Langlands dual

In mathematics , the Langlands dual of a group is important in the context of the Langlands program , a series of far-reaching conjectures that link number theory and group representation theory.

definition

Be a splittable reductive group over a global body . The Langlands dual is the fissile reductive group whose weights and roots the Kogewichte and Kowurzeln of are. ${\displaystyle G}$ ${\displaystyle F}$ ${\displaystyle {\widehat {G}}}$ ${\displaystyle G}$

Langlands dual of semi-simple complex lie groups

Let be a simple complex Lie group with Lie algebra . Be the Langlands dual with Lie algebra . ${\displaystyle G}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle {\widehat {G}}}$ ${\displaystyle {\hat {\mathfrak {g}}}}$

Then the Dynkin diagram is from dual to the Dynkin diagram from . (The Dynkin diagram of is dual to the Dynkin diagram of and vice versa. All other Dynkin diagrams are dual to themselves.) ${\displaystyle {\hat {\mathfrak {g}}}}$ ${\displaystyle {\mathfrak {g}}}$ ${\displaystyle B_{n}}$ ${\displaystyle C_{n}}$

For semisimple Lie groups , the Lie algebra is isomorphic to . ${\displaystyle G_{1},G_{2}}$ ${\displaystyle {\widehat {G_{1}\times G_{2}}}}$ ${\displaystyle {\hat {\mathfrak {g}}}_{1}\oplus {\hat {\mathfrak {g}}}_{2}}$

Furthermore, this is the center of isomorphic to the fundamental group of and vice versa. ${\displaystyle {\widehat {G}}}$ ${\displaystyle G}$

Examples

• The Langlands dual of is . ${\displaystyle SL_{n}}$ ${\displaystyle PGL_{n}}$
• The Langlands dual of is and vice versa. ${\displaystyle SO_{2n+1}}$ ${\displaystyle Sp_{2n}}$
• The Langlands dual of is . ${\displaystyle Spin_{2n}}$ ${\displaystyle SO_{2n}/\left\{\pm 1\right\}}$
• For is . ${\displaystyle G\in \left\{GL_{n},SO_{2n},E_{6},E_{7},E_{8},F_{4},G_{2}\right\}}$ ${\displaystyle {\widehat {G}}=G}$

motivation

Be the adelering too . The goal of the Langlands program is the presentation of on in by Galois representations for parameterized summands disassemble. ${\displaystyle A}$ ${\displaystyle F}$ ${\displaystyle G(A)}$ ${\displaystyle L^{2}(G(F)\backslash G(A),\mathbb {C} )}$ ${\displaystyle {\widehat {G}}}$

literature

• JW Cogdell: Dual groups and Langlands functoriality in An introduction to the Langlands program , Birkhäuser, 2004, ISBN 978-0-8176-8226-2