Primitive ideals in quantum Schubert cells: Dimension of the strata
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Abstract
Abstract. The aim of this paper is to study the representation theory of quantum Schubert cells. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> $\mathfrak {g}$ be a simple complex Lie algebra. To each element w of the Weyl group W of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝔤</m:mi> </m:math> $\mathfrak {g}$ , De Concini, Kac and Procesi have attached a subalgebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mi>w</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> $U_q[w]$ of the quantised enveloping algebra <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>𝔤</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $U_q(\mathfrak {g})$ . Recently, Yakimov showed that these algebras can be interpreted as the (quantum) Schubert cells on quantum flag manifolds. In this paper, we study the primitive ideals of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mi>w</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> $U_q[w]$ . More precisely, it follows from the Stratification Theorem of Goodearl and Letzter, and from recent works of Mériaux–Cauchon and Yakimov, that the primitive spectrum of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>U</m:mi> <m:mi>q</m:mi> </m:msub> <m:mrow> <m:mo>[</m:mo> <m:mi>w</m:mi> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:math> $U_q[w]$ admits a stratification indexed by those elements <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>∈</m:mo> <m:mi>W</m:mi> </m:mrow> </m:math> $v \in W$ with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>≤</m:mo> <m:mi>w</m:mi> </m:mrow> </m:math> $v \le w$ in the Bruhat order. Moreover each stratum is homeomorphic to the spectrum of maximal ideals of a torus. The main result of this paper gives an explicit formula for the dimension of the stratum associated to a pair <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>v</m:mi> <m:mo>≤</m:mo> <m:mi>w</m:mi> </m:mrow> </m:math> $v \le w$ .
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