We show that the Braden-MacPherson algorithm computes the stalks of parity sheaves. As a consequence we deduce that the Braden-MacPherson algorithm may be used to calculate the characters of tilting modules for algebraic groups and show that the $p$-smooth locus of a (Kac-Moody) Schubert variety coincides with the rationally smooth locus, if the underlying Bruhat graph satisfies a GKM-condition.

Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need...

Soient $F$ un corps $p$-adique, $G={\mathrm{GL}}_3\left(F\right)$. Pour $\chi$ un caractère de l’algèbre de Hecke sphérique de $G$ sur un anneau commutatif $k$, on introduit à la suite de Serre une représentation lisse $M_{\chi }$ de $G$ sur $k$ qui gouverne la théorie des représentations non ramifiées de $G$ sur $k$. Nous prouvons que $M_{\chi }$ est plat sur $k$ et que si $p$ est inversible dans $k$, alors pour tout sous-groupe compact ouvert suffisament petit $U$ de $G$, le module $M_{\chi }^U$ est libre de rang fini sur $k$. Ceci était conjecturé par Lazarus. Comme corollaire, nous obtenons...

Let $G$ be a finite group and $k$ a field of characteristic $p>0$. In this paper, we obtain several equivalent conditions to determine whether the principal block $B_0$ of a finite $p$-solvable group $G$ is $p$-radical, which means that $B_0$ has the property that $e_0{\left(k_P\right)}^G$ is semisimple as a $kG$-module, where $P$ is a Sylow $p$-subgroup of $G$, $k_P$ is the trivial $kP$-module, ${\left(k_P\right)}^G$ is the induced module, and $e_0$ is the block idempotent of $B_0$. We also give the complete classification of a finite $p$-solvable group $G$ which has not more than three...