Commuting families of symmetric differential operators
Commuting families of symmetric differential operators
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对称微分算子的通勤族
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发表时间：
1994
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影响因子：
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通讯作者：
Hideko Sekiguchi
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作者：
Hiroyuki Ochiai;T. Oshima;Hideko Sekiguchi
许多微分算子的通勤族或完全可积的量子系统已经被构造为与根系统相关（参见[OP2]和其中的参考文献）。这样的族在坐标上往往具有一定的对称性。对称空间上不变微分算子的径向部分给出了微分算子交换族的一个很好的例子（参见[HC]）。在这种情况下，一些参数仅采用由对称空间的根空间的维数确定的一些离散值。另一方面，[Sj]将它们概括为 An 类型根系统的复杂参数。对于一般根系统，[H1]、[H2]、[HO]、[Op1]、[Op2] 给出了相同的概括。如果根系统是经典类型，则其算子会给出本说明中研究的通勤族的示例（参见备注 3 iii））。也就是说，我们将在坐标对称的假设下确定所有族。令 W 为 An−1 型（n ≥ 3）或 Bn 型（n ≥ 2）或 Dn 型（n ≥ 4）的 Weyl 群。我们用 R 的坐标变换 (x1,...,xn)7→(ε1xσ(1),...,εnxσ(n)) 的群来识别 W，其中 σ 是第 n 个对称群 Sn 的元素， ε1 = · · · = εn = 1 如果 W 为 An−1 类型，则 ε1 = ±1, · · · , εn = ±1 如果 W 为 Bn 类型，则 ε1 = ±1, · · · , εn = ±1 且 #{i ;如果 W 的类型为 Dn，则 εi = −1} 为偶数。
Many commuting families of differential operators or completely integrable quantum systems have been constructed in connection with root systems (cf. [OP2] and references therein). Such families often have a certain symmetry in coordinates. The radial parts of invariant differential operators on symmetric spaces give a good example of a commuting family of differential operators (cf. [HC]). In this case some parameters take only some discrete values determined by the dimensions of the root spaces for the symmetric spaces. On the other hand, [Sj] generalized them to complex parameters for the root system of type An. The same generalization was given by [H1], [H2], [HO], [Op1], [Op2] for general root systems. If the root system is of classical type, their operators give examples of the commuting families studied in this note (cf. Remark 3 iii)). Namely we shall determine all the families under the assumption of a symmetry in coordinates. Let W be the Weyl group of type An−1 with n ≥ 3 or of type Bn with n ≥ 2 or of type Dn with n ≥ 4. We identify W with the group of the coordinate transformations (x1, . . . , xn) 7→ (ε1xσ(1), . . . , εnxσ(n)) of R, where σ are the elements of the n-th symmetric group Sn and ε1 = · · · = εn = 1 if W is of type An−1, ε1 = ±1, · · · , εn = ±1 if W is of type Bn, ε1 = ±1, · · · , εn = ±1 and #{i ; εi = −1} is even if W is of type Dn.