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つづき

Applications

The Geometric Satake equivalence establishes an equivalence between representations of the Langlands dual group {}^{L}G} of a reductive group G and certain equivariant perverse sheaves on the affine Grassmannian associated to G.
This equivalence provides a non-combinatorial construction of the Langlands dual group. It is proved by showing that the mentioned category of perverse sheaves is a Tannakian category and identifying its Tannaka dual group with {}^{L}G}.

Extensions
Wedhorn (2004) has established partial Tannaka duality results in the situation where the category is R-linear, where R is no longer a field (as in classical Tannakian duality),
but certain valuation rings. Duong & Hai (2017) showed a Tannaka duality result if R is a Dedekind ring.

Iwanari (2014) has initiated the study of Tannaka duality in the context of infinity-categories.