Commutativity of the invariant differential operators on a symmetric space
Commutativity of the invariant differential operators on a symmetric space
复制标题
对称空间上不变微分算子的交换性
DOI：
10.1090/s0002-9939-1968-0218494-5
复制
发表时间：
1968
影响因子：
0.9
通讯作者：
W. Smoke
中科院分区：
文献类型：
--
作者：
W. Smoke
已知（见Helgason[1]）黎曼对称空间上不变微分算子的代数是可交换的。代数可以用代数来定义。我们给出了它的交换性的代数证明。设g是特征为0的域上的李代数，设f是g的子代数。将g上f的伴随表示推广到g的全称包络代数U(g)上，使x xrf作为U(g)的导数。则（ad x） (u) = xu ux, u u (g)。因为这是u属于g的情况，g生成u (g)由公式可知，U(g)的不变元素——那些被f的作用湮灭的元素——是U(g)中与f的元素交换的元素。设U(g)为不变元素的子代数。现在设U(g)f是f在U(g)中生成的左理想。而且，[U(g)flt= U(g)f]中U(g)f是U(g)f中的双面理想。设U（g, f）是商代数。若g是李群g的李代数，f是连通子群K的李代数，则U（g, f）是齐次空间g /K （Smoke[2]）上不变微分算子的李代数。如果K是紧的，并且G的对合自同构的不动点集，则G/K是黎曼对称空间，并且代数U（G, f）是可交换的。
It is known (see Helgason [1]) that the algebra of invariant differential operators on a Riemannian symmetric space is commutative. The algebra may be defined algebraically. We give here an algebraic proof of its commutativity. Let g be a Lie algebra over a field of characteristic zero and let f be a subalgebra of g. Extend the adjoint representation of f on g to the universal enveloping algebra U(g) of g so that ad x, xrf, acts as a derivation of U(g). Then (ad x) (u) = xu ux for u E U(g). For this is the case if u belongs to g, and g generates U(g). It follows from the formula that the invariant elements of U(g)-those annihilated by the action of f-are the elements of U(g) which commute with the elements of f. Let U(g) be the subalgebra of invariant elements. Now let U(g)f be the left ideal generated by f in U(g). This left ideal is preserved by the action of f. Moreover, [U(g)flt= U(g)f n U(g)f is a two-sided ideal in U(g)f. Let U(g, f) be the quotient algebra. If g is the Lie algebra of a Lie group G and f that of a connected subgroup K then U(g, f) is the algebra of invariant differential operators on the homogeneous space G/K (Smoke [2]). If K is compact and the fixed point set of an involutive automorphism of G then G/K is a Riemannian symmetric space and the algebra U(g, f) is commutative.