SL2 Seminar

The decomposition of the reduction of a Deligne--Lusztig character as a sum of Weyl modules is given by Jantzen's formula. In the case of $SL_2(F_q)$ the formula can be proved by a direct computation, and as a result we will see that the entries in the decomposition matrix are all 0 or 1. This multiplicity-one property (or rather, its analogue for $GL_2(F_q))$ has applications in number theory through the formulation of Breuil's lattice conjecture, which gives a connection with Emerton's talk in this study group.