Hessenberg varieties and hyperplane arrangements

Abstract

Abstract Given a semisimple complex linear algebraic group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> {{G}} and a lower ideal I in positive roots of G , three objects arise: the ideal arrangement <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒜</m:mi> <m:mi>I</m:mi> </m:msub> </m:math> {\mathcal{A}_{I}} , the regular nilpotent Hessenberg variety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Hess</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>N</m:mi> <m:mo>,</m:mo> <m:mi>I</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{Hess}(N,I)} , and the regular semisimple Hessenberg variety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>Hess</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>I</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {\operatorname{Hess}(S,I)} . We show that a certain graded ring derived from the logarithmic derivation module of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝒜</m:mi> <m:mi>I</m:mi> </m:msub> </m:math> {\mathcal{A}_{I}} is isomorphic to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>Hess</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>N</m:mi> <m:mo>,</m:mo> <m:mi>I</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{*}(\operatorname{Hess}(N,I))} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>Hess</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>I</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>W</m:mi> </m:msup> </m:mrow> </m:math> {H^{*}(\operatorname{Hess}(S,I))^{W}} , the invariants in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>H</m:mi> <m:mo>*</m:mo> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>Hess</m:mi> <m:mo>⁡</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mi>S</m:mi> <m:mo>,</m:mo> <m:mi>I</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> {H^{*}(\operatorname{Hess}(S,I))} under an action of the Weyl group W of G . This isomorphism is shown for general Lie type, and generalizes Borel’s celebrated theorem showing that the coinvariant algebra of W is isomorphic to the cohomology ring of the flag variety <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow>