Cyclotomic nil-Brauer and singular Soergel bimodules of type D

<p> We study a new family of strict monoidal categories, which are cyclotomic quotients of the nil-Brauer category. We construct a monoidal functor from the cyclotomic nil-Brauer category of level <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to another monoidal category constructed from singular Soergel bimodules of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper A Subscript l minus 1 Baseline minus normal upper D Subscript l slash normal upper A Subscript l minus 1"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant="normal">A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo> ⁡ </mml:mo> <mml:mi class="MJX-variant" mathvariant="normal"> ∖ </mml:mi> <mml:msub> <mml:mi mathvariant="normal">D</mml:mi> <mml:mi>l</mml:mi> </mml:msub> <mml:mo> ⁡ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:msub> <mml:mi mathvariant="normal">A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>l</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\operatorname {A}_{l-1} \backslash \operatorname {D}_l / \operatorname {A}_{l-1}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We conjecture that our functor is an equivalence of categories. Although we can prove neither fullness nor faithfulness at this point, we are able to show that the functor induces an isomorphism at the level of Grothendieck rings. We compute these rings and their canonical bases, and give diagrammatic descriptions of the corresponding primitive idempotents. </p>