2:00 pm Tuesday, February 11, 2003
Number Theory Seminar : Mirror Symmetry, Langlands duality and representations of finite groups of Lie type by Tamas Hausel (UT Auatin) in RLM 9.166
This talk reports on an on-going project with Fernando Rodriguez-Villegas. For any complex Lie group G and compact Riemannian surface C of genus g, the character variety M_B(G) is defined as the variety of representations of the fundamental group of C to G modulo conjugation. Motivated by mirror symmetry and previous work with Michael Thaddeus (math.AG/0205236), we conjecture that the stringy Hodge polynomials of M_B(SL(n,C)) and M_B(PGL(n,C) agree. The method of attacking this conjecture is to ``approximate'' our varieties by replacing the complex Lie groups by their analouges over a finite field. Then we count the points on our varieties over these these finite fields which will lead us to an arithmetic calculation of the stringy Hodge polynomials of the character varieties. The calculation will reduce to checking a formula using only the character tables for SL(n,q) and PGL(n,q). As explicit character tables for finite groups of Lie type, like ours, are available in the literature for small rank, we can prove our mirror symmetry conjecture for SL(2,q) and SL(3,q). Moreover we can verify other general consequences of the conjecture, which imply some new results even about the original mirror symmetry conjecture of Thaddeus and the speaker. In summary our mirror symmetry proposal can roughly be formulated as follows: ``The differences between the character tables of SL(n,q) and PGL(n,q) are governed by mirror symmetry.''
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