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Let $G$ be a simple algebraic group over the complex field $\mathbb C$, $P$ a parabolic subgroup containing $B$ its Borel subgroup, $P'$ its derived group and $\mathfrak m$ the Lie algebra of its nilradical. The nilfibre $\mathscr N$ for this action is the zero locus of the augmentation $\mathscr I_+$ of the semi-invariant algebra $\mathscr I=\mathbb C[\mathfrak m]^{P'}$. For $G=SL(n)$ practically nothing was known previously. The only result of comparable, but lesser complexity, is for $\mathscr V:=\mathscr O\cap \mathfrak n$, with $\mathscr O$ a nilptent $G$ orbit and $\mathfrak n$ the set of strictly upper triangular matrices. Then $\mathscr V$ is equidimensional with components known as orbital varieties, parameterised by standard tableaux whose shape is dictated by $\mathscr O$. Here the components of $\mathscr N$ are studied for $G=SL(n)$. They increase exponentially in $n$ with no a priori discernable pattern. For each choice of numerical data $\mathcal C$, a semi-standard tableau $\mathscr T^\mathcal C$, is constructed from $\mathscr T$. A \textit{delicate and tightly interlocking} analysis constructs a set of excluded root vectors from $\mathfrak m$ such that the complementary space $\mathfrak u^\mathcal C$ has the following properties. First it is a subalgebra of $\mathfrak m$. Secondly $\mathscr C:=\overline{B.\mathfrak u^\mathcal C}$ lies in $\mathscr N$ to which, thirdly, a Weierstrass section can be associated. Fourthly $\dim \mathscr C = dim \mathfrak m-\textbf{g}$, where \textbf{g} is the number of generators of the polynomial algebra $\mathscr I$. Fifthly the Weierstrass section, is shown to imply that $\mathscr C$ an irreducible component of $\mathscr N$, yet $\mathscr C$ is \textit{ only sometimes} an orbital variety closure. The resulting Component Map $\mathscr T^\mathcal C\mapsto\mathscr C$ is shown to be injective. Evidence for its surjectivity is given.

A "Composition map" is constructed, leaning heavily on earlier work [Y. Fittouhi and A. Joseph, Parabolic adjoint action, Weierstrass sections and components of the nilfibre in type $A$, Indag Math. and Y. Fittouhi and A. Joseph, The canonical component of the nilfibre for parabolic adjoint action, Weierstrass sections in type $A$, preprint, Weizmann, 2021]. It defines a composition tableau which recovers the "canonical" Weierstrass section $e+V$ described in the first paper above. Moreover \textit{without reference to this earlier work}, it is then shown that $e+V$ is indeed a Weierstrass section. This results in a huge simplification. Moreover one may read off from the composition tableau the "VS quadruplets'' of the second of the above papers, thereby describing the "canonical component" of the nil-fibre in which $e$ lies but does not of itself determine.

This work is a continuation of [Fittouhi and Joseph, Parabolic adjoint action, Weierstrass Sections and components of the nilfibre in type $A$]. Let $P$ be a parabolic subgroup of an irreducible simple algebraic group $G$, $P'$ its derived group and $\mathfrak m$ be the nilradical to its Lie algebra. A theorem of Richardson implies that the subalgebra $\mathbb C[\mathfrak m]^{P'}$, spanned by the $P$ semi-invariants in $\mathbb C[\mathfrak m]$, is polynomial. A linear subvariety $e+V$ of $\mathfrak m$ is is called a Weierstrass section for the action of $P'$ on $\mathfrak m$, if the restriction map induces an isomorphism of $\mathbb C[\mathfrak m]^{P'}$ onto $\mathbb C[e+V]$. Thus a Weierstrass section can exist only if the latter is polynomial, but even when this holds its existence is far from assured. The existence of a Weierstrass section $e+V$ in $\mathfrak m$ was established by a general combinatorial construction. Notably $e \in \mathscr N$ and is a sum of root vectors with linearly independent roots. The Weierstraass section $e+V$ looks very different for different choices of parabolics but nevertheless has a uniform construction and exists in all cases. It is called the "canonical Weierstrass section". It was announced in [Fittouhi and Joseph, loc. cit.] that one may augment $e$ to an element $e_{VS}$ by adjoining root vectors. Then the linear span $E_{VS}$ of these root vectors lies in $\mathscr N^e$ and its closure is just $\mathscr N^e$. Yet this result shows that $\mathscr N^e$ need not admit a dense $P$ orbit. However this theorem was only verified in the special case needed to obtain the example showing that $\mathscr N^e$ may fail to admit a dense $P$ orbit. Here a general proof is given. Finally a map from compositions to the set of distinct non-negative integers is defined. Its image is shown to determine the canonical Weierstrass section.

