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Spherical Hecke algebra and classical Satake isomorphism

The Satake isomorphism is a way to identify the spherical Hecke algebra of a reductive group G with the invariant part (under the action of the Weyl group) of cocharacters (also called one-parameter subgroups) and both are isomorphic to the Grothendieck ring of the category of the Langlands dual group G^{\vee}. This fits perfectly with the so-called Langlands duality philosophy, suggesting that algebraic objects associated with G^{\vee} should be captured by topological objects associated with G. More concretely, let F be a non-archimedian local field, and O the ring of integers, and let G be a split reductive group over \mathcal{O}. Let T \subset G be a maximal torus and X_{\ast}(T) = \operatorname{Hom}(\mathbb{G}_m,T) the group of cocharacters. The group ring \mathbb{Z}[X_{\ast}(T)] is endowed with an action of the Weyl group W. Let q be the cardinality of the residue field of K, the Satake isomorphism reads $$\mathbb{Z}_c[G(\mathcal{O}) \setminus G(K)/G(\...
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Drinfeld curves and geometric Mackey Formula

This blog serves as my preparation for my talk at Bloch's seminar workshop on Deligne-Lusztig varieties. I will present a talk about \ell-adic cohomology of Drinfeld curves, which are easiest examples of Degline-Lusztig varieties. To begin with, the story dates back to representation theory of finite group of Lie type (by a finite group of Lie type, I mean G^F with G a reductive group over a finite field and F the Frobenious endomorphism). To say why these groups are important, it is the classification of finite simple groups saying that any finite simplegroup belongs to four classes: cyclic, alternating, 26 sporadic groups, and the others are all of Lie type.  The central object of this blog is SL_2(\mathbb{F}_q) with q=p^r a prime power. It is a finite group of Lie type whose cardinality is q(q-1)(q+1) (it's a fun exercise to show this in case you do not know). Not only SL_2(\mathbb{F}_q), but the representation theory of any finite group over complex numbe...
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Concrete counting points on elliptic curves of the form y^2 = x^3 + D

In this post, I present explicit computations with elliptic curves of form y^2=x^3+D (similar trick leads to curves y^2 = x^3 + Dx; the common point is they have complex multiplication). This was my TA for Ariane Mezard. at the Summer School on Galois Representations and Reciprocity last summer. The talk is based on the master thesis of Matteo Tamiozzo.  Reminder on zeta functions For a prime p and let f(x,y,z) \in \mathbb{F}_p[x,y,z] be a homogeneous polynomial so that C = V(f) =  \left \{[x:y:z] \in \mathbb{P}^2_{\mathbb{F}_p} \mid f(x,y,z) = 0 \right \} is a smooth, projective curve. We define N_m = N_m(f) = \left \{P \ \text{has coordinates in} \ \mathbb{F}_{p^m} \right \} to be the cardinality of solutions defined over the \mathbb{F}_{p^m}. We package these numbers N_1,N_2,...,N_m,... into a zeta function Z_C(t) = \exp \left(\sum_{m=1}^{\infty} \frac{N_m}{m}t^m \right). The celebrated result due to Weil and the school of Grothendieck gives us the (...
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Affine Grassmannians of reductive groups

This serves as a starting point for my self-study of geometric Langlands. Currently, I am interested in geometric Satake equivalence, which establishes an equivalence between the category of representation of the Langland dual group (of a given reductive group) and the category of perverse sheaves on affine Grassmannians. Affine grassmannians are interesting objects, and appear naturally when studying stack of principal bundles. Let us now study them, and the story, of reductive groups, is often with \mathrm{GL}_n.  Affine Grassmannians of general linear groups via lattices Let R be a commutative ring and denoteby R[[t]] and R((t)) the formal power series and Laurent series, respectively.  Definition . A lattice L \subset R((t))^n is a finite locally free R[[t]]-submodule of R((t))^n such that the canonical morphism L \otimes_{R[[t]]} R((t)) \longrightarrow R((t))^n is an isomorphism. Note that locally free means that a vector bundle at the level of schemes...
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Potential semi-stability of p-adic representations coming from algebraic geometry

Last week, I participated in a summer school on arithmetic geometry and Langlands program. Although this is not my field of interests, I always give some respect those who work in those fields. I also did gave a motivational TA talk about couting points on elliptic curves. To be honest, I do not expect to learn much from this school but I got enough inspiration to read arithmetic stuff, which I am going to write. Since I am an algebraic geometer, I'd like to present things in a geometric way. Let p be a prime number and denote \mathbb{Q}_p the p-adic local field. Let K be a finite extension of \mathbb{Q}_p. Let G_K = \mathrm{Gal}(\overline{K}/K) the absolute Galois group. A \ell- adic representation of G_K is a continuous morphism (of groups) G_K \longrightarrow \mathrm{GL}_n(V), where V is a finite-dimensional \mathbb{Q}_{\ell}-vector space with \ell \neq p a prime. We denote by K^{unr} the maximal unramified extension and $I_K = \mathrm{Gal}(\overline...
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From Fourier analysis on finite groups to l-adic Fourier transform

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In this post, I'd like to give a rapid introduction to the theory of l-adic Fourier transform developed by Laumon-Deligne-... My goal is not to how can we apply l-adic Fourier transform to prove the Weil conjectures but rather to see why their definitions are natural in comparison with the classical theory. My feeling is that it is easier to present Fourier transforms on finite fields than on measurable spaces (which require a lot of work and details) and the l-adic one is formally adapted from the one for finite fields. Fourier analysis on finite abelian groups Given a finite abelian group G, written additively, what we want to do here is to define a space L^2(G) similar to the Hilbert space L^2(X) of square integrable functions X \longrightarrow \mathbb{C} (modulo equal almost everywhere relation) for X being a measurable space. Then it is possible to develop a Fourier transform on L^2(G). The finiteness seems to be a technical condition that you can see to be u...
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