Parabolic Induction and Geometry of Orbital Varieties for GL(n)

Abstract : Ariki and Ginzburg, after the previous work of Zelevinsky on orbital varieties,proved that multiplicities in a total parabolically induced representations aregiven by the value at q = 1 of Kazhdan-Lusztig Polynomials associated to thesymmetric groups. In my thesis I introduce the notion of partial derivativewhich refines the Zelevinsky derivative and show that it can be identified withthe formal exponential of the q-derivative of Kashiwara with q=1. With thehelp of this notion, I exploit the geometry of the nilpotent orbital varietiesto construct a symmetrization process for the multi-segments, which allowsme to proove a conjecture of Zelevinsky on the property of the independenceof the total parabolic induction. On the other hand, I develop a strategyto calculate the multiplicity in a general parabolic induction by using theLusztig product of perverse sheaves.
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Taiwang Deng. Parabolic Induction and Geometry of Orbital Varieties for GL(n). Algebraic Geometry [math.AG]. Université Sorbonne Paris Cité, 2016. English. ⟨NNT : 2016USPCD070⟩. ⟨tel-02136337⟩

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