Andrey Mudrov: Pseudo-parabolic categories over HP_q^n

Abstract: With every point in a maximal torus T of a simple algebraic group G one can associate a full subcategory 𝒪(t) in the category 𝒪 of modules over the corresponding quantum group U_q(𝔤). It is closed under tensoring with finite-dimensional modules. If this category is semisimple, then it is equivalent to the category of equivariant quantum vector bundles over the conjugacy class Ad_G(t) passing through t. Then the modules from 𝒪(t) play a role of their “representations”. A case of special interest is when the centralizer subgroup of t is not Levi, e.g. when Ad_G(t) is an orthogonal or symplectic Grassmannian. Then the stabilizer subgroup is not quantized as a quantum subgroup in U_q(𝔤). In our talk we consider quaternionic projective space HPⁿ as a conjugacy class of the symplectic compact group G = Sp(n). We prove that 𝒪(t) is semi-simple and describe its simple objects, for one of two points t ∈ T ∩ HPⁿ.