An Iwahori-Whittaker model for the Satake category

Abstract

In this paper we prove, for G a connected reductive algebraic group satisfying a technical assumption, that the Satake category of G (with coefficients in a finite field, a finite extension of Q ℓ , or the ring of integers of such a field) can be described via Iwahori-Whittaker perverse sheaves on the affine Grassmannian. As an application, we confirm a conjecture of Juteau-Mautner-Williamson describing the tilting objects in the Satake category.

A geometric Steinberg formula

“…In case (2), the functor Φ tς w•,S is the main object of study of [BGMRR]; in this setting we have C tς w•,S = Perv (I S u ,XS) (Gr, k) by Lemma 2.5, t ς w • is minimal for the Bruhat order because it has minimal length in S W S ext (by the same statement and Lemma 2.7), and the main result of [BGMRR] states that this functor is an equivalence of categories. (Note that revisiting the arguments in [BGMRR,§4.3] involving parity complexes, one can prove directly that Φ tς w•,S is essentially surjective once we know that it is fully faithful.) Finally, in case (3), parity considerations imply that the category C y,A is semisimple, which of course implies the statement.…”

“…This result is a geometric version of a theorem on tensor products of modules with good filtrations (for reductive algebraic groups over fields of positive characteristic) first due to Mathieu [Ma] in full generality. It can be deduced from this result using the geometric Satake equivalence; a direct geometric proof can also be obtained from [BGMRR,Theorem 4.16], see [JMW] for some details. 4.3.…”

“…It is known that this bifunctor is again t-exact, in the sense that for F in Perv (I A u ,XA) (Gr, k) and G in Perv L + G (Gr, k) the convolution F ⋆ L + G G is perverse; see [BGMRR,Lemma 2.3] for details and references.…”

“…χ + , and the second one is the "residue" morphism sending a Laurent series to the coefficient of z −1 (see e.g. [BGMRR,§3.4]). Then for any µ ∈ Y there exist unique morphisms…”

“…Recall that for any β ∈ R and n ∈ Z we have a "root subgroup" U β,n ⊂ LG defined as in [BGMRR,Proof of Lemma 3.10]. If we set…”