Universal Weil module

8 Mar 2021  ·  Justin Trias ·

The classical construction of the Weil representation, that is with complex coefficients, has been expected to be working for more general coefficient rings. This paper exhibits a minimal ring $\mathcal{A}$, being the integral closure of $\mathbb{Z}[\frac{1}{p}]$ in a cyclotomic field, and carries the construction of the Weil representation over $\mathcal{A}$-algebras... As a leitmotiv all along the work, most of the problems can actually be solved over the based ring $\mathcal{A}$ and transferred for any $\mathcal{A}$-algebra by scalar extension. The most striking fact consists in these many Weil representations arising as the scalar extension of a single one with coefficients in $\mathcal{A}$. In this sense, the Weil module obtained is universal. Building upon this construction, we are speculating and making prognoses about an integral theta correspondence. read more

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Representation Theory