n)}$
  5. Automorphic Form for $SL(n,\mathbb Z)\ltimes \mathbb Z^{(m,n)}$

Key Result

Theorem 2.1

A geodesic $\alpha (t)$ joining $I_n$ and $Y\in {\mathscr P}_n$ has the form where is the spectral decomposition of $Y$, where $k\in O(n,\mathbb R),\ A={\rm diag} (a_1,\cdots,a_n)$ with all $a_j\in \mathbb R.$ The distance of $\alpha (t) \ (0\leq t\leq 1)$ between $I_n$ and $Y$ is

Theorems & Definitions (42)

  • Theorem 2.1
  • proof
  • Theorem 2.2
  • proof
  • Proposition 2.1
  • proof
  • Theorem 2.3
  • proof
  • Proposition 2.2
  • proof
  • ...and 32 more