Atlas of Lie Groups and Representations

Documentation contents

atlas Library · Script File

representations.at

<nilpotent.at
Source
atlas-scripts/representations.at (123 lines)
Definitions
16
Loads
Loaded by
test_unitarity.at finite_dimensional_signature.at K_highest_weights.at extended.at weyl_character_formula.at

Definitions in source order

some common representations
L9

finite_dimensional

L16finite_dimensional (RealForm G,vec lambda) = Param
finite dimensional with highest weight gamma in X^*
param(last KGB element,lambda=gamma+rho,nu=gamma+rho)
consequently every simple root is in the tau-invariant,
and the infinitesimal character is gamma+rho

also defined in finite_dimensional.at

finite_dimensional_fundamental_weight_coordinates

L19finite_dimensional_fundamental_weight_coordinates (RealForm G,vec tau) = Param

trivial_block

L27trivial_block (RealForm G) = [Param]
the trivial block

large_fundamental_series 2 overloads

L30large_fundamental_series (RealForm G,ratvec lambda) = Param
return a large fundamental series if G is quasisplit (not necessarily unique)
L35large_fundamental_series (RealForm G) = Param

large_discrete_series 2 overloads

L40large_discrete_series (RealForm G,ratvec lambda) = Param
return a large discrete series if G is quasisplit and equal rank
(not necessarily unique)
L44large_discrete_series (RealForm G) = Param

minimal_principal_series 2 overloads

L49minimal_principal_series (RealForm G,ratvec lambda,ratvec nu) = Param
warning: p may fail to be final
minimal principal series of split group with given lambda,nu
L54minimal_principal_series(RealForm G) = Param

minimal_spherical_principal_series 2 overloads

L56minimal_spherical_principal_series (RealForm G,ratvec nu) = Param
L58minimal_spherical_principal_series (RealForm G) = Param

spherical_unipotent_representations

L64spherical_unipotent_representations (RealForm G) = [Param]
G should be absolutely simple, SL(n,R), Sp(2n,R), S(n,n) or SO(n+1,n) or
simple complex, and have standard coordinates (as we use |nilpotent_lambdas|)
let lietype=Lie_type(root_datum(G)) in
if #lietype=1 then  { assert(@:is_split(G),"Group is not split");}
  for gamma in nilpotent_lambdas(G)
  do minimal_spherical_principal_series(G,gamma) od
elif #lietype=2 then { check that it looks like a simple complex group }
  let (first,second)=let arr=%lietype in (arr[0],arr[1]) in
  assert(@:Cartan_matrix(first)=Cartan_matrix(second),"not a complex group");
  let lambdas=nilpotent_lambdas(simply_connected(first))
  , x=x(trivial(G)), zero=null(rank(G))
  in {parameter might not be normal}
  for lambda in lambdas do dominant(param(x, zero, lambda##lambda)) od
else []
fi
L80

hc_parameter 2 overloads

L99hc_parameter (Param p,KGBElt x_b) = ratvec
Harish-Chandra parameter of (relative) discrete series, with respect to
given x_b
L108hc_parameter (Param p) = ratvec
Harish-Chandra parameter of (relative) discrete series, with respect to
x_b=KGB(G,0)

discrete_series 2 overloads

L112discrete_series (KGBElt x,ratvec lambda) = Param
discrete series parameter with given HC parameter lambda, wrt given x
L121discrete_series (RealForm G,ratvec lambda) = Param
discrete series parameter with given HC parameter lambda, wrt KGB(G,0)

Generated from atlas-scripts at commit 648506c (2026-10-13).