Congruences of parahoric group schemes

Parameters of Hecke algebras for Bernstein components of p-adic groups

“…, and similarly for G. Since ζ m comes from the isomorphism X * (S) ∼ = X * ( S), it induces a linear bijection X * (A G ) → X * (A G), and hence an isomorphism from (3.9) to its counterpart for G. The A G -characters of representations in Rep(G, G x,m ) are precisely the characters of (3.9), and ζ G m pushes them forward along ζ m . (c) This can be shown as in [ABPS1, Theorem 5.1.c], using [Gan1,Gan2]. We only have to note that by [AMS1, Lemma 6.6], the groups M and M have the same parameter in Irr(Z(M * ∨ sc ) W F ), via (3.4).…”

“…For classical groups (symplectic, special orthogonal, unitary) we run into the problem that some representation theoretic results have been proven over p-adic fields but not (yet) over local function fields. We overcome this with the method of close fields [Kaz], which Ganapathy recently generalized to arbitrary connected reductive groups [Gan1,Gan2].…”

“…The following descriptions are based on [Gan1,Gan2]. They work when F and F are -close with large enough.…”

“…For every special vertex x ∈ A S , we get a special vertex x ∈ A S . For m ∈ Z ≥0 , there is a refined version of the Moy-Prasad group G x,m , see [Gan1]. It is a compact open normal subgroup of G x , the G-stabilizer of x.…”