L-packet
In mathematical representation theory, an L-packet is a collection of (isomorphism classes of) irreducible representations of a reductive group over a local field, that are L-indistinguishable, meaning they have the same Langlands parameter, and so have the same L-function and ε-factors. The letter "L" stands for either L-function or Robert Langlands, who introduced them in (Langlands 1989), (Langlands & Labesse 1979).
The classification of irreducible representations splits into two parts: first classify the L-packets, which the Langlands conjectures predict correspond to certain representations of the Langlands group, then classify the representations in each L-packet.
For general linear groups over local fields, the L-packets have just one representation in them (up to isomorphism). An example of an L-packet is the discrete series representations with a given infinitesimal character.
References
- Arthur, James (2006), "A note on L-packets", Pure and Applied Mathematics Quarterly 2 (1): 199–217, ISSN 1558-8599, MR 2217572
- Labesse, Jean-Pierre; Langlands, R. P. (1979), "L-indistinguishability for SL(2)", Canadian Journal of Mathematics 31 (4): 726–785, doi:10.4153/CJM-1979-070-3, ISSN 0008-414X, MR 540902
- Labesse, Jean-Pierre (2008), "Introduction to endoscopy", in Arthur, James; Schmid, Wilfried; Trapa, Peter E., Representation theory of real reductive Lie groups, Contemp. Math. 472, Providence, R.I.: American Mathematical Society, pp. 175–213, ISBN 978-0-8218-4366-6, MR 2454335
- Langlands, Robert P. (1989) [1973], "On the classification of irreducible representations of real algebraic groups", in Sally, Paul J.; Vogan, David A., Representation theory and harmonic analysis on semisimple Lie groups, Math. Surveys Monogr. 31, Providence, R.I.: American Mathematical Society, pp. 101–170, ISBN 978-0-8218-1526-7, MR 1011897