Cite as: doi:10.26081/K6WG7D

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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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