Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity
Commutativity of invariant differential operators on nilpotent homogeneous spaces with finite multiplicity
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有限重数幂零齐次空间上不变微分算子的交换性
DOI：
10.1002/cpa.3160450603
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发表时间：
1992
影响因子：
3
通讯作者：
F. Greenleaf
中科院分区：
文献类型：
--
作者：
L. Corwin;F. Greenleaf
设g是幂零李代数，t是任意子代数，G和K是单连通李群，xEk是任意特征标。则x（expY）= exp {2ai（fo，Y）}对于YEt，其中fo Eg * 和folt是李同态.对于z= ind（KtG，x），文[8]和Lipsman [24]给出了z= Jfm（a）adp（a）中不可约表示aEGs及其重数m（a）的轨道描述.如果m（a）z+0 0，则存在一个有限界m（a）2 N pa。e.在G上;我们关心的是后一种“有限多重性”的情况。本文将z视为L2（K\G）中的上圈作用，研究了K\G上的C”微分算子代数D_i（K\G）与包络代数u（g）中的算子zg，g ∈ G或等价于z（A），A可交换.关于这些不变算子的第一个结果是由Benoist（见[1]）得到的，他研究了一个非常特殊的情况：x= 1，（g，e）对称空间-t ={g：a（X）= X}，其中g的某个对合自同构CT。他证明了m（a）= 1 on spec（z）= supp（p），D，（K\G）是可交换的，并且D，（K\G）C [tLIK= Ad* K-不变多项式在tL c g* 上的代数。当m（a）= 1时，交换性并不那么令人惊讶，但越来越多的证据表明，即使在正测度集上m（a）> 1，当重数是有限的时，交换性也会发生。
Let g be a nilpotent Lie algebra, t any subalgebra, G and K the simply connected Lie groups, and x E k any character. Then x (expY)= exp {2ai (fo, Y)} for Y E t, where fo E g* and folt is a Lie homomorphism. For z= ind (K t G, x) the authors in [8] and Lipsman in [24] gave an orbital description of the irreducible representations a E G appearing in z= Jf m (a) adp (a) and also their multiplicities m (a). It happens that m (a) z+ oo or else there is a finite bound m (a) 2 N pa. e. on G; we are concerned with the latter “finite multiplicity” case. We view z as a cocycle action in L2 (K\G) and study the algebra D,(K\G) of C” differential operators on K\G that commute with the operators zg, g E G, or equivalently with z (A), A in the enveloping algebra u (g). The first results on these invariant operators were obtained by Benoist (see [l]), who studied a very special situation: x= 1 and (g, e) a symmetric space-t={Xe g: a (X)= X} for some involutive automorphism CT of g. He showed that m (a)= 1 on spec (z)= supp (p), that D,(K\G) is commutative, and that D,(K\G) C [tLIK= the algebra of Ad* K-invariant polynomials on tL c g*. Commutativity is not so surprising when m (a)= 1, but evidence has accumulated that it can occur when the multiplicity is finite even if m (a)> 1 on a set of positive measure.