In mathematics, a unipotent representation 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 direct ...
is a
representation that has some similarities with
unipotent
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''.
In particular, a square matrix ''M'' is a unipoten ...
conjugacy class
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wo ...
es of groups.
Informally,
Langlands philosophy suggests that there should be a correspondence between representations of a reductive group and conjugacy classes of a
Langlands dual group
In representation theory, a branch of mathematics, the Langlands dual ''L'G'' of a reductive algebraic group ''G'' (also called the ''L''-group of ''G'') is a group that controls the representation theory of ''G''. If ''G'' is defined over a fie ...
, 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 characters ''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.
Non-archimedean local fields
classified the unipotent characters over non-archimedean local fields.
Archimedean local fields
discusses several different possible definitions of unipotent representations of real Lie groups.
See also
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Deligne–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 .
used these representations to find all representations of all ...
References
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*{{Citation , last1=Vogan , first1=David A. , title=Unitary representations of reductive Lie groups , url=https://books.google.com/books?id=0O-9c_kImJYC , publisher=
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large.
The press was founded by Whitney Darrow, with the financia ...
, series=Annals of Mathematics Studies , isbn=978-0-691-08482-4 , year=1987 , volume=118
Representation theory