Schedule Nov 05, 2020
Harmonic weak Maass forms and extensions
Stephen Kudla, U. Toronto
Cite as: doi:10.26081/K6WG7D

The fact that classical holomorphic modular forms of weight at least 2 are associated to holomorphic discrete series representations of SL_2(R) plays a basic role in the passage from such modular forms to automorphic representations. On the other hand, given that the mock modular forms, their modular completions, and the many variants arising in physics and elsewhere can be seen as exotic versions of classical modular forms, it is natural to ask for an explanation of such objects in representation theoretic language. In this talk I will review some old joint work with Kathrin Bringmann on the case of harmonic weak Maass forms. Elementary representation theory shows that there are 9 isomorphism classes of indecomposable (g,K)-modules that could arise, and we show that all of them are actually occur by giving explicit examples. The most interesting cases involve indecomposable modules that are non-trivial extensions, a structure that reflects the relation between a Mock modular form and its shadow, in classical language. In the second part of the talk I will discuss various fragmentary results concerning the extension of this theory to Siegel modular forms. For example, there is a Siegel modular form of genus 2 and weight 3/2, constructed in joint work with Rapoport and Yang, which can be viewed as the modular completion of a genus 2 mock modular form. Its shadow, i.e., its image under a genus 2 \xi operator, can also be described in terms of Eisenstein series. It is an interesting question as to whether this representation theoretic aspect has any relevance when such exotic modular forms play a role in physics.

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