Invariant generalized functions on sl(2,R) with values in a sl(2,R) -module
Abstract
Let
g
be a finite dimensional real Lie algebra. Let
r:g→End(V)
be a representation of
g
in a finite dimensional real vector space. Let $C_{V}=(End(V)\tens S(g))^{g}$ be the algebra of
End(V)
-valued invariant differential operators with constant coefficients on
g
. Let
U
be an open subset of
g
.
We consider the problem of determining the space of generalized functions
ϕ
on
U
with values in
V
which are locally invariant and such that
C
V
ϕ
is finite dimensional.
In this article we consider the case
g=sl(2,R)
. Let
N
be the nilpotent cone of
sl(2,R)
. We prove that when
U
is
SL(2,R)
-invariant, then
ϕ
is determined by its restriction to
U∖N
where
ϕ
is analytic. In general this is false when
U
is not
SL(2,R)
-invariant and
V
is not trivial. Moreover, when
V
is not trivial,
ϕ
is not always locally
L
1
. Thus, this case is different and more complicated than the situation considered by Harish-Chandra where
g
is reductive and
V
is trivial.
To solve this problem we find all the locally invariant generalized functions supported in the nilpotent cone
N
. We do this locally in a neighborhood of a nilpotent element
Z
of
g
and on an
SL(2,R)
-invariant open subset
U⊂sl(2,R)
. Finally, we also give an application of our main theorem to the Superpfaffian.