arXiv subjects
Publications and source records attributed to S. Senthamarai Kannan.
Let $\overline G$ be the wonderful compactification of a simple affine algebraic group $G$ defined over $\mathbb C$ such that its center is trivial and $G\not= {\rm PSL}(2,\mathbb{C})$. Take a maximal torus $T \subset G$, and denote by $\overline T$ its closure in $\overline G$. We prove that $T$ coincides with the connected component, containing the identity element, of the group of automorphisms of the variety $\overline T$.
Let $T$ be a maximal torus of ${\rm PSL}(n, \mathbb C)$. For $n\,\geq\, 4$, we construct a smooth compactification of ${\rm PSL}(n, \mathbb C)/T$ as a geometric invariant theoretic quotient of the wonderful compactification $\overline{{\rm PSL}(n, \mathbb C)}$ for a suitable choice of $T$--linearized ample line bundle on $\overline{{\rm PSL}(n, \mathbb C)}$. We also prove that the connected component, containing the identity element, of the automorphism group of this compactification of ${\rm PSL}(n, \mathbb C)/T$ is ${\rm PSL}(n, \mathbb C)$ itself.
Let $X$ be the wonderful compactification of a complex symmetric space $G/H$ of minimal rank. For a point $x\,\in\, G$, denote by $Z$ be the closure of $BxH/H$ in $X$, where $B$ is a Borel subgroup of $G$. The universal cover of $G$ is denoted by $\widetilde{G}$. Given a $\widetilde{G}$ equivariant vector bundle $E$ on $X,$ we prove that $E$ is nef (respectively, ample) if and only if its restriction to $Z$ is nef (respectively, ample). Similarly, $E$ is trivial if and only if its restriction to $Z$ is so.
Let $G$ be a simple algebraic group of adjoint type over $\mathbb C$, and let $M$ be the wonderful compactification of a symmetric space $G/H$. Take a $\widetilde G$--equivariant principal $R$--bundle $E$ on $M$, where $R$ is a complex reductive algebraic group and $\widetilde G$ is the universal cover of $G$. If the action of the isotropy group $\widetilde H$ on the fiber of $E$ at the identity coset is irreducible, then we prove that $E$ is polystable with respect to any polarization on $M$. Further, for wonderful compactification of the quotient of $\text{PSL}(n,{\mathbb C})$, $n\,\neq\, 4$ (respectively, $\text{PSL}(2n,{\mathbb C})$, $n \geq 2$) by the normalizer of the projective orthogonal group (respectively, the projective symplectic group), we prove that the tangent bundle is stable with respect to any polarization on the wonderful compactification.
Let $G$ be a complex linear algebraic group which is simple of adjoint type. Let $\overline G$ be the wonderful compactification of $G$. We prove that the tangent bundle of $\overline G$ is stable with respect to every polarization on $\overline G$.
Let $G$ be a simple, simply connected algebraic group over the field of complex numbers. We give a necessary and a sufficient condition for a Schubert variety $X(τ)$ for which all the higher cohomologies $H^{i}(X(τ), E)$ vanish for the restriction $E$ of the tangent bundle of $G/B$ to X(τ)$. We further show that the global sections $H^{0}(X(τ), E)$ is the adjoint representation of $G$ when $G$ is simply laced.
In this article, we prove that for any indecomposable dominant character of a maximal torus $T$ of a simple adjoint group $G$ such that there is a Coxeter element $w \in W$ for which $X(w)^{ss}_T(\mathcal L_χ) \neq \emptyset$. If further, for any dominant character $χ_1$ of $T$ such that $χ_1\lneqq χ$ with respect to the dominant ordering, $dim(H^0(G/B, \mathcal L_{χ_1})^T) < dim (H^0(G/B, \mathcal L_χ)^T)$, then the graded algebra $\oplus_{d \in \mathbb Z_{\geq 0}}H^0(G/B, \mathcal L_χ^{\otimes d})^T$ is a polynomial ring in $r$ variables where $r\geq 2$.
In this paper, we prove that for any semisimple simply connected algebraic group $G$, for any regular dominant character $λ$ of a maximal torus $T$ of $G$ and for any element $τ$ in the Weyl group $W$, the character $e^ρ\cdot char(H^{0}(X(τ), \mathcal{L}_{λ-ρ}))$ is equal to the sum $\sum_{w\leq τ}char(H^{l(w)}(X(w),\mathcal{L}_{-λ}))^{*})$ of the characters of dual of the top cohomology modules on the Schubert varieties $X(w)$, $w$ running over all elements satisfying $w\leq τ$. Using this result, we give a basis of the intersection of the Kernels of the Demazure operators $D_α$ using the sums of the characters of $H^{l(w)}(X(w),\mathcal{L}_{-λ})$, where the sum is taken over all elements $w$ in the Weyl group $W$ of $G$.