The space of cuspidal automorphic forms is the set of all (complex-valued) functions such that
is smooth (i.e. locally constant in the finite places and smooth in the infinite places);
is -finite/admissible (The space of right translates under right translates by product of maximal compact subgroups (speficied as follows) is finite-dimensional. For the finite places, we use where is the profinite completion of , so it is . For the infinite places, we use where is a maximal compact subgroup of , required to be when is real and when is complex.);
is -finite (should be treated together with the previous condition at infinite place; here is the center of the universal envelopping algebra where and the action is by and extended to by universal property, c.f. the Harish-Chandra isomorphism);
is slowly increasing (polynomial growth);
is cuspidal (integral of the left-translates along every unipotent radical of the standard parabolic subgroup vanishes);
The space is not quite a representation of , because -finite is not preserved under right translation by (instead it is -finite; note that there is no problem at finite places). However it does admits an action by and an action by , and they are related by
Another remark is that in the non-Archimedean case requring -finite is the same as admissibility (the space of fixed vectors of any compact open subgroup is finite-dimensional), and the latter is more convenient since we don’t need to keep track of isotypic components (see Getz, Intro to Automorphic representations, Prop. 5.3.11).
A third remark is that automorphic representations are factorizable, i.e. an irreducible (Flath’s theorem, see Theorem 5.7.1 for a proof).
The center of the universal envelopping algebra at infinite places act by by scalars. By the Harish-Chandra isomorphism, we have , so each gives us complex numbers. We make the following definition: If for each , then it is algebraic. If it has distinct elements for each , then it is regular. The regular algebraic representations are accessible via topology since they appear in the Betti cohomology of the symmetric space for some choice of compact subgroup .
The case of Global Langlands is a reformulation of class field theory. A cuspidal automorphic representation of is just a continuous character . The algebraicity condition says that looks like for some integers . From we would like to produce a Galois representation. First we define by (note this no longer trivial on , but it takes to ). By continuity of the character, it is invariant by some open compact subgroup and also on by construction. A fundamental fact is that any quotient is finite, so is valued in on the entire .
We can now use the isomorphism (restricted to ) to make valued in and then modify it at the places above to make it invariant by by undoing the integral twist: Since this involves places above , the character will factor through .
Similarly, starting from an algebraic -adic Hecke character, we can get a complex valued algebraic Hecke character. One thing to note is that the image of lies in a number field. First, the image of under lies in , the Galois closure of (the image of is independent of choice of the embedding ). At other places the image of is unchanged. For the infinite place, we must have since the target is totally disconnected. For finite places , the incompatibility of the profinite topologies implies that there is an open neighborhood of such that is trivial. Hence by compactness of , the image is finite, hence it has image in roots of unity . Since where is the open subgroup . Thus is an open compact subgroup, so it has finite index by compactness of . Thus for all but finitely many places, the restriction of is trivial (more generally any automorphic representation is unramified almost everywhere, see Flath’s theorem mentioned above). By putting the behaviour at , , we see that is locally constant with open kernel . Since contains , we see that the double coset is finite since replacing by the double coset is compact. That means there exists finitely many such that the value of is determined by its restriction to , this implies that the image of the character lies in some number field .
This means that the Hecke character differs from a character taking values in number field by a very simple algebraic character. Without algebraicity, automorphic representations naturally form families in real or complex topology, e.g. twisting by , on the other hand -adic Galois characters form families in -adic topology. In order to state Langlands reciprocity, we need to either impose such algebraicity condition or introduce more general objects on both sides.
If is a number field, then for each place of , recall , and . Define and similarly, and . The difference is that this is not discretely valued and also .
For a (not necessarily finite) extension , we say it is unramified at if has image in . More generally, a continuous homomorphism where is a topological group is unramified at if , i.e. is defined (depending on the emdedding of the local Galois group into the global Galois group, but the conjugacy class of is well-defined).
For any subset of places of , We say has density if . By Prime Number theorem, the denominator is .
Recall Cebotarev density theorem, if is a union of conjugacy classes in , the set of places whose Frobenius lies in has density . The first corollary is that each is the Frobenius elements of infinitely many unramified places of . The second corollary is the for Galois but not necessarily finite, Frobenius elements of unramified places of are dense in .
If is a characteristic zero field, then for any , two irreducible semisimple (direct sum of irreducible) finite-dimensional representations of -algebra with , then (Bourbaki, Ch 8, chapter 12, section 1, prop. 3). Combined with Cebotarev density theorem, we get that if are continuous semisimple representations such that both are unramified outside a given finite subset of places, then iff .
We say is tamely ramified if . There is a maximal unramified (resp. tamely ramified) extension and of in . If we let be any uniformizer of , then . Similarly, . Let be the wild inertia (whose retriction to is trivial), it is a pro--group, and the quotient of is (the identification is via the Kummer map). Under this identification, we have (by considering the action of geometric Frobenius on roots of unity).
Let denote the projection to the -th place. To compare representations with coefficient in different characteristic, we need to remove the influence of topology. If we are considering -adic representations of the local Galois group at and , then the wild inertia will have finite image since the characteristic doesn’t match. But the tame inertia (projection to the -th place) could have infinite image. Grothendieck’s -adic monodromy theorem allows us to describe what this image look like and leads to the notion of Weil-Deligne representation.
Given continuous representation or where for some . Then there exists open subgroup (of finite index) such that is unipotent for all .
Let or . As explained before there exists finite extension and a finite index subgroup such that . Since is a pro- group, factors through . Since depends only on , and is conjugate to for . Let (which converges by the choice of ). Then is conjugate to . If is the -th coefficeint of the characteristic polynomial, then . If for some we have for all , then has finite order, but this contradict the key assumption that satisfies. Hence we must have for for every . Thus the characteristic polynomial is just , i.e. is nilpotent.
Thus we can write for some nilpotent representation , i.e. a nilpotent matrix such that for any lift of we have .
Recipe: Choose any lift of and (i.e. a system of -th root of unity) inducing . Define and . Such a pair is a Weil-Deligne representation. The formal definition is as follows:
Let be any field. A Weil-Deligne representation of is a pair where is a continuous representation where (equipped with the discrete topology) and nilpotent such that for all lifts of , we have .
(Deligne) There is an equivalence of categories:
Intuition: Think of as a punctured disk with the puncture. The Galois group is the fundamental group. The puncture has fundamental group generated by the Frobenius. The kernel control the monodromy. The key assumption satisfied by is that no finite extension of contains all -th power of unity.
A -representation is unramified if and . We say is Frobenius-semisimple if is simisimple. Any has a Frob-semisimplification : Pick any lift of , and write . Then we define with (so we only semisimplify the action of the Frobenius, but has finite order because of continuity hence also semisimple; it is useful since most of the time we only know the characteristic polynomial).
Note that we use the fact that the coefficient field has characteristic zero. Otherwise we need to impose semisimplicity assumption on .
Reference:
For topology on adelic point of algebraic groups: Brian conrad’s paper https://math.stanford.edu/~conrad/papers/adelictop.pdf
For restriction of scalar: appendix of Conrad-Gabber-Prasad