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A. A. Kirillov University of Pennsylvania Philadelphia, United States kirillov@math.upenn.edu

Generalized exp onents for the children of the spinor representation of Bn
To any representation (, V ) of a semisimple Lie algebra g. Author introduced in [1] a socalled classical family algebra C l (g) = S (g)End V . One of the goals was the computation of generalized exponents for irreducible g-submodules of End V , which are called "children" of . In my talk I'll speak about recent joint results with A.Rupinski, which suggests an explicit formula for the generalized exponents of all children of the spinor representation of the simple Lie group g of type Bn so(2n + 1). These children include all fundamental representations of g except the last one and one representation with the highest weight 2n . Note, that the known results allow only obtain a very beautiful but even more impractical formula for the generalized exponents (Hesselink formula), which reduced the problem to the summation of about 2(n-1) · n! polynomials in q with are defined by rather involved combinatorial procedure. References [1] Kirillov A.A. Introduction to Family Algebras, Moscow Math.J., vol.1, No1 (2001), 27-41.

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