From Archimedean $L$-factors to Topological Field Theories

Dmitry Lebedev

ITEP (Moscou)

Mon, Jun. 22nd 2009, 11:00

Salle Claude Itzykson, Bât. 774, Orme des Merisiers

Archimedean local $L$-factors were introduced to simplify functional equations of global $L$-functions. From the point of view of arithmetic geometry these factors complete the Euler product representation of global $L$-factors by taking into account Archimedean places of the compactified spectrum of the global field. A construction of non-Archimedean local $L$-factors is rather transparent and uses characteristic polynomial of the image of the Frobenius homomorphism in finite-dimensional representations of the local Weil-Deligne group closely related to the local Galois group. On the other hand, Archimedean $L$-factors are expressed through products of $\Gamma$-functions and thus are analytic objects avoiding simple algebraic interpretation. In a series of papers [1,2,3,4] we approach the problem of the proper interpretation of Archimedean $L$-factors using various methods developed to study quantum integrable systems and low-dimensional topological field theories. As a result we produce several interesting explicit representations for Archimedean $L$-factors and related special functions revealing some hidden structures that might be relevant to the Archimedean (also known as $\infty$-adic ) algebraic geometry. \\
\par \noindent The talk is based on common with A.Gerasimov and S. Kharchev papers: \par \noindent [1] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {\it Baxter operator and Archimedean Hecke algebra}, Comm. Math. Phys. DOI 10.1007/s00220-008-0547-9; {\tt [arXiv:0706.347]}, 2007. \par \noindent [2] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {\it Baxter Q-operators and their Arithmetic implications}, Lett. Math. Phys. DOI 10.1007/911005-008-0285-0; {\tt [arXiv:0711.2812]}. \par \noindent [3] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {\it On q-deformed $\mathfrak{gl}_{\ell+1}$-Whittaker functions I,II,III}, {\tt [arXiv:0803.0145]}, {\tt [arXiv:0803.0970]}, {\tt [arXiv:0805.3754]}. \par \noindent [4] A.~Gerasimov, D.~Lebedev, S.~Oblezin, {\it Archimedean L-factors and Topological Field Theories}, {\tt [arXiv:0906.1065]}.

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