Let \(G\) be a connected reductive linear algebraic group over an algebraically closed field \(\mathbb{F}\) of positive characteristic. Let \(k\) be a finite field of characteristic \(\ell\not=\) char\((\mathbb{F})\), or a finite extension of \(\mathbb{Q}_\ell\), or the ring of integers of such an extension. Put \(K= \mathbb{F}((z))\), \(O= \mathbb{F}[z]\). One has a group subscheme \(G_O\) of the group Ind-scheme \(G_K\), and the affine Grassmanian \(\mathrm{Gr}=G_K/G_O\). The Satake category is the category Perv\(_{G_{O}}\) (Gr, \(k\)) of \(G_O\)-equivariant (étale) \(k\)-perverse sheaves on Gr. This category is a fundamental object in Geometric Representation Theory. It appears in the geometric Satake equivalence. This equivalence claims that the Satake category admits a natural convolution product (-)\(\star^{G_O}\)(-) that endows it with a monoidal structure. Moreover, there exists an equivalence of monoidal categories \(S:(\mathrm{Perv}_{G_O} (\mathrm{Gr},k),\star^{G_O})\) \(\to\) \((\mathrm{Rep}(G_k^\vee),\otimes)\). The right side here is the category of algebraic representations of the split reductive \(k\)-group scheme that is Langlands dual to \(G\) on finitely generated \(k\)-modules; see [I. Mirković and K. Vilonen, Ann. Math. (2) 166, No. 1, 95–143 (2007; Zbl 1138.22013)] for the original proof of this equivalence in full generality, and [R. Bezrukavnikov and S. Riche, “A topological approach to Soergel theory”, Preprint, arXiv:1807.07614] for a more detailed exposition. (In these references, what is explicitly treated is the analogous equivalence for a complex group \(G\), in which case \(k\) can be any Noetherian commutative ring of finite global dimension. The étale setting is similar; see [I. Mirković and K. Vilonen, Ann. Math. (2) 166, No. 1, 95–143 (2007; Zbl 1138.22013), §14] and [R. Bezrukavnikov and S. Riche, “A topological approach to Soergel theory”, Preprint, arXiv:1807.07614, §1.1.4] for a few comments.)
This category already has another avatar since (as proved by Mirković-Vilonen) the forgetful functor \(\mathrm{Perv}_{G_O}(\mathrm{Gr},k)\) \(\to\) \(\mathrm{Perv}_{(G_O)}(\mathrm{Gr},k)\) from the Satake category to the category of perverse sheaves on Gr that are constructible with respect to the stratification by \(G_O\)-orbits, is an equivalence of categories.
The first main result of the present paper provides a third incarnation of the Satake category, as a category \(\mathrm{Perv}_{IW}(\mathrm{Gr},k)\) of Iwahori-Whittaker perverse sheaves on Gr. More precisely the paper shows that a natural functor \(\mathrm{Perv}_{G_O}(\mathrm{Gr},k)\) \(\to\) \(\mathrm{Perv}_{IW}(\mathrm{Gr},k)\) is an equivalence of categories (see Theorem 3.9).
This result is useful because computations in \(\mathrm{Perv}_{IW}(\mathrm{Gr},k)\) are much easier than in the categories \(\mathrm{Perv}_{G_O} (\mathrm{Gr},k)\) or \(\mathrm{Perv}_{(G_O)}(\mathrm{Gr},k)\), in particular due to the facts that standard and costandard objects have more explicit descriptions and that the “realization functor” \(D^b \mathrm{Perv}_{IW}(\mathrm{Gr},k) \to D_{IW}^b(\mathrm{Gr},k)\) is an equivalence of triangulated categories.
In the analogous setting of Whittaker \(D\)-modules over a field of characteristic 0, this statement is already known.
This interesting and well-written paper continues with a discussion of the relation with the Finkelberg-Mirković conjecture, with results of Lusztig, and those of Frenkel-Gaitsgory-Kazhdan-Vilonen, as well as applications to tilting objects.