Budmiger, Jonas. Deformation of orbits in minimal sheets. 2010, PhD Thesis, University of Basel, Faculty of Science.

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Official URL: http://edoc.unibas.ch/diss/DissB_9037
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Abstract
Keywords: Algebraic group theory, algebraic geometry, classical invariant theory
The main object of study of this work are orbits in socalled minimal sheets in irreducible representations of semisimple groups. Let $G$ be a semisimple group. The notion of sheets goes back to Dixmier: Given a $G$module $V$, the union of all orbits in $V$ of a fixed dimension is a locally closed subset. Its irreducible components are called sheets of $V$. We call a sheet minimal if it contains an orbit in $V$ of minimal strictly positive dimension among all orbits in $V$.
In Chapter I, some notation is fixed and some basic results are proved.
In Chapter II, we describe minimal sheets in simple $G$modules, and study $G$stable deformations of orbits in minimal sheets by means of an invariant Hilbert scheme. Invariant Hilbert Schemes have been introduced by Alexeev and Brion in 2005. These are quasiprojective schemes representing functors of families of $G$schemes with prescribed Hilbert function. The discussion in Chapter II is closely related to the work of Jansou in the following way: Choose once and for all a highest weight vector $v_\lambda \in V(\lambda)$ for each dominant weight $\lambda \in \Lambda^+$, and let $X_\lambda = \overline{G v_\lambda} \subset V(\lambda)$ be the closure of the orbit $G v_\lambda$ of $v_\lambda$ in $V(\lambda)$. In his thesis Jansou investigates $G$stable deformations of $X_\lambda$ in $V(\lambda)$. If $h_\lambda$ denotes the Hilbert function of $X_\lambda$, then Jansou proves that the invariant Hilbert scheme $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine space of dimension 0 or 1, depending on $G$ and $\lambda$. Furthermore, he gives a complete list of all pairs $(G,\lambda)$ such that $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine line. In the sequel, we call these weights Jansouweights.
The orbit $Gv_\lambda$ is of minimal strictly positive dimension among all $G$orbits in $V(\lambda)$. There exist other orbit of the same dimension as $Gv_\lambda$ in $V(\lambda)$ if and only if $\lambda$ is an integral multiple of a Jansouweight. Here, we start with a general orbit $X$ of minimal strictly positive dimension in a fixed simple $G$module $V(\lambda)$, and we study $G$stable deformations of $X$. In particular, we conjecture that the invariant Hilbert scheme parametrizing the $G$stable deformations of $X$ in the closure of the sheet of $X$ is an affine space of dimension either 0 or 1. This will stand in contrast to the fact that the invariant Hilbert scheme parametrizing the $G$stable deformations of $X$ in $V(\lambda)$ can look much more complicated. This is the content of Chapter III, in which we will focus on the group $\SL_2$, and compute some corresponding invariant Hilbert schemes. In particular, we study deformations of orbits of the form $SL_2 \cdot x^{d/2}y^{d/2}$ in the space $k[x,y]_d = V(d)$ of binary forms of degree $d$. It turns out that easiest accessible case is when $d$ is a multiple of 4, and even in this case the corresponding invariant Hilbert scheme can become very complicated. This reflects the principle that even in `simple' cases for invariant Hilbert schemes all possible sort of `bad' things (different irreducible components, nonreduced points, singularities) occur. (This `bad' behavior is also encountered in the case of the classical Grothendieck Hilbert scheme parametrizing closed subschemes of projective space with a given Hilbert polynomial.) In Chapter III Classical Invariant Theory is often used, and some computations are computerbased.
Finally, in Chapter IV we turn our attention to not necessarily simple modules. In the multiplicityfree case important work has been done by Bravi and CupitFoutou. We translate some of their results to the case of not necessarily multiplicityfree modules. This corrects a result by Alexeev and Brion. Chapter IV is independent from the preceding chapters.
