On the category 𝒪 of a hybrid quantum group

We study the representation theory of a hybrid quantum group at root of unity <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta"> <mml:semantics> <mml:mi>ζ</mml:mi> <mml:annotation encoding="application/x-tex">\zeta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> introduced by Gaitsgory. After discussing some basic properties of its category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we study deformations of the category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For subgeneric deformations, we construct the endomorphism algebra of big projective object and compute it explicitly. Our main result is an algebra isomorphism between the center of deformed category <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper O"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">O</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {O}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the equivariant cohomology of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="zeta"> <mml:semantics> <mml:mi>ζ</mml:mi> <mml:annotation encoding="application/x-tex">\zeta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-fixed locus on the affine Grassmannian attached to the Langlands dual group.