In the field of mathematics known as

representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, an L-packet is a collection of (isomorphism classes of) irreducible representations of a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...

over a

local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...

, that are L-indistinguishable, meaning they have the same Langlands parameter, and so have the same

L-function In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give ...

and ε-factors. L-packets were introduced by

Robert Langlands Robert Phelan Langlands, (; born October 6, 1936) is a Canadian mathematician. He is best known as the founder of the Langlands program, a vast web of conjectures and results connecting representation theory and automorphic forms to the study ...

in , . The classification of irreducible representations splits into two parts: first classify the L-packets, then classify the representations in each L-packet. The

local Langlands conjectures In mathematics, the local Langlands conjectures, introduced by , are part of the Langlands program. They describe a correspondence between the complex representations of a reductive algebraic group ''G'' over a local field ''F'', and representatio ...

state (roughly) that the L-packets of a reductive group ''G'' over a local field ''F'' are conjecturally parameterized by certain homomorphisms of the

Langlands group In mathematics, the Langlands group is a conjectural group ''L'F'' attached to each local or global field ''F'', that satisfies properties similar to those of the Weil group. It was given that name by Robert Kottwitz. In Kottwitz's formulati ...

of ''F'' to the L-group of ''G'', and Arthur has given a conjectural description of the representations in a given L-packet.

The elements of an L-packet

For irreducible representations of connected complex reductive groups, Wallach proved that all the L-packets contain just one representation. The L-packets, and therefore the irreducible representations, correspond to quasicharacters of a

Cartan subgroup In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate. Exam ...

, up to conjugacy under the

Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections ...

. For general linear groups over

s, the L-packets have just one representation in them (up to isomorphism). An example of an L-packet is the set of

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(''G''). In the Plancherel mea ...

s with a given infinitesimal character and given

central character A protagonist () is the main character of a story. The protagonist makes key decisions that affect the plot, primarily influencing the story and propelling it forward, and is often the character who faces the most significant obstacles. If a st ...

. For example, the discrete series representations of SL₂(R) are grouped into L-packets with two elements. gave a conjectural parameterization of the elements of an L-packet in terms of the connected components of ''C''/''Z'', where ''Z'' is the center of the L-group, and ''C'' is the centralizer in the L-group of Im(φ), and φ is the homomorphism of the Langlands group to the L-group corresponding to the L-packet. For example, in the general linear group, the centralizer of any subset is Zariski connected, so the L-packets for the general linear group all have 1 element. On the other hand, the centralizer of a subset of the projective general linear group can have more than 1 component, corresponding to the fact that L-packets for the special linear group can have more than 1 element.

References

* * * *{{Citation , last1=Langlands , first1=Robert P. , editor1-last=Sally , editor1-first=Paul J. , editor2-last=Vogan , editor2-first=David A. , title=Representation theory and harmonic analysis on semisimple Lie groups , orig-year=1973 , url=http://publications.ias.edu/rpl/paper/16 , publisher=

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings ...

, location=Providence, R.I. , series=Math. Surveys Monogr. , isbn=978-0-8218-1526-7 , mr=1011897 , year=1989 , volume=31 , chapter=On the classification of irreducible representations of real algebraic groups , pages=101–170 Langlands program