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Spherical harmonics on Grassmannians. (English) Zbl 1194.22015

Let \(\mathcal P(\mathbb C^n)=\mathbb C[x_1,x_2,\cdots, x_n]\) be the space of polynomial functions on \(\mathbb C^n\). The multiplication by \(r^2=x_1^2+x_2^2+\cdots+x_n^2\) and the Laplacian \(\Delta=\partial^2/\partial x_1^2+\partial^2/\partial x_2^2+\cdots+\partial^2/\partial x_n^2\) generate an algebra of operators on \(\mathcal P(\mathbb C^n)\) that commute with the action of the orthogonal group \(\text{O}_n(\mathbb C)\). As said by the authors, one can consider the theory of spherical harmonics as a description of the action of this algebra.
In this paper, the authors generalize the theory of spherical harmonics to the case of Grassmannians. Precisely, denote by \(\mathbb G^n_k\) the projective variety of all \(k\)-dimensional subspaces of \(\mathbb C^n\). Denote by \(\mathcal R(\mathbb G_k^n)\) the homogeneous coordinate ring of \(\mathbb G_k^n\). This is equal to \(\mathcal P(\mathbb C^n)\) if \(k=1\). The authors find two operators on \(\mathcal R(\mathbb G_k^n)\), namely \(\gamma_0\) and \(L\), which generalize \(r^2\) and \(\Delta\) respectively. They give a thorough study of the actions of these two operators. In particular, by using these two operators, an explicit decomposition of \(\mathcal R(\mathbb G_k^n)\) into irreducible \(\text{O}_n(\mathbb C)\)-modules is obtained.

MSC:

22E46 Semisimple Lie groups and their representations
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