The "klbasis" command prints out the list of all the non-zero
Kazhdan-Lusztig polynomials in the block. These are polynomials
P_{\gamma,\nu} with \gamma,\nu in the block. Necessarily
length(\gamma)<=length(\nu) (in fact gamma<=\nu in the Bruhat order).
Here \gamma,\nu are represented by numbers 0,...,n-1 from the output
of the block command.
The polynomials have the following meaning. Associated to each block B
is a family of regular integral infinitesimal characters. For any such
infinitesimal character lambda the block corresponds to a set of
irreducible representations with infinitesimal character lambda. See
the help file for the block command.
Fix a block B and a a regular infinitesimal character for B. Then
associated to each block element 0<=x<=n is a standard module I(x),
with unique irreducible quotient J(x). The Grothendieck group of B
consists of formal sums \sum_x a_x J(x) with each a_x an integer. This
Z-module is also spanned by the standard modules I(x). In other words
there is an identity in the Grothendieck group
J(y)=sum_x M(x,y)I(x)
where the sum is over the block (only x<=y in the Bruhat order
contribute), and the M(x,y) are integers. These are given by the
Kazhdan-Lusztig polynomials P_{x,y} evaluated at q=1:
M(x,y)=(-1)^{length(x)-length(y)}P_{x,y}(1).
See [The Kazhdan-Lusztig Conjecture for Real Reductive Groups, David
Vogan, Proceedings of the Park City Conference, Progr. Math., 40,
Birkhauser, 1983]
The output of klbasis is of the form:
y: x1: P_{x1,y}
x2: P_{x2,y}
x3: P_{x3,y}
...
For example
8: 0: 1
1: 1
2: 1
4: 1
6: 1
8: 1
means
P_{0,8}=P_{1,8}=P_{2,8}=P_{4,8}=P_{6,8}=P_{8,8}=1
and all other P_{x,8}=0.
The final information is the number of non-zero polynomials, the
number of polynomials P_{x,y}=0 even though where x<=y in the Bruhat
order, and the number of pairs (x,y) with x<=y in the Bruhat order.
Example: the block of the trivial representation of SL(2,R):
block:
0(0,1): 0 0 [i1] 1 (2,*)
1(1,1): 0 0 [i1] 0 (2,*)
2(2,0): 1 1 [r1] 2 (0,1) 1
I(0)=J(0)=pi_+=holomorphic discrete series, length=0
I(1)=J(1)=pi_-=anti-holomorphic discrete series, length=0
I(0)=spherical principal series, J(0)=C=trivial representation, length=1
klbasis:
Full list of non-zero Kazhdan-Lusztig-Vogan polynomials:
0: 0: 1
1: 1: 1
2: 0: 1
1: 1
2: 1
The last three lines tell us:
P_{0,2)=P_{1,2}=P_{2,2}=1
which with the length information gives:
M(0,2)=-1
M(1,2)=-1
M(2,2)=1
i.e.
J(2)=-I(0)-I(1)+I(2)
or in more familiar terms an identity in the Grothendieck group:
C= (spherical principal series) - pi_+ - pi_-
This is an inversion of the familiar decomposition of the spherical
principal series into irreducible Jordan Holder factors:
spherical principal series = pi_+ + pi_- + C
Example: Sp(4,R)
Here is part of the output of klbasis for the block of the trivial
representation of Sp(4,R):
10: 0: 1
1: 1
2: 1
3: 1
4: 1
5: 1
6: 1
7: 1
8: 1
9: 1
10: 1
J(10) is the trivial representation C, and we conclude
C=I(10)-I(9)-I(8)-I(7)+I(6)+I(5)+I(4)-I(3)-I(2)-I(1)-I(0)