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| Real loop Grassmannians and Langlands duality |
| Abstract: It is known that the category of perverse sheaves on the loop Grassmannian of a reductive complex algebraic group G is equivalent to the category of finite-dimensional representations of the dual group G^. The result can be viewed as a geometric version of the Satake isomorphism, and is a basic part of the geometric Langlands program. It turns out that a similar statement is true for the loop Grassmannian of a reductive real algebraic group G_R. I would like to illustrate this result in the case when G_R=PSO(1,n). The loop Grassmannian of SO(1,n) may be identified with the based loop space of the real projective space RP^n, and a certain category of perverse sheaves on it is equivalent to the category of finite-dimensional representations of SL_2(C). |