The main object of study of this work are orbits in socalled minimal sheets in irreducible representations of semisimple groups. Let $G$ be a semisimple group. The notion of sheets goes back to Dixmier: Given a $G$module $V$, the union of all orbits in $V$ of a fixed dimension is a locally closed subset. Its irreducible components are called sheets of $V$. We call a sheet minimal if it contains an orbit in $V$ of minimal strictly positive dimension among all orbits in $V$.
In Chapter I, some notation is fixed and some basic results are proved.
In Chapter II, we describe minimal sheets in simple $G$modules, and study $G$stable deformations of orbits in minimal sheets by means of an invariant Hilbert scheme. Invariant Hilbert Schemes have been introduced by Alexeev and Brion in 2005. These are quasiprojective schemes representing functors of families of $G$schemes with prescribed Hilbert function. The discussion in Chapter II is closely related to the work of Jansou in the following way: Choose once and for all a highest weight vector $v_\lambda \in V(\lambda)$ for each dominant weight $\lambda \in \Lambda^+$, and let $X_\lambda = \overline{G v_\lambda} \subset V(\lambda)$ be the closure of the orbit $G v_\lambda$ of $v_\lambda$ in $V(\lambda)$. In his thesis Jansou investigates $G$stable deformations of $X_\lambda$ in $V(\lambda)$. If $h_\lambda$ denotes the Hilbert function of $X_\lambda$, then Jansou proves that the invariant Hilbert scheme $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine space of dimension 0 or 1, depending on $G$ and $\lambda$. Furthermore, he gives a complete list of all pairs $(G,\lambda)$ such that $Hilb^G_{h_\lambda}(V(\lambda))$ is an affine line. In the sequel, we call these weights Jansouweights.
The orbit $Gv_\lambda$ is of minimal strictly positive dimension among all $G$orbits in $V(\lambda)$. There exist other orbit of the same dimension as $Gv_\lambda$ in $V(\lambda)$ if and only if $\lambda$ is an integral multiple of a Jansouweight. Here, we start with a general orbit $X$ of minimal strictly positive dimension in a fixed simple $G$module $V(\lambda)$, and we study $G$stable deformations of $X$. In particular, we conjecture that the invariant Hilbert scheme parametrizing the $G$stable deformations of $X$ in the closure of the sheet of $X$ is an affine space of dimension either 0 or 1. This will stand in contrast to the fact that the invariant Hilbert scheme parametrizing the $G$stable deformations of $X$ in $V(\lambda)$ can look much more complicated. This is the content of Chapter III, in which we will focus on the group $\SL_2$, and compute some corresponding invariant Hilbert schemes. In particular, we study deformations of orbits of the form $SL_2 \cdot x^{d/2}y^{d/2}$ in the space $k[x,y]_d = V(d)$ of binary forms of degree $d$. It turns out that easiest accessible case is when $d$ is a multiple of 4, and even in this case the corresponding invariant Hilbert scheme can become very complicated. This reflects the principle that even in `simple' cases for invariant Hilbert schemes all possible sort of `bad' things (different irreducible components, nonreduced points, singularities) occur. (This `bad' behavior is also encountered in the case of the classical Grothendieck Hilbert scheme parametrizing closed subschemes of projective space with a given Hilbert polynomial.) In Chapter III Classical Invariant Theory is often used, and some computations are computerbased.
Finally, in Chapter IV we turn our attention to not necessarily simple modules. In the multiplicityfree case important work has been done by Bravi and CupitFoutou. We translate some of their results to the case of not necessarily multiplicityfree modules. This corrects a result by Alexeev and Brion. Chapter IV is independent from the preceding chapters.
Advisors:  Kraft, Hanspeter 

Committee Members:  Brion, Michel 
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Algebra (Kraft) 
Item Type:  Thesis 
Thesis no:  9037 
Bibsysno:  Link to catalogue 
Number of Pages:  97 S. 
Language:  English 
Identification Number: 

Last Modified:  30 Jun 2016 10:41 
Deposited On:  07 May 2010 08:52 
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