Remark on a Theorem of Puig

Corollaire 3.6 of Puig's paper "Algèbres de source de certains blocs des groupes de Chevalley" [Pui4] gives a description of the source algebras of some blocks of finite groups having subgroups H and I satisfying certain properties.

Let G=G^F be the group of rational points of a connected reductive group G defined over the field F_q of q elements, and let B be a Borel subgroup of the finite reductive group G containing a quasi-split maximal torus T and unipotent radical U.

If p is a prime dividing q-1 but not dividing the order of the Weyl group W=N_G(T)/T of G, then T contains a Sylow p-subgroup P of G, and in characteristic p the conditions of Puig's result are satisfied in the following cases.

The principal block of G. In this case we take H=T and I=U.
The principal block of N_G(P)=N_G(T). In this case we take H=T as before, and I={1}.

(The technical condition 1.3.1 of Puig's paper, which imposes conditions on elements of G-I.N_G(H).I, is vacuously satisfied in both of these cases, since G=U.N_G(T).U by the Bruhat decomposition).

Moreover, the description of the source algebra is identical in the two cases, so the principal blocks of G and N_G(P) have isomorphic source algebras, and so in particular they are Morita equivalent.