In mathematical representation theory, an L-packet is a collections of (isomorphism classes of) irreducible representations of a reductive group

Reductive group

In mathematics, a reductive group is an algebraic group G over an algebraically closed field such that the unipotent radical of G is trivial . Any semisimple algebraic group is reductive, as is any algebraic torus and any general linear group...

 over a local field

Local field

In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...

, 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

L-function

The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...

 or Robert Langlands

Robert Langlands

Robert Phelan Langlands is a mathematician, best known as the founder of the Langlands program. He is an emeritus professor at the Institute for Advanced Study...

, who introduced them in , .

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

Langlands group

In representation theory, a branch of mathematics, the Langlands dual LG of a reductive algebraic group G is a group that controls the representation theory of G. If G is a group over a field k, LG is an extension of the absolute Galois group of k by a complex Lie 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 representation

Discrete series representation

In mathematics, a discrete series representation is an irreducible unitary representation of a locally compact topological group G that is a subrepresentation of the left regular representation of G on L²...

s with a given infinitesimal character

Infinitesimal character

In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group G on a vector space V is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation...

.