Pseudo-parabolic Category over Quaternionic Projective Plane

Abstract

AbstractQuaternionic projective plane $\mathbb {H} P^{2}$ ℍ P 2 is the next simplest conjugacy class of a complex symplectic group with pseudo-Levi stabilizer subgroup after the sphere $\mathbb {S}^{4}\simeq \mathbb {H} P^{1}$ S 4 ≃ ℍ P 1 . Its quantization gives rise to a module category $\mathcal {O}_{t}\bigl (\mathbb {H} P^{2}\bigr )$ O t ( ℍ P 2 ) over finite-dimensional representations of the symplectic quantum group $U_{q}\bigl (\mathfrak {s}\mathfrak {p}(6)\bigr )$ U q ( s p ( 6 ) ) , a full subcategory in the BGG category $\mathcal {O}$ O . We prove that $\mathcal {O}_{t}\bigl (\mathbb {H} P^{2}\bigr )$ O t ( ℍ P 2 ) is semi-simple and equivalent to a category of quantized equivariant vector bundles on $\mathbb {H} P^{2}$ ℍ P 2 .