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Certain L2-norms on automorphic representations of SL(2)

Hongyu He

Abstract

Let $Γ$ be a non-uniform lattice in $SL(2, \mathbb R)$. In this paper, we study various $L^2$-norms of automorphic representations of $SL(2, \mathbb R)$. We bound these norms with intrinsic norms defined on the representation. Comparison of these norms will help us understand the growth of $L$-functions in a systematic way.

Certain L2-norms on automorphic representations of SL(2)

Abstract

Let be a non-uniform lattice in . In this paper, we study various -norms of automorphic representations of . We bound these norms with intrinsic norms defined on the representation. Comparison of these norms will help us understand the growth of -functions in a systematic way.

Paper Structure

This paper contains 21 sections, 27 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Certain $L^2$-norm of $\Gamma$-invariant functions
  3. setup
  4. Estimates on $\| f \|_{T_1, a_1^{-},\epsilon}$
  5. Estimate on $\| f \|_{T_1, a_1^{+}, \epsilon}$
  6. Estimates of $\|f\|_{T_1, \epsilon}$
  7. Matrix Coefficients and Analysis on $P_0/N_{\mathfrak{p}}$
  8. Principal series representations of $SL(2, \mathbb R)$
  9. Matrix coefficients with respect to periodical distribution with zero constant term
  10. $L^2$-norms on $P_0/N_{\mathfrak{p}}$
  11. Estimates of Fourier coefficients $b_{n}$
  12. $L^2$-norms of Bounded Periodical Matrix coefficients
  13. $K$-invariant Norms and complementary series
  14. Intertwining operator and complementary series
  15. Normalizing $(*,*)_u$
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\pi=\mathcal{P}(u, \pm)$ be a unitary representation in the principal series (see Section psr for the definition). Let $\mathcal{H}$ be a cuspidal representation in $L^2(G/\Gamma)_{\pi}$. Then for any $\epsilon >0$, there exists a $C_{\epsilon} >0$ such that For any $\epsilon <0$, there exists a $C_{\epsilon}>0$ such that Here $u_0=\Re(u)$ and the norm ${\left\vert\left\vert\left\vert f \ri

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Theorem 2.1
  • Proposition 2.2
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 3.1
  • Proposition 3.1
  • Theorem 3.1
  • ...and 17 more