Unipotent representation
Encyclopedia
In mathematics, a unipotent representation of a reductive group
is a representation
that has some similarities with unipotent
conjugacy class
es of groups.
Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes a Langlands dual group, and the unipotent representations should be roughly the ones corresponding to unipotent classes in the dual group.
Unipotent representations are supposed to be the basic "building blocks" out of which one can construct all other representations in the following sense.
Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that all irreducible representations can be obtained from unipotent representations of possibly smaller groups by some sort of systematic process, such as (cohomological or parabolic) induction.
R of the trivial representation 1 of a torus T . They were classified by .
Some examples of unipotent representations over finite fields are the trivial 1-dimensional representation, the Steinberg representation
, and θ10
.
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...
is a representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
that has some similarities with unipotent
Unipotent
In mathematics, a unipotent element r of a ring R is one such that r − 1 is a nilpotent element, in other words such that some power n is zero....
conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
es of groups.
Informally, Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes a Langlands dual group, and the unipotent representations should be roughly the ones corresponding to unipotent classes in the dual group.
Unipotent representations are supposed to be the basic "building blocks" out of which one can construct all other representations in the following sense.
Unipotent representations should form a small (preferably finite) set of irreducible representations for each reductive group, such that all irreducible representations can be obtained from unipotent representations of possibly smaller groups by some sort of systematic process, such as (cohomological or parabolic) induction.
Finite fields
Over finite fields, the unipotent representations are those that occur in the decomposition of the Deligne–Lusztig charactersDeligne–Lusztig theory
In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by ....
R of the trivial representation 1 of a torus T . They were classified by .
Some examples of unipotent representations over finite fields are the trivial 1-dimensional representation, the Steinberg representation
Steinberg representation
In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field...
, and θ10
Θ10
In representation theory, a branch of mathematics, θ10 is a cuspidal unipotent complex irreducible representation of the symplectic group Sp4 over a finite, local, or global field....
.