The srtest command is part of the K-type suite of commands: KTypeform,
Ktypemat, srtest and branch. It describes "K-types", the irreducible
representations of K. See [David Vogan, Branching to a Maximal Compact
Subgroup, in proceedings of the conference for Roger Howe, Singapore
2007].
The srtest command takes as input a parameter for a (continued) standard
representation restricted to K, and writes this representation as a sum
of normal final standard modules. This is primarily intended as an aid
to the user in understanding these parameters for use in the other
K-type commands.
See the help file for Ktypeform for an overview of K-types and normal
final standard modules.
The user is prompted for a kgb element and sequence of integers a1,
..., an just as in the Ktypeform command. The resulting parameter is
not required to be final (the corresponding representation may be 0)
or standard (the corresponding representation may be a virtual
"continued" representation).
The software then converts the representation to a sum of normal final
standard limit representations.
This is useful for understanding when a standard module fails to be
final.
Example 1: holomorphic discrete series of SL(2,R) and continuations:
real: srtest
Choose KGB element: 0
2rho = [ 2 ]
Give lambda-rho: 0
Weight (1/2)[ 2 ] converted to (1/2)[ 2 ]
Height is 2
Standard normal final limit representations:
R0: [ 2 ]@(0)#0 [2]
Standardized expression for [ 2 ]@(0)#0:
+ R0
real: srtest
Choose KGB element: 0
2rho = [ 2 ]
Give lambda-rho: -1
Weight (1/2)[ 0 ] converted to (1/2)[ 0 ]
Height is 0
Standard normal final limit representations:
R0: [ 0 ]@(0)#0 [0]
Standardized expression for [ 0 ]@(0)#0:
+ R0
real: srtest
Choose KGB element: 0
2rho = [ 2 ]
Give lambda-rho: -1
Weight (1/2)[ 0 ] converted to (1/2)[ 0 ]
Height is 0
Standard normal final limit representations:
R0: [ 0 ]@(0)#0 [0]
Standardized expression for [ 0 ]@(0)#0:
+ R0
real: srtest
Choose KGB element: 0
2rho = [ 2 ]
Give lambda-rho: -2
Weight (1/2)[-2 ] converted to (1/2)[-2 ]
Height is 2
Intermediate representations:
N0: [-2 ]@(0)#0 [2], non Standard, witness [1]
Standard normal final limit representations:
R0: [ 2 ]@(1)#0 [2]
R1: [ 2 ]@(0)#1 [0]
Standardized expression for [-2 ]@(0)#0:
+ R1 - R0
The first case (lambda-rho =0) is the holomorphic discrete series
representation of SL(2,R) at infinitesimal character rho. The second
(lambda-rho=0), is this representation continued to infinitesimal
character 0, i.e. the limit of discrete series.
The final case (lambda-rho=-2) is this representation continued to -rho.
This continued standard module is virtual. Its restriction to K is
(spherical principal series) - (antiholomorphic discrete series at
infinitesimal character rho)
Example: non-spherical principal series of SL(2,R)
real: srtest
Choose KGB element: 2
2rho = [ 2 ]
Give lambda-rho: -1
Weight (1/2)[ 0 ] converted to (1/2)[ 4 ]
Height is 0
Intermediate representations:
N0: [ 4 ]@(0)#1 [0], non Final, witness [1]
Standard normal final limit representations:
R0: [ 0 ]@(0)#0 [0]
R1: [ 0 ]@(1)#0 [0]
Standardized expression for [ 4 ]@(0)#1:
+ R0 + R1
The non-spherical principal series of SL(2,R) is not final (since we're
only considering the restriction to K, we are essentially considering
nu-parameter, i.e. infinitesimal character, 0). The Ktypeform commands
will not return a result for this parameter (see the Ktypeform help).
The srtest command shows that this representation is the sum of the two
limits of discrete series.