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The third Russian-Chinese conference on complex analysis, algebra, algebraic geometry and mathematical physics
May 15, 2016 16:00, Moscow, Steklov Mathematical Institute of RAS, Gubkina, 8

Distribution spaces of two-dimensional local fields and representations of the discrete Heisenberg group

D. V. Osipovab

a Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
b National University of Science and Technology «MISIS»
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D. V. Osipov
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Abstract: The talk is based on joint paper with A.N. Parshin: arXiv:1510.02423.
The discrete Heisenberg group $Heis(3, \mathbb{Z})$ is the group of integer upper-triangular $ 3 \times 3$ matrices with units on the diagonal. This group is the simplest non-Abelian nilpotent group of class $2$. The classical theory of unitary representations of locally compact groups in a Hilbert space does not work smoothly in this case. But in our case of the group $Heis(3, \mathbb{Z})$, if we change the category of representation spaces and consider, instead of the Hilbert spaces, vector spaces of countable dimension and without any topology, then the situation will be much better. For irreducible nonunitary representations, the new approach yields a moduli space which is a complex manifold.
We consider a two-dimensional local field $K$ that is isomorphic to the field of iterated Laurent series $\mathbb{F}_q((u))((t))$. This field naturally appears from a flag of subvarieties on an algebraic surface over a finite field $\mathbb{F}_q$: a point and an irreducible curve on this surface, via localization and completion procedure. D.V. Osipov and A.N. Parshin have previously constructed the infinite-dimensional $\mathbb{C}$-vector space of distributions $\mathcal{D}'_{\mathcal{O}}(K)$ on $K$ (note that $K$ is not locally compact group).
We construct an explicit family of pair-wise non-isomorphic irreducible infinite-dimensional representations of $Heis(3, \mathbb{Z})$ inside of $\mathcal{D}'_{\mathcal{O}}(K)$ such that this family is parametrized by points of an elliptic curve $\mathbb{C}^*/ q^{\mathbb{Z}}$. To calculate the traces on these representations, we consider an action of the extended discrete Heisenberg group, which is isomorphic to $ Heis(3, \mathbb{Z}) \rtimes \mathbb{Z}$ and it is a discrete nilpotent group of class $3$. The traces which we obtain are classical Jacobi theta functions.

Language: English

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