Joseph polynomials and Springer representation. (Polynômes de Joseph et représentation de Springer.) (French) Zbl 0718.22009
Let \(G\) be a complex connected semisimple Lie group, let \(\mathfrak g\) be the Lie algebra of \(G\) and let \(D\) be the flag manifold of all Borel subalgebras of \(\mathfrak g\). If \(\mathfrak b\in D\), \(\mathfrak b=\mathfrak h\oplus \mathfrak n\) and \(u\in\mathfrak n\) is a nilpotent element, consider its orbit \(O_ u\) in \(\mathfrak g\), related to the action of the adjoint group. Denote by \(C\) an irreducible component of the manifold \(O_ u\cap \mathfrak n\) and consider \(D(C)=\{x\in D: x\in gx_ 0,\;g^{-1}u\in C\}\).
Using a variant of the method given by A. Joseph [Ann. Sci. Ec. Norm. Supér. (4) 22, No. 4, 569–603 (1989; Zbl 0695.17003)] and some ideas contained in the papers of W. Rossmann [J. Funct. Anal. 96, No. 1, 130–154 (1991; Zbl 0755.22004), 155–193 (1991; Zbl 0755.22005)] the author obtains a new proof for the proportionality of the Joseph and Springer polynomials, i.e. the following formula: \[ \bigl(\prod_{\alpha >0}(\alpha,\rho)\bigr)^{-1}J_ C(X)=m(u)S_{D(C)}(X) \] where \(m(u)=\int_{O_ u}\exp (-\| y\|^ 2/4)\,d\beta_ u\). Note that in the above equality the proportionality coefficient is really determined.
Using a variant of the method given by A. Joseph [Ann. Sci. Ec. Norm. Supér. (4) 22, No. 4, 569–603 (1989; Zbl 0695.17003)] and some ideas contained in the papers of W. Rossmann [J. Funct. Anal. 96, No. 1, 130–154 (1991; Zbl 0755.22004), 155–193 (1991; Zbl 0755.22005)] the author obtains a new proof for the proportionality of the Joseph and Springer polynomials, i.e. the following formula: \[ \bigl(\prod_{\alpha >0}(\alpha,\rho)\bigr)^{-1}J_ C(X)=m(u)S_{D(C)}(X) \] where \(m(u)=\int_{O_ u}\exp (-\| y\|^ 2/4)\,d\beta_ u\). Note that in the above equality the proportionality coefficient is really determined.
Reviewer: Dorin Andrica (Cluj-Napoca)
MSC:
| 22E46 | Semisimple Lie groups and their representations |
| 22E10 | General properties and structure of complex Lie groups |
| 17B08 | Coadjoint orbits; nilpotent varieties |
| 17B10 | Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) |
Keywords:
complex connected semisimple Lie group; Lie algebra; Borel subalgebras; proportionality; Joseph and Springer polynomialsReferences:
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