Local L-factors and perverse sheaves on certain loop spaces

We present a conjectural construction of local L-factors for representations of a reductive group G over a local non-archimedian field of positive characteristic (associated with arbitrary algebraic representation of the Langlands dual group).The construction involves perverse sheaves on loop spaces into some algebraic semi-groups.

We prove that our construction gives the right answer for those representations of G which have an Iwahori-fixed vector. This is a joint work with R.Bezrukavnikov, M.Finkelberg and D.Kazhdan.

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