2012/03/27 by Vinoth Nandakumar
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Algorithm #Artificial intelligence #Computer science #math.AG #math.QA #math.RT
paper · pdf · doi:10.1090/s1088-4165-2013-00444-2
published as Represent. Theory 17 (2013), 663-681
arxiv created 2012/03/27 · openalex publication_date 2013/12/23 · arxiv updated 2020/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G equals upper S p Subscript 2 n Baseline left-parenthesis double-struck upper C right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>=</mml:mo> <mml:mi>S</mml:mi> <mml:msub> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">G=Sp2n(\mathbb C)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper N"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be Kato’s exotic nilpotent cone. Following techniques used by Bezrukavnikov in 2003 to establish a bijection between <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper Lamda Superscript plus"> <mml:semantics> <mml:msup> <mml:mi mathvariant="bold"> Λ </mml:mi> <mml:mo>+</mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">\boldsymbol Λ +</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the dominant weights for an arbitrary simple algebraic group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="bold upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mtext mathvariant="bold">O</mml:mtext> </mml:mrow> <mml:annotation encoding="application/x-tex">\textbf O</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , the set of pairs consisting of a nilpotent orbit and a finite-dimensional irreducible representation of the isotropy group of the orbit, we prove an analogous statement for the exotic nilpotent cone. First we prove that dominant line bundles on the exotic Springer resolution <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper N overTilde"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:mo> ~ </mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding="application/x-tex">\widetilde \mathfrak N</mml:annotation> </mml:semantics> </mml:math> </inline-formula> have vanishing higher cohomology, and compute their global sections using techniques of Broer. This allows us to show that the direct images of these dominant line bundles constitute a quasi-exceptional set generating the category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript b Baseline left-parenthesis normal upper C normal o normal h Superscript upper G Baseline left-parenthesis German upper N right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Db(\mathrm CohG(\mathfrak N))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and deduce that the resulting <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -structure on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper D Superscript b Baseline left-parenthesis normal upper C normal o normal h Superscript upper G Baseline left-parenthesis German upper N right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>D</mml:mi> <mml:mi>b</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">C</mml:mi> <mml:mi mathvariant="normal">o</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> <mml:mi>G</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">N</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">Db(\mathrm CohG(\mathfrak N))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coincides with the perverse coherent <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="t"> <mml:semantics> <mml:mi>t</mml:mi> <mml:annotation encoding="application/x-tex">t</mml:annotation> </mml:semantics>