Some stratifications on algebraic varieties

In this thesis we study stratifications on algebraic varieties in two different contexts. The first one concerns the relation between the conjugacy classes in a reductive algebraic group and the irreducible representations of its associated Weyl group, with a focus on non connected algebraic groups. Let $G$ be a non-connected reductive algebraic group over an algebraically closed field and let $D$ be a connected component of $G$. The connected component of $G$ containing the identity is denoted by $G^\circ$, and $W$ is its Weyl group. In a work of 2020, G. Lusztig defines a map from $D$ to the set of isomorphism classes of irreducible representations of a subgroup of $W$ depending on $D$. In this thesis we study the geometry of the fibers of this map, namely the Lusztig strata. In particular we prove that they are locally closed and we describe their irreducible components. In order to do that we study the stratification of $G$ into Jordan classes. Another approach to study the connection between conjugacy classes in $G$ and irreducible representations of $W$ is established by G. Lusztig through a map $\Psi$ from the set of unipotent conjugacy classes in $D$ to the set of twisted conjugacy classes of $W$. For $G^\circ$ simple, we give a combinatorial description of the restriction of $\Psi$ to the set of unipotent spherical conjugacy classes in $D$ and we give a classification of spherical unipotent conjugacy classes of $G$. The second part of the thesis is devoted to the application of the theory of Seshadri stratifications to matrix Schubert varieties, namely varieties of matrices defined by conditions on the rank of some their submatrices. These varieties were first introduced by Fulton in 1992 and they are useful in the study of combinatorics of determinantal ideals and Schubert polynomials. The theory of Seshadri stratifications has been introduced in a recent work of R. Chirivì, X. Fang and P. Littelmann. One of the aim of this theory is to provide a geometric setup for standard monomial theory. A Seshadri stratification of an embedded projective variety $X$ is the datum of a suitable collection of subvarieties $X_\tau$ that are smooth in codimension one, and a collection of suitable homogeneous functions $f_\tau$ on $X$ indexed by the same finite set. In this thesis we provide a Seshadri stratification for the matrix Schubert varieties.