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Zbl 0919.22003
Torossian, Charles
Harish-Chandra homomorphism for orthogonal symmetric pairs and radial parts of invariant differential operators on symmetric spaces. (L'homomorphisme de Harish-Chandra pour les paires symétriques orthogonales et parties radiales des opérateurs différentiels invariants sur les espaces symétriques.) (French)
[J] Bull. Soc. Math. Fr. 126, No.3, 295-354 (1998). ISSN 0037-9484

Let ${\germ g}$ be a finite-dimensional Lie algebra over $\bbfC$, $\sigma$ an involution of ${\germ g}$ and ${\germ g}= {\germ k}+{\germ p}$ the associated decomposition of ${\germ g}$ into $\pm 1$ eigenspaces of $\sigma$. Let $U[{\germ g}]^{\germ k}$ be the subalgebra of $\text{ad }{\germ k}$-invariants in the universal enveloping algebra $U[{\germ g}]$ of ${\germ g}$. Let $\delta\in{\germ k}^*$ be defined by $\delta(x)= 1/2$ and let $\text{tr}_{{\germ g}/{\germ k}}\text{ad}(x)$ for $x\in {\germ k}$ and ${\germ k}^{-\delta}$ be the subspace of $U[{\germ g}]$ formed by the elements $x-\delta(x)$.\par A fundamental problem in harmonic analysis is to understand the nature of the commutative algebra $$A_{\germ k}({\germ g})= U[{\germ g}]^{\germ k}/ U[{\germ g}]^{\germ k}\cap U[{\germ g}]\cdot {\germ k}^{-\delta},$$ which is directly related with the algebra of invariant differential operators on the symmetric space $G/K$.\par In his previous paper [J. Funct. Anal. 117, 174-214 (1993; Zbl 0803.43004)], the author constructed an injective homomorphism $\nu$ from the algebra $A_{\germ k}({\germ g})$ into the quotient field of $S[{\germ p}]^{\germ k}$ -- the algebra of $\text{ad }{\germ k}$-invariants in the symmetric algebra of ${\germ p}$. When ${\germ g}$ is semisimple the homomorphism $\nu$ reduces to Harish-Chandra's isomorphism. One can guess that for every $u\in U[{\germ g}]^{\germ k}$ the element $\nu(u)$ is polynomial valued, i.e., $\nu(u)\in S[{\germ p}]^{\germ k}$. This is the so-called polynomial conjecture.\par By using the recurrence procedure, the author first proves that the polynomial conjecture is true for orthogonal symmetric pairs. Secondly, he describes connections between radial techniques in the semisimple case and global techniques in the nilpotent case and proves that the homomorphism $\nu$ coincides with Rouvière's isomorphism in the case of solvable symmetric pairs. Finally, he proves that the polynomial conjecture is also valid in the case, where $u$ is an element of the enveloping algebra of a solvable ideal of ${\germ g}$.
[Zhu Fulin (Hubei)]

MSC 2000:: *22E30 Analysis on real and complex Lie groups
43A85 Analysis on homogeneous spaces
22E27 Representations of nilpotent and solvable Lie groups

Keywords: symmetric space; differential operator; radial part; solvable pair; orthogonal pair; solvable ideal; Harish-Chandra homomorphism; polynomial conjecture; differential operators; orthogonal symmetric pairs; solvable symmetric pairs

Citations: Zbl 0803.43004

Cited in: Zbl 1062.22027

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