In [OOS] we announce a classification of integrable systems invariant
under simple classical Weyl groups.
The precise discussion has already been given by [OS] and [O]
except for the case of type .
As is shown in [OS], the classification problem for type
is
reduced to a functional differential equation (1.4).
In §2 we give a complete list of solutions of this functional equation. Some solutions have already been obtained, after [OP2], by Inozemtsev [IM], [I] (See also [P]). The main result of §2 is Theorem 2.9, which is stated in §1.3 in a different form.
In §3 we examine the reducibility of the system obtained in §2. We note that if the system coincides with the system satisfied by zonal spherical functions of a semisimple Lie group, the reducibility is related to degenerate series representations.
The final draft of this paper was completed when the authors were visiting University of Leiden in the fall of 1994. The authors express their sincere gratitude to Profdrvan Dijk for his hospitality during their stay there.
1.2.
Now we give a quick review of the results in [OS, §6]
concerning with type .
Let
be the Weyl group of type
, which is identified with
the group of coordinate transformations of
generated by
and
.
Consider
-invariant differential operators
The operators are proved to be expressed by even functions and
of
one variable as follows ([OS, Proposition 6.3]):
Conversely for any solution of (1.4) and the pair
of
the operators which are given by (1.2) with
1.3.
We give a complete list of solutions of the functional equation (1.4).
Remind that the Schrödinger operator is explicitly expressed
as in (1.2) using
and
.
(Trivial case)
constant,
an arbitrary even function,
an arbitrary even function,
constant.
Let and
denote the primitive half periods of the Weierstrass elliptic function
and put
and
.
(Elliptic case) For
,
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1.4.
Although we deal with the commuting differential operators of type
with the Weyl group symmetry
in the main body of this paper,
we will give a brief summary of the related works.
The commuting differential operators of type
have been studied very well.
The commuting differential operators of type
with the Weyl group invariant condition
are classified in [OS].
This work is generalized to
the commuting differential operators of type
without Weyl group invariant condition
For type ,
the classification of the commuting differential operators (1.1)
without the Weyl group symmetry has not been done yet.
It is known that the similar functional differential equation
(see (2.3)) is related to such operators.
The following results are obtained in [Oc]:
(i)
We have the expression of the (non Weyl group invariant) operators
and
by using four functions
,
,
and
with one-variable.
Actually, if we replace
by
,
by
, and so on,
the formula (1.2) is also valid for non-invariant operators.
These functions satisfy the functional differential equation like (1.4).
(ii)
Suppose be non-trivial (cf. Lemma 2.4 i)).
If
is holomorphic at some point,
then
and
can be meromorphically continued to
whole plane
.
The orders of poles of
are at most two.
(iii)
Suppose, moreover, that has poles at three points
such that
and
are
linearly independent over
.
Then the function
can be expressed as