The Ktypemat command is part of the K-type suite of commands:
KTypeform, Ktypemat, srtest and branch. It describes the irreducible
representations of K "K-types". See [David Vogan, Branching to a
Maximal Compact Subgroup, in proceedings of the conference for Roger
Howe, Singapore 2007].
Ktypemat computes the K-types of a given final standard module, up to
a given height. Each such K-type is the lowest K-type of a final
standard module, and the K-types of these are computed also, giving a
square matrix.
See the help file for Ktypeform for an overview of K-types and normal
final standard modules.
In response to the Ktypemat command the user enters a standard module,
as in the Ktypeform command. The software tests if this is both
"standard" and "final" (for Ktypeform it is only required to be
final); if not it prints a message and quits. If so, the user is told
the height of this representation, and asked for a height bound.
The software first converts the paramter into normal final standard
form. It then computes all K-types of this representation up to the
given height. Each of these K-types is parametrized by a normal final
standard module, and each of the K-types of these are also computed,
up to the given height. This gives a matrix, with rows corresponding
to standard modules (restricted to K) and columns to K-types. Both
rows and columns are parametrized by the same set of normal final
standard modules. An entry m (an integer) in a row corresponding to
standard module I_K and column corresponding to K-type mu means:
I_K occurs with multiplicity m in the formula for mu
(see the help file for the Ktypeform command). Thus each column of the
matrix gives a formula for a K-type as a formal linear combination of
standard modules restricted to K (truncated at the given height). In
particular the first column of the matrix gives the same information
as the Ktypeform command for this K-type (truncated by height).
The inverse of this matrix gives the matrix of K-multiplicities in the
given normal standard representations. Now each column of the matrix
correpsponds to a standard module, and each row a K-type. The entries
in the column (positive integers) corresponding to standard module I_K
gives the multiplicities of K-types in this stanard module, up to the
given height.
*** Input and output ***
The user is prompted for a kgb element and sequence of integers a1
... an just as in the Ktypeform command. If the given parameter is
standard and final, the user also gives a height, a nonnegative
integer which should be at least as big as the height of the given
standard module.
The software then converts this parameter to normal final standard
(form, and computes an "Initial Formula" for its lowest K-type, as a
sum of normal final standard modules. This is the information computed
by the Ktypeform command.
It then computes all K-types of the given representation up to the
given height, and lists them. Denote the I_K(0), ..., I_K(n). Each
I_K(n) has a lowest K-type mu(n).
The software then prints a matrix labelled "triangular system". This
is a lower triangular matrix, with possibly negative entries and 1s on
the diagonal. This is the first matrix discussed above; and entry m in
row i and column j means
I_K(i) occurs with multiplicity m in the formula for mu(j).
Thus if the matrix is a(i,j):
mu(j) = sum_i a(i,j)I_K(i)
(possibly plus other terms of greater height).
The software then prints the "Matrix of K-type multiplicities". An
entry m in row i and column j means
mu(i) has multipicity m in I_K(j)
i.e. if the matrix is b(i,j):
I_K(j)= sum_i b(i,j)mu(i)
(plus other K-types of greater height).
*** Examples ***
Example 1: K-types of the holomorphic discrete series of SL(2,R):
real: Ktypemat
Choose KGB element: 0
2rho = [ 2 ]
Give lambda-rho: 0
Height of representation [ 2 ]@(0)#0 is 2.
Give height bound: 20
Name an output file (return for stdout, ? to abandon):
Initial formula: mu([ 2 ]@(0)#0) =
+ [ 2 ]@(0)#0 - [ 6 ]@(0)#0
Ordering of representations/K-types:
[ 2 ]@(0)#0, height 2
[ 6 ]@(0)#0, height 6
[10 ]@(0)#0, height 10
[14 ]@(0)#0, height 14
[18 ]@(0)#0, height 18
Triangular system:
1 0 0 0 0
-1 1 0 0 0
0 -1 1 0 0
0 0 -1 1 0
0 0 0 -1 1
Matrix of K-type multiplicites:
1 0 0 0 0
1 1 0 0 0
1 1 1 0 0
1 1 1 1 0
1 1 1 1 1
Write pi_k for the holomorphic discrete series representation with
infinitesimal character k*rho. The given standard module is pi_1. The
first column of the first matrix says that
(lowest K-types of pi_1) = pi_1 - pi_3 (restricted to K).
See Example 1 in the help file for Ktypeform.
The first column of the second matrix says that pi_1 restricted to K
contains the given K-types (the lowest K-types of pi_1, pi_3, ...,
pi_9) with multiplicity 1.
Example 2: K-types of the spherical principal series of SL(2,R)
real: Ktypemat
Choose KGB element: 2
2rho = [ 2 ]
Give lambda-rho: 0
Height of representation [ 2 ]@(0)#1 is 0.
Give height bound: 10
Name an output file (return for stdout, ? to abandon):
Initial formula: mu([ 2 ]@(0)#1) =
+ [ 2 ]@(0)#1 - [ 2 ]@(0)#0 - [ 2 ]@(1)#0
Ordering of representations/K-types:
[ 2 ]@(0)#1, height 0
[ 2 ]@(0)#0, height 2
[ 2 ]@(1)#0, height 2
[ 6 ]@(0)#0, height 6
[ 6 ]@(1)#0, height 6
[10 ]@(0)#0, height 10
[10 ]@(1)#0, height 10
Triangular system:
1 0 0 0 0 0 0
-1 1 0 0 0 0 0
-1 0 1 0 0 0 0
0 -1 0 1 0 0 0
0 0 -1 0 1 0 0
0 0 0 -1 0 1 0
0 0 0 0 -1 0 1
Matrix of K-type multiplicites:
1 0 0 0 0 0 0
1 1 0 0 0 0 0
1 0 1 0 0 0 0
1 1 0 1 0 0 0
1 0 1 0 1 0 0
1 1 0 1 0 1 0
1 0 1 0 1 0 1
The first column of the second matrix says that the K-types of the
spherical principal series are the following with multiplicity one:
the trivial representation, and the lowest K-types of the two discrete
series representations with infinitesimal character 1, 3, 5, ...
Example 3: Spherical principal series of Sp(4,R):
real: Ktypemat
Choose KGB element: 10
2rho = [ 2, 2 ]
Give lambda-rho: 0 0
Height of representation [ 2, 2 ]@(0,0)#3 is 0.
Give height bound: 10
Name an output file (return for stdout, ? to abandon):
Initial formula: mu([ 2, 2 ]@(0,0)#3) =
+ [ 2, 2 ]@(0,0)#3 - [ 2, 2 ]@(0,0)#1 - [ 2, 0 ]@(0,0)#0 - [ 2, 0 ]@(0,1)#0
- [ 2, 0 ]@(1,0)#0 - [ 2, 0 ]@(1,1)#0
Ordering of representations/K-types:
[ 2, 2 ]@(0,0)#3, height 0
[ 2, 2 ]@(0,0)#1, height 4
[ 2, 0 ]@(0,0)#0, height 6
[ 2, 0 ]@(0,1)#0, height 6
[ 2, 0 ]@(1,0)#0, height 6
[ 2, 0 ]@(1,1)#0, height 6
Triangular system:
1 0 0 0 0 0
-1 1 0 0 0 0
-1 -1 1 0 0 0
-1 0 0 1 0 0
-1 -1 0 0 1 0
-1 0 0 0 0 1
Matrix of K-type multiplicites:
1 0 0 0 0 0
1 1 0 0 0 0
2 1 1 0 0 0
1 0 0 1 0 0
2 1 0 0 1 0
1 0 0 0 0 1