Twisted geometric Satake equivalence. (English) Zbl 1316.22019
Summary: Let \(k\) be an algebraically closed field and \(O = \text{k}[[t]] \subset F = \text{k}((t))\). For an almost simple algebraic group \(G\) we classify central extensions \(1 \rightarrow \mathbb G_{m} \rightarrow E \rightarrow G(F) \rightarrow 1\); any such extension splits canonically over \(G(O)\). Fix a positive integer \(N\) and a primitive character \(\zeta : \mu_N(k) \rightarrow \bar{\mathbb Q}^*_{\ell}\) (under some assumption on the characteristic of k). Consider the category of \(G(O)\)-bi-invariant perverse sheaves on \(E\) with \(\mathbb G_{m}\)-monodromy \( \zeta\). We show that this is a tensor category, which is tensor equivalent to the category of representations of a reductive group \(\check G_{E,N}\). We compute the root datum of \(\check G_{E,N}\).
MSC:
22E57 | Geometric Langlands program: representation-theoretic aspects |
14D24 | Geometric Langlands program (algebro-geometric aspects) |
11R39 | Langlands-Weil conjectures, nonabelian class field theory |
References:
[1] | DOI: 10.1007/s10240-001-8192-2 · Zbl 1093.20027 · doi:10.1007/s10240-001-8192-2 |
[2] | Bourbaki, Groupes et algèbres de Lie (1968) |
[3] | DOI: 10.1007/BF01456050 · Zbl 0644.22007 · doi:10.1007/BF01456050 |
[4] | Lysenko, Annales Scient. Éc. Norm. Sup. 39 pp 415– (2006) |
[5] | Deligne, Lecture Notes in Mathematics 900 pp 101– (1982) |
[6] | DOI: 10.1098/rspa.1994.0058 · Zbl 0829.17018 · doi:10.1098/rspa.1994.0058 |
[7] | Kapustin, Commun. Num. Theory Phys. 1 pp 1– (2007) · Zbl 1128.22013 · doi:10.4310/CNTP.2007.v1.n1.a1 |
[8] | DOI: 10.1007/s00029-008-0053-0 · Zbl 1160.17009 · doi:10.1007/s00029-008-0053-0 |
[9] | DOI: 10.1007/s10097-002-0045-x · Zbl 1020.14002 · doi:10.1007/s10097-002-0045-x |
[10] | DOI: 10.4007/annals.2007.166.95 · Zbl 1138.22013 · doi:10.4007/annals.2007.166.95 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.