Root data with group actions

Suppose $k$ is a field, $G$ is a connected reductive algebraic $k$-group, $T$ is a maximal $k$-torus in $G$, and $\Gamma$ is a finite group that acts on $(G,T)$. From the above, one obtains a root datum $\Psi$ on which $\text{Gal}(k)\times\Gamma$ acts. Provided that $\Gamma$ preserves a positive system in $\Psi$, not necessarily invariant under $\text{Gal}(k)$, we construct an inverse to this process. That is, given a root datum on which $\text{Gal}(k)\times\Gamma$ acts appropriately, we show how to construct a pair $(G,T)$, on which $\Gamma$ acts as above. Although the pair $(G,T)$ and the action of $\Gamma$ are canonical only up to an equivalence relation, we construct a particular pair for which $G$ is $k$-quasisplit and $\Gamma$ fixes a $\text{Gal}(k)$-stable pinning of $G$. Using these choices, we can define a notion of taking "$\Gamma$-fixed points" at the level of equivalence classes, and this process is compatible with a general "restriction" process for root data with $\Gamma$-action.Comment: v2: one word inserted, one citation inserted, one reference updated, one misspelling correcte

Formulae relating the Bernstein and Iwahori-Matsumoto presentations of an affine Hecke algebra

We give explicit formulae for certain elements occurring in the Bernstein presentation of an affine Hecke algebra, in terms of the usual Iwahori- Matsumoto generators. We utilize certain minimal expressions for said elements and we give a sheaf-theoretic interpretation for the existence of these minimal expressions.Comment: To appear, J. of Algebr

The base change fundamental lemma for central elements in parahoric Hecke algebras

Clozel and Labesse proved the base change fundamental lemma for spherical Hecke algebras attached to an unramified group over a p-adic field. This paper proves an analogous fundamental lemma for centers of parahoric Hecke algebras attached to the same class of groups. This provides an ingredient needed for the author's program to study Shimura varieties with parahoric level structure at p.Comment: No figures; 53 pages. Statement of Lemma 4.2.1 modified; minor corrections in section 5; minor expositional changes; final version -- to appear in Duke Math

Cellular pavings of fibers of convolution morphisms

This article proves, in the case of split groups over arbitrary fields, that all fibers of convolution morphisms attached to parahoric affine flag varieties are paved by products of affine lines and affine lines minus a point. This applies in particular to the affine Grassmannian and to the convolution morphisms in the context of the geometric Satake correspondence. The second part of the article extends these results over $\mathbb Z$. Those in turn relate to the recent work of Cass-van den Hove-Scholbach on the geometric Satake equivalence for integral motives, and provide some alternative proofs for some of their results.Comment: 18 pages. Comments welcom