On endomorphism algebras of Gelfand-Graev representations

Abstract

For a connected reductive group G G defined over F q \mathbb {F}_q and equipped with the induced Frobenius endomorphism F F , we study the relation among the following three Z \mathbb {Z} -algebras: (i) the Z \mathbb {Z} -model E G \mathsf {E}_G of endomorphism algebras of Gelfand-Graev representations of G F G^F ; (ii) the Grothendieck group K G ∗ \mathsf {K}_{G^\ast } of the category of representations of G ∗ F ∗ G^{\ast F^\ast } over F q ¯ \overline {\mathbb {F}_q} (Deligne-Lusztig dual side); (iii) the ring B G ∨ \mathsf {B}_{G^\vee } of the scheme ( T ∨ / / W ) F ∨ (T^\vee /\!\!/ W)^{F^\vee } over Z \mathbb {Z} (Langlands dual side). The comparison between (i) and (iii) is motivated by recent advances in the local Langlands program.