Twisted geometric Satake equivalence

Source abstract

Abstract Let k be an algebraically closed field and O = k[[ t ]] ⊂ F = k(( t )). For an almost simple algebraic group G we classify central extensions 1 → $\mathbb{G}_m\to E\to G(\bm{F})\$ m → E → G ( F ) → 1; any such extension splits canonically over G ( O ). Fix a positive integer N and a primitive character ζ : μN (K) →$\mathbb{Q}}_\ell^*}$(under some assumption on the characteristic of k). Consider the category of G ( O )-bi-invariant perverse sheaves on E with$\mathbb{G}_m$ m -monodromy ζ. We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group Ǧ E,N . We compute the root datum of Ǧ E,N .