**Speaker:** Olivier Dudas (Paris)

**Title:** *Decomposition Numbers and Deligne-Lusztig Theory*

**Abstract:**
(Joint with Gunter Malle)

Two different problems arise when one tries to compute decomposition numbers
for groups of Lie type:

(1) Determine cuspidal modules and their projective cover

(2) Compute the endomorphism algebra of the induced module

In this talk I will explain some recent progress on step (1). The
construction of the missing projective modules is achieved using the
cohomology of Deligne-Lusztig varieties, yielding new decomposition
numbers for various unipotent blocks and various groups. The example of
unitary groups is very instructive and I will use it to illustrate some (yet
unexplained) patterns and conjectures on the decomposition numbers.