Let G be a connected reductive algebraic group with Lie algebra g defined over an algebraically closed field, k, with char k = 0. Fix a parabolic subgroup of G with Levi decomposition P = LU where U is the unipotent radical of P . Let u = Lie(U) and let z denote the center of Lie(L). Let T be a maximal torus in L with Lie algebra t. Then the root system of (g, t) is a subset of t∗ and by restri...

The Atlas of Lie Groups and Representations is a project in representation theory of real reductive groups. The main goal of the atlas computer software, currently under development, is to compute the unitary dual of any real reductive Lie group G. As a step in this direction it currently computes the admissible representations of G. The underlying mathematics of the software is described in Al...

We prove that any reductive group G over a non-Archimedean local field has a cuspidal complex representation.

One disclaimer is necessary: while in principle this factorization result makes it clear that representation theory is relevant to study of automorphic forms and L-functions, in practice there are other things necessary. In effect, one needs to know that the representation theory of reductive linear p-adic groups is tractable, so that conversion of other issues into representation theory is a c...

We show that every admissible representation of a real reductive group has a canonical system of Sobolev norms parametrized by positive characters of a minimal parabolic subgroup. These norms are compatible with morphisms of representations. Similar statement also holds for representations of reductive p-adic groups. 1991 Mathematics Subject Classification: 22E46 43A70 46E39

We show that a reductive group scheme over base S admits faithful linear representation if and only its radical torus is isotrivial, is, it splits after finite etale cover.

‎‎in cite{fr 1‎, ‎fr 2} using faithful complex representations of cyclically pinched and conjugacy pinched one-relator groups we proved that any limit group has a faithful representation in $psl(2,c)$‎. ‎further this representation can be effectively constructed using the jsj decomposition‎. ‎in this note we show that any hyperbolic cyclically pinched one-relator group with maximal amalgamated ...

Let G be a connected reductive linear algebraic group over C and let (; V) be a regular representation of G. There is a locally-nite representation (^ ; C V ]) on the aane algebra C V ] of V deened by ^ (g)f(v) = f(g ?1 v) for f 2 C V ]. Since G is reductive, (^ ; C V ]) decomposes as a direct sum of irreducible regular representations of G. The representation (; V) is said to be multiplicity f...

An action of an algebraic reductive groupG on an affine varietyM is calledmultiplicity-free if the multiplicity of any particular irreducible representation of G in the space C [M ] of regular functions on M is at most one: In [1], Kac provides a complete list of multiplicity-free actions for the case when G is a connected reductive algebraic group and M is a finite-dimensional vector space upo...

Let F be either ℝ or a finite extension of ℚ p , and let G central the group F-points reductive defined over F. Also π smooth representation (Fréchet moderate growth if F=ℝ). For each nilpotent orbit