RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
Video Library
Archive
Most viewed videos

Search
RSS
New in collection





You may need the following programs to see the files






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»
Video records:
Flash Video 268.6 Mb
Flash Video 1,601.5 Mb
MP4 268.6 Mb

Number of views:
This page:172
Video files:32

D. V. Osipov
Photo Gallery


Видео не загружается в Ваш браузер:
  1. Установите Adobe Flash Player    

  2. Проверьте с Вашим администратором, что из Вашей сети разрешены исходящие соединения на порт 8080
  3. Сообщите администратору портала о данной ошибке

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

SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru
 
Contact us:
 Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2017