Download Discrete series representations and K multiplicities for U(p, q). User's

Let us explain the behavior of the partition function NA+ on the domain c − Z.
We first explain the case of an unimodular system.
Definition 26 The system A+ is unimodular if each σ ∈ Bases(A+ ) is a Z-basis
of VZ .
Example 27 It is easy to see that A+
r is unimodular, so is any subsystem.
Thus if A+ is unimodular, the set F = {0} satisfies the condition (C) and we
choose this set F .
Proposition 28 If A+ is unimodular, the function NcA+ (h) is a polynomial function on V .
Proof. We have just to consider K(G, h) = K(0, h) and we can write
QN
hαi , ui
ehh,ui
ehh,ui
= QN
× QN i=1
K(0, h)(u) = QN
−hα
,ui
−hαi ,ui )
i
)
i=1 (1 − e
i=1 hαi , ui
i=1 (1 − e
where
QN
hαi ,ui
QN i=1 −hα ,ui
i
(1−e
)
i=1
=
P+∞
k=0 ψk (u)
is a holomorphic function of u in a neighbor-
hood of 0 with ψ0 (u) = 1.
It follows that NcA+ (h) is given by the following polynomial function of h
NcA+ (h)
(8)
= vol (V /VZ , dh) JKc
= vol (V /VZ , dh)
N
−r
X
k=0
ehh,ui
QN
i=1 hαi , ui
×
+∞
X
k=0
1
JKc
(N − r − k)!
!
ψk (u)
hh, uiN −r−k ψk (u)
QN
i=1 hαi , ui
!
.
Note that the function NcA+ is a polynomial function of degree N − r whose
c (h), that is the volume
homogeneous component of degree N −r is the function YA
+
of the polytope.
Let us now consider the general case where F is no longer reduced to {0}.
For example for parabolic root systems of Br , Cr , Dr , the set F satisfying the
condition (C) cannot longer be taken as equal to {0}.
We recall that an exponential polynomial function is a linear combination of
exponential functions multiplied by polynomials.
Proposition 29 The function NcA+ (h) is an exponential polynomial function on
V and the restriction of NcA+ (h) to VZ is a quasipolynomial function on VZ .
26