We prove cases of Rietsch mirror conjecture that the quantum
connection for projective homogeneous varieties is isomorphic to the
pushforward D-module attached to Berenstein-Kazhdan geometric crystals.
The idea is to recognize the quantum connection as Galois and the
geometric crystal as automorphic. In particular we link the purity of
Berenstein-Kazhdan crystals to the Ramanujan property of certain Hecke
eigensheaves.
The isomorphism of D-modules comes from global rigidity results where a
Hecke eigenform is determined by its local ramification. We reveal
relations with the works of Gross, Frenkel-Gross, Heinloth-Ngo-Yun and
Zhu on Kloosterman sheaves. The talk will keep the algebraic geometry
prerequisite knowledge to a minimum by introducing the above concepts of
"mirror" and "crystal" with the examples of CP^1, projective spaces and
Grassmannians. Work with Thomas Lam.
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