# Mathematics Colloquia and Seminars

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### Symmetries of the generalized hypergeometric function mFm-1

**Mathematical Physics & Probability**

Speaker: | Oleg Gleizer, UCLA |

Location: | 693 Kerr |

Start time: | Tue, Jun 7 2005, 3:10PM |

In one sentence, what I will present in this talk is an elliptization of the results of Beukers and Heckman on the monodromy of mFm-1. mFm-1 is a solution of the generalized hypergeometric equation (g.h.g.eq.) which is the closest relative of the famous hypergeometric equation of Gauss-Riemann. It is known that when all the local exponents of the g.h.g.eq. are generic real numbers, there exists a (unique up to a constant multiple) monodromy invariant hermitian form on the space of solutions H_trig. The m-hypergeometric system (m-h.g.s.) is a Fuchsian system equivalent to the g.h.g.eq. as a flat connection. When all its local exponents are generic real numbers, there exists a (unique up to a constant multiple) complex symmetric form on the residue space H_0 such that the residue matrices are self-adjoint with respect to it. The formulae for the symmetric product on H_0 and for the hermitian product on H_trig look very similar to each other despite the different nature of the products. It was the initial goal of the investigation to understand reasons for such a similarity. It turns out that there exists a Hilbert space H naturally associated with the problem. This space has three two-parameter families of both hermitian and complex symmetric forms on it. In particular, it is hyperkahler. H_trig and H_0 are m-dimensional subspaces of H. The hyperkahler structure on H explains why both H_0 and H_trig have complex symmetric and hermitian forms on them. Moreover, there exists another m-dimensional subspace H(omega1, omega2) of H such that the space of solutions of the m-h.g.s. H_trig is its trigonometric limit as omega2 -> infinity. The residue space H_0 is the rational limit of H(omega1, omega2) as both omega1, omega2 -> infinity. This explains the similarity between the formulae for the two spaces. The main technique is to realize solutions of the m-h.g.s. as Fermionic fields. Analytic continuation is then replaced by the vacuum expectation value pairing. If time permits and depending on the choice of the audience, applications to Integrable Systems and/or to Representation Theory and/or to Algebraic Geometry could be discussed.