This work is a continuation of [Y. Fittouhi and A. Joseph, Weierstrass Sections for Parabolic adjoint action in type $A$]. Let $G$ be an irreducible simple algebraic group and $B$ a Borel subgroup of $G$. Let $\mathfrak n$ be the Lie algebra of the nilradical of $B$. Consider an irreducible subgroup $P$ of $G$ containing $B$. Let $P'$ be the derived group of $P$. Let $\mathfrak m$ be the Lie algebra of the nilradical of $P$. A theorem of Richardson asserts that the algebra $\mathbb C[\mathfrak m]^{P'}$ of $P$ semi-invariants is multiplicity-free. A linear subvariety $e+V$ such that the restriction map induces an isomorphism of $\mathbb C[\mathfrak m]^{P'}$ onto $\mathbb C[e+V]$ is called a Weierstrass section for the action of $P'$ on $\mathfrak m$. Here in type $A$ such a section is constructed, but in better form than that given in Sect. 4, loc cit. Yet the main difference is a complete change of emphasis from the construction of a Weierstrass section, to its application. Let $\mathscr N$ be the nilfibre relative to this action. From the construction of a Weierstrass section $e+V$, it is shown that $e \in \mathscr N$. Then $P.e$ is contained in a unique irreducible component $\mathscr C$ of $\mathscr N$. The structure of $e+V$ is used to give a rather explicit description of $\mathscr C$ as a $B$ saturation set, that is of the form $\overline{B.\mathfrak u}$, where $\mathfrak u$ is a subalgebra of $\mathfrak n$ . This algebra is not necessarily complemented by a subalgebra in $\mathfrak n$ and so $\overline{B.\mathfrak u}$ is not necessarily an orbital variety closure (hence Lagrangian) but it can be. It is shown that $\mathscr C$ need not contain a dense $P$ orbit and this by a purely theoretical analysis. This occurs for an appropriate parabolic in $A_{10}$ and is possibly the simplest example. In this particular case $\mathscr C$ is not an orbital variety closure.

The notion of "Weierstrass Section", comes from Weierstrass canonical form for elliptic curves. In celebrated work [B. Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327-404] constructed such a section for the action of a semisimple Lie algebra on its dual using a principal s-triple. Actually it is enough to have an "adapted pair" and indeed the construction in [A. Joseph and D. Shafrir, Polynomiality of invariants, unimodularity and adapted pairs, Transform. Groups 15 (2010), no. 4, 851-882] works rather well for the coadjoint action of an algebraic, but not necessarily reductive Lie algebra. In the present work a Weierstrass section is constructed for the adjoint action of the derived algebra of a parabolic subalgebra on its nilradical in type $A$. The starting point is Richardson's theorem which implies the polynomiality of the invariant sub-algebra. Here adapted pairs seldom exist. A new construction is developed and this is mainly combinatorial based on joining boxes in the Young tableau associated to the "Richardson component". Indications are given for extending this construction in other types. The construction has relations to quivers [T. Brustle, L. Hille, Lutz, C.M. Ringel and G. Rohrle, The $\delta$-filtered modules without selfextensions for the Auslander algebra of $k[T]/Tn$. Algebr. Represent. Theory 2 (1999), no. 3, 295-312] and to hypersurface orbital varieties [A. Joseph and A. Melnikov, Quantization of hypersurface orbital varieties in $sl(n)$. The orbit method in geometry and physics (Marseille, 2000), 165-196, Progr. Math., 213].

Many finite dimensional integrable systems qre expressed with the help of the Lax equation which highlights a spectral parameter and therefore a spectral curve. These spectral curves are the starting point of an algebro-geometric investigation. The algebraic aspect of this investigation will be expressed using the Jacobians of smooth spectral curves and generalized jacobians of singular spectral curves and in the other hand the geometrical aspect will be attributed to the vector fields which are defined by the Lax equation. In this article will be dedicated only to singular hyperelliptic spectral curves that are associated to Mumford systems. We will study the complementarity of these two mathematical approaches to describe the same object and revealing their characteristics., Comment: in French

An integrable system is a dynamic system characterized by the existence of constants of motion and the existence of algebraic invariants, having an origin in algebraic geometry. In the 1970s, Mumford introduced a new completely integrable system defined on a smooth hyperelliptic curve. In the 2000s, Vanhaecke completed the description of the Munford integrable system by defining a Poisson structure on the phase space of the Mumford system. In this article we will study the singular Mumford system. The starting point is to determine when and why the Mumford system is singular. For this we will do an in-depth study to understand what happens to singularities, using the concept of stratification. We will define two stratifications of the phase space, one algebraic stratification and the other geometric stratification. We will prove that these stratifications are identical, and they will allow us to define a finer stratification on each fiber of the Mumford system. We will conclude this article with the following surprising result: each stratum of a fiber is a partition of equidimensional quasi-affine submanifolds., Comment: in French