Description of the Mathematica notebook accompanying

Representations of Fourier term modules

Roelof Bruggeman and Roberto Miatello

Lecture Notes in Mathematics 2340, Springer Nature

In this Mathematica notebook we carry out computations that are to complicated to carry out by hand, and we check many other results.

The most convenient way to use the Mathematica notebook rFtm.nb is to consult it in Mathematica, provided one has access to this application.
Here we give a pdf version of the various section and subsections with the commands and their output.

pdf-version of the sections of the notebook

• Initial information
• Chapter 2
  • 1 The group G=SU(2,1); §2.1.
  • 2 Symmetric space ; §2.1.1.
  • 3 Lie algebra; §2.3, §2.4, §3.1, §3.2
  • 4 Functions on K; §2.2
  • 5 Functions on N; §2.3, §3.3.2
  • 6 Discrete subgroup and lattice; §2.4
• Chapter 3
  • 7 Shift operators, construction; §3.1
  • 8 Explicit right differentiation of functions on G; §3.1
  • 9 Large abelian Fourier term modules; §3.3.1
  • 10 Large non-abelian Fourier term modules; §3.3.2
  • 11 Eigenfunctions of the center of the enveloping algebra; §3.4
  • 12 One-dimensional K-types; §3.4.2
  • 13 Kernel of downward shift operators; Proposition 3.21
  • 14 Kernel of upward shift operators; Proposition 3.24
  • 15 Principal series; §3.5.1, §4.2
  • 16 Evaluation at zero; §3.5.3
• Chapter 4
  • 17 Intersection of kernels of downward shift operators (principal series); §4.1, §4.2
  • 18 Intersection of kernels of downward shift operators in the generic abelian case; §4.3
  • 19 Kernel relations in the nonabelian case; §4.4.2
  • 20 Extremal families in the non-abelian case; §4.4.2
  • 21 Intersection of kernels of downward shift operators, non-abelian case; §4.4.2, §4.4.6
  • 22 Lemma 4.18 on dimensions
  • 23 Comparison of M with W and V; computations for Proposition 4.33
  • 24 Structure of non-abelian Fourier term modules; Table 4.12 in §4.4.7
  • 25 Sesquilinear forms and unitarity; §4.5
• Chapter 5
  • 26 Holomorphic automorphic forms; §5.1.2
  • 27 General lattices; §5.4
• Appendix
  • A1 Modified Bessel functions; §A1
  • A2 Whittaker functions; §A2

August 30, 2023