Interwining operators and automorphic forms for the metaplectic group.

Area of Science:

• Number Theory
• Representation Theory
• Harmonic Analysis

Background:

• The metaplectic group is a central object in the study of automorphic forms.
• Weil's representation provides a fundamental link between quadratic forms and group representations.

Purpose of the Study:

• To investigate the analytic continuation of intertwining operators for the metaplectic covering group of SL(2).
• To analyze the decomposition of Weil's representation attached to a specific quadratic form.
• To explore global results concerning Eisenstein series and their relation to theta functions.

Main Methods:

• Analytic continuation techniques applied to intertwining operators.
• Decomposition of representations of the metaplectic group.
• Study of Eisenstein series and their residues.
• Application of Siegel-Weil type formulas.

Main Results:

• Established local results concerning the analytic continuation of intertwining operators.
• Determined the decomposition of Weil's representation for the given quadratic form.
• Obtained global results on Eisenstein series on the metaplectic group.
• Derived a Siegel-Weil type formula connecting series residues to generalized theta functions.

Conclusions:

• The study provides a comprehensive analysis of local and global properties of representations for the metaplectic group.
• The derived Siegel-Weil formula offers new insights into the relationship between automorphic forms and theta functions.