42 research outputs found

### Explicit Demazure character formula for negative dominant characters

In this paper, we prove that for any semisimple simply connected algebraic
group $G$, for any regular dominant character $\lambda$ of a maximal torus $T$
of $G$ and for any element $\tau$ in the Weyl group $W$, the character
$e^{\rho}\cdot char(H^{0}(X(\tau), \mathcal{L}_{\lambda-\rho}))$ is equal to
the sum $\sum_{w\leq \tau}char(H^{l(w)}(X(w),\mathcal{L}_{-\lambda}))^{*})$ of
the characters of dual of the top cohomology modules on the Schubert varieties
$X(w)$, $w$ running over all elements satisfying $w\leq \tau$. Using this
result, we give a basis of the intersection of the Kernels of the Demazure
operators $D_{\alpha}$ using the sums of the characters of
$H^{l(w)}(X(w),\mathcal{L}_{-\lambda})$, where the sum is taken over all
elements $w$ in the Weyl group $W$ of $G$.Comment: 11 page

### Syzygies of some GIT quotients

Let $X$ be flat scheme over $\mathbb{Z}$ such that its base change, $X_p$, to
$\bar{\mathbb{F}}_p$ is Frobenius split for all primes $p$. Let $G$ be a
reductive group scheme over $\mathbb{Z}$ acting on $X$. In this paper, we prove
a result on the $N_p$ property for line bundles on GIT quotients of
$X_{\mathbb{C}}$ for the action of $G_{\mathbb{C}}$. We apply our result to the
special cases of (1) an action of a finite group on the projective space and
(2) the action of a maximal torus on the flag variety of type $A_n$.Comment: 11 pages; improved bounds in main results; new references adde

### On the automorphism of a smooth Schubert variety

Let $G$ be a simple algebraic group of adjoint type over the field
$\mathbb{C}$ of complex numbers. Let $B$ be a Borel subgroup of $G$ containing
a maximal torus $T$ of $G$. Let $w$ be an element of the Weyl group $W$ and let
$X(w)$ be the Schubert variety in $G/B$ corresponding to $w$. Let $\alpha_{0}$
denote the highest root of $G$ with respect to $T$ and $B.$ Let $P$ be the
stabiliser of $X(w)$ in $G.$ In this paper, we prove that if $G$ is simply
laced and $X(w)$ is smooth, then the connected component of the automorphism
group of $X(w)$ containing the identity automorphism equals $P$ if and only if
$w^{-1}(\alpha_{0})$ is a negative root ( see Theorem 4.2 ). We prove a partial
result in the non simply laced case ( see Theorem 6.6 ).Comment: 23 Pages. Sections are divided in to

### An analogue of Bott's theorem for Schubert varieties-related to torus semistable points

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 $X(\tau)$ for which all the higher cohomologies $H^{i}(X(\tau), E)$
vanish for the restriction $E$ of the tangent bundle of $G/B$ to X(\tau)$. We
further show that the global sections$H^{0}(X(\tau), E)$is the adjoint
representation of$G$when$G$ is simply laced.Comment: 36 pages. arXiv admin note: text overlap with arXiv:1211.354

### On equivariant principal bundles over wonderful compactifications

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

### Equivariant vector bundles on complete symmetric varieties of minimal rank

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

### Automorphisms of $\overline{T}$

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 $\overline T$.Comment: Final versio

### On a smooth compactification of PSL(n, C)/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

### Torus quotients of Richardson varieties in the Grassmannian

We study the GIT quotient of the minimal Schubert variety in the Grassmannian
admitting semistable points for the action of maximal torus $T$, with respect
to the $T$-linearized line bundle ${\cal L}(n \omega_r)$ and show that this is
smooth when $gcd(r,n)=1$. When $n=7$ and $r=3$ we study the GIT quotients of
all Richardson varieties in the minimal Schubert variety. This builds on
previous work by Kumar \cite{kumar2008descent}, Kannan and Sardar
\cite{kannan2009torusA}, Kannan and Pattanayak \cite{kannan2009torusB}, and
recent work of Kannan et al \cite{kannan2018torus}. It is known that the GIT
quotient of $G_{2,n}$ is projectively normal. We give a different combinatorial
proof

### Rigidity of Bott-Samelson-Demazure-Hansen variety for $PSO(2n+1, \mathbb{C})$

Let $G=PSO(2n+1, \mathbb{C}) (n \ge 3)$ and $B$ be the Borel subgroup of $G$
containing maximal torus $T$ of $G.$ Let $w$ be an element of Weyl group $W$
and $X(w)$ be the Schubert variety in the flag variety $G/B$ corresponding to
$w.$ Let $Z(w, \underline{i})$ be the Bott-Samelson-Demazure-Hansen variety
(the desingularization of $X(w)$) corresponding to a reduced expression
$\underline{i}$ of $w.$ In this article, we study the cohomology modules of the
tangent bundle on $Z(w_{0}, \underline{i}),$ where $w_{0}$ is the longest
element of the Weyl group $W.$ We describe all the reduced expressions of
$w_{0}$ in terms of a Coxeter element such that all the higher cohomology
modules of the tangent bundle on $Z(w_{0}, \underline{i})$ vanish (see Theorem
\ref{theorem 8.1}).Comment: 35 pages. arXiv admin note: substantial text overlap with
arXiv:1610.00812, arXiv:1908.0559