Local Langlands Conjectures For Other Groups
Borel (1979) and Vogan (1993) discuss the Langlands conjectures for more general groups. As of 2011, the Langlands conjectures for arbitrary reductive groups G are not as precise as the ones for general linear groups, and it is unclear what the correct way of stating them should be. Roughly speaking, admissible representations of a reductive group are grouped into disjoint finite sets called L-packets, which should correspond to some classes of homomorphisms, called L-parameters, from the Weil–Deligne group to the L-group of G.
Langlands (1989) proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible admissible representations (up to infinitesimal equivalence), or, equivalently, of their irreducible -modules.
Gan & Takeda (2011) proved the local Langlands conjectures for the symplectic similitude group GSp(4) and used that in Gan & Takeda (2010) to deduce it for the symplectic group Sp(4).
Read more about this topic: Local Langlands Conjectures
Famous quotes containing the words local, conjectures and/or groups:
“The improved American highway system ... isolated the American-in-transit. On his speedway ... he had no contact with the towns which he by-passed. If he stopped for food or gas, he was served no local fare or local fuel, but had one of Howard Johnson’s nationally branded ice cream flavors, and so many gallons of Exxon. This vast ocean of superhighways was nearly as free of culture as the sea traversed by the Mayflower Pilgrims.”
—Daniel J. Boorstin (b. 1914)
“After all, it is putting a very high price on one’s conjectures to have a man roasted alive because of them.”
—Michel de Montaigne (1533–1592)
“Trees appeared in groups and singly, revolving coolly and blandly, displaying the latest fashions. The blue dampness of a ravine. A memory of love, disguised as a meadow. Wispy clouds—the greyhounds of heaven.”
—Vladimir Nabokov (1899–1977)