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Bessel Functions on GL(n), II -- The case $n=4$

Jack Buttcane

Abstract

The purpose of this article is to verify the conjectures of the previous paper in the particular case of $GL(4)$. We accomplish this in general, but observe two failures of the conjectures: First, that the Strong Interchange of Integrals conjecture is perhaps false for a single Weyl element $w_{2,2}$, though we prove the Weak Interchange of Integrals still holds. Second, again for a single Weyl element $w_{2,1,1}$ and its conjugate $w_{1,1,2}$, it appears that the space of solutions to the Bessel differential equations may not be spanned by the Frobenius series solutions. We discuss what refinements, namely to the Asymptotics Theorem, would be necessary to uniquely identify the Bessel functions for such Weyl elements, and prove them in the exceptional cases for $GL(4)$.

Bessel Functions on GL(n), II -- The case $n=4$

Abstract

The purpose of this article is to verify the conjectures of the previous paper in the particular case of . We accomplish this in general, but observe two failures of the conjectures: First, that the Strong Interchange of Integrals conjecture is perhaps false for a single Weyl element , though we prove the Weak Interchange of Integrals still holds. Second, again for a single Weyl element and its conjugate , it appears that the space of solutions to the Bessel differential equations may not be spanned by the Frobenius series solutions. We discuss what refinements, namely to the Asymptotics Theorem, would be necessary to uniquely identify the Bessel functions for such Weyl elements, and prove them in the exceptional cases for .

Paper Structure

This paper contains 53 sections, 9 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. The Kuznetsov formula
  3. Results
  4. Some notation
  5. The conjectures
  6. Caveats
  7. Background
  8. Characters
  9. Gamma Functions
  10. Hypergeometric Series
  11. Mellin-Barnes Integrals
  12. The $J$-Bessel Power Series
  13. The Asymptotics Theorem
  14. Matrix Decompositions
  15. Iwasawa Decompositions
  16. ...and 38 more sections

Key Result

Theorem 1

The Bessel functions at $w=w_{31},w_{22},w_{121},w_{211},w_l$ are annihilated by the differential operators eq:w31DE, eq:w22DE, eq:w121DE1 and eq:w121DE2, eq:211DEs1 and eq:211DEs2, and eq:1111DEs1-eq:1111DEs3, respectively.

Theorems & Definitions (11)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 9
  • proof
  • Lemma 10
  • Corollary 11
  • ...and 1 more