arXiv subjects
Publications and source records attributed to Boming Jia.
Let $\mathcal O$ be a nonzero nilpotent orbit in a complex simple Lie algebra $\mathfrak g$. For $\mathfrak g$ of classical type, we classify the orbit closures $\overline{\mathcal O}$ that are isomorphic to $(T^*X)^{\mathrm{aff}}$ for a smooth quasi-affine variety $X$. In types $G_2$, $F_4$, and $E_8$, we show that no nonzero nilpotent orbit closure admits such a cotangent model.
Let $\mathfrak g$ be a complex simple Lie algebra of type $G_2$, $F_4$, or $E_8$. We prove that the closure $\overline{\mathcal O}_{\min}(\mathfrak g)$ of its minimal nilpotent orbit is not isomorphic to $(T^*X)^{\mathrm{aff}}$ for any smooth quasi-affine variety $X$. We do not assume that the isomorphism is equivariant or Poisson. We also prove the same statement in types $A_1$ and $A_2$.
Let $\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)$ be the collection of elements of $\mathfrak{sl}_{n+1}(\mathbb C)$ with rank less than or equal to $1$ and with all diagonal entries equal to zero. We show that the coordinate ring $\mathbb C[\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)]$ of the scheme-theoretic intersection $\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)$ has a flat degeneration to the ring of $(\mathbb C^{\times})^n$-equivariant cohomology of the projective toric variety associated with the fan of compatible subsets of almost positive roots of type $C_n$. Then we compute the Hilbert series of $\mathbb C[\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)]$ and prove that $\overline{\mathcal{O}}_\textrm{min} \cap (\mathfrak n^+ \oplus \mathfrak n^-)$ is reduced and Gorenstein. Moreover, our proof method allows us to prove that the scheme-theoretic intersection $\overline{\mathcal{O}}_\textrm{min} \cap \mathfrak n^+$, of which the irreducible components are known as the ``orbital varieties'', is reduced and Cohen-Macaulay.
We first show the closure of the minimal nilpotent adjoint orbit Omin^{D_n} in so_{2n} is isomorphic to the affinization of T^*(SL_{n-1}/[P,P]) where P is the parabolic subgroup P_{(1,1,n-3)} of SL_{n-1}(C). Then we prove that the closure of the minimal nilpotent adjoint orbit Omin^{E_6} of the complex simple Lie algebra E_6 is isomorphic to the affinization of T^*(SL_4/P^u) where P^u is the unipotent radical of the parabolic subgroup P_{(2,2)} of SL_4(\C). In the end we will formulate a similar result for type E_7.
We compute the Hilbert series of the coordinate ring of some highest weight varieties. We also explain why Narayana numbers (and their generalizations) appear naturally in the numerator of the Hilbert series of the homogeneous coordinate ring of the Grassmannian $Gr(d,n+d+1)$ and of the minimal nilpotent adjoint orbit in $\mathfrak{sl}_\mathrm{n+1}(\mathbb{C})$.
We show that the affine closure of T^*(SL_n/U) has symplectic singularities, in the sense of Beauville. In the special case n=3, we show that the affine closure of T^*(SL_3/U) is isomorphic to the closure of the minimal nilpotent adjoint orbit in so(8,C). Moreover, the quasi-classical Gelfand-Graev action of the Weyl group W on the affine closure of T^*(SL_3/U) can be identified with the restriction to the closure of the minimal nilpotent adjoint orbit of the triality action on so(8,C).