Main reference: Oliver Taibi’s note
Characterization of supercuspidal representation in terms of matrix coefficients: Theorem 5.3.1 of Casselman’s book
Langlands classficiation: classify irreducible representations of a reductive Lie group in terms of tempered representations of smaller groups. Since tempered representations are in turn given as certain representations induced from discrete series or limit of discrete series representations, one can do both inductions at once and get a Langlands classification parameterized by discrete series or limit of discrete series representations instead of tempered representations. The problem with doing this is that it is tricky to decide when two irreducible representations are the same.
Different decompositions of -adic groups:
Cartan decompsition of Lie algebra of a real Lie group, generalizing the polar decomposition:
Iwasawa decomposition of Lie algebra of a real semisimple Lie group or over a non-Archimedean field, generalizing the Gram–Schmidt orthogonalization:
Langlands decomposition: where reductive and is the split center of . The unique minimal parabolic (Borel) is .
Representation of : Using the fact that it is an eigenvector of the Casimir operator and has an eigenvector for , it follows easily that any irreducible admissible representation is a subrepresentation of a parabolically induced representation. (This also is true for more general reductive Lie groups and is known as Casselman’s subrepresentation theorem.) The irreducible unitary representations can be found by checking which of the irreducible admissible representations admit an invariant positively definite Hermitian form.
Reference on real group representations
Classical fact: For , any smooth representation is either one-dimensional or infinite dimensional (the kernel of the representation is open normal; and it contains the standard unipotent subgroup so it contains ; this argument fails for for a quaternion algebra since has no unipotent and is compact).
Langlands dual group:
Let be a connected reductive group over . Take a finite separable extension s.t. admits a Killing/Borel pair . Let be the associated (reduced) based root datum. Here is the group of characters, is the set of roots of in , is the set of coroots, which is a subset of and is a set of simple roots (we should also include a bijection ). All other choices of Killing pair in yield based root data canonically isomorphic to , and so do other choices for .
There is also a continuous action of on this based root datum. First acts on the set of closed subgroups of , and there is a unique s.t. . Then we define where to be .
Inner twist/forms of a reductive group over :
A reductive group over is an inner form of if there is an isomorphism such that for any , we have is inner.
We can ‘classify’ connected reductive groups over as follows: - Fix a representative in each isomorphism class of based root datum with continuous action of ;
For each such representative , fix a quasi-split connected reductive group over together with an isomorphism ;
For each element of choose an inner twist of representing it.
Construction of and :
The choice of a pinning of induces a splitting of the extension because the subgroup of maps bijectively onto . Since we also have , we can form .
For two connected reductive groups and their Langlands dual groups and are isomorphic as extensions of if and only if and are inner forms of each other, and in this case they are even isomorphic as extensions endowed with conjugacy classes of distinguished splittings. The construction of the Langlands dual group is not functorial for arbitrary morphisms between connected reductive groups.
We can also define an analogue of parabolic subgroup of . It is such that a parabolic subgroup such that its normalizer in maps onto . For Levi factor, we also have an embedding of extensions , see page 13 of the above note. It is well-defined up to conjugation by , and in particular independent of the choice of parabolics containing . They are called the -relevant Levis of .
We can now talk about Langlands parameters or Weil-Deligne Langlands parameters. There are three versions of it, see page 16 of the note. We denote by the -conjugacy class of Langlands parameters.
For a semisimple elements, we can consider polar decomposition. This leads to the decomposition of a Langlands parameter and it’s related to the Langlands classification. It is then natural to consider ‘essentially discrete’ parameters (-irreducible parameters)
Relationship between Langlands parameters and Weil-Deligne Langlands parameters
Requirement on the Local Langlands correspondence:
Let be the set of isomorphism classes of admissible representations of over . The tentative conjecture is that there should exist a map from to (Langlands parameter to be defined) satisfying the following property:
If is a torus, then it should be the bijection deduced from class field theory.
The fiber should be finite (-packets) and the image should contain all essentially discrete parameters.
Compatibility with product
Compatibility with central isogeny ( induce map between -group )
If then the -map for and can be identified.
is square-integrable iff is essentially discrete.
Compatibility with paraoblic induction
Compatibility with Langlands classification
Compatibility with Harish-Chandra character
They should imply compatibility with central character (we have a surjection ) and compatibility with twist by character . They also tell us we can reduce to the discrete case. This also suggests the image of should be -relevant parameters.
The unramified case: If is unramified (extend to reductive group scheme over ) and is a hyperspecial compact open subgroup, then on -unramified irreps of the map is given by Satake isomorphism (See also here). On one hand, we know unramified representations are parametrized by rational Weyl orbits orbits under the rational Weyl group of continuous characters of where is an unramified torus (take the unique unramified constituent of ) for any Borel subgroup of containing . We then have by compatibility with parabolic induction when is unitary. In other words is the parameter associated to by the Satake isomorphism. The hard part is to construct the explicit construction of an equivalence between and where is an element of the (relative) Weyl group.
Semisimplified correspondence: We should have one more property of the map (actually the semisimplified version ); It is essentially compatibility with supercuspidal support. The LLC evidently implies semisimplified LLC, but the reverse direction still seems open. What is missing is the fact that for any essentially square-integrable irreducible smooth representation of , the semi-simplified parameter comes from an essentially discrete Langlands parameter (which as above is automatically unique up to conjugation by the centralizer of in ). The notion of essentially discrete Langlands parameter is purely algebraic (it does not rely on the topology of the coefficient field) so there ought to be a purely algebraic characterization of essentially squareintegrable representations. We check the validity of this intuition in the proposition 6.3 in Taibi’s note.
We also see that from the requirement on if is essentially discrete and trivial on , then is supercuspidal. But the converse is not true, this is related to the the classification of essentially square-integrable representations in terms of supercuspidal representations (of Levi subgroups) is much more complicated in general than in the case of . In some applications having just the map LL is too crude, e.g. to formulate the global multiplicity formula for the automorphic spectrum of a connected reductive group over a global field, and so we would like to understand the fibers . See Conjecture 6.4.
A Whittaker datum is , where is the unipotent radical of a Borel and is a generic character. The adjoint group acts transitively on the set of such pairs. Shahidi’s conjecture is that there should be a unique -generic representation in every -packet. The conjectural embedding should map this unique generic representation to the trivial representation.
To characterize the embedding we introduce endoscopic data . If a semisimple element, we can construct , a a virtual character on (take the weighted sum of the characters for ). For we write it as and dropping . We are going to perform the same trick as introduced by Lusztig, by introducing , a quasi-split connected reductive group over (dual to the connected component of the centralizer of ) an embedding and a unique Langlands parameter for which factors. The virtual characters and are related by the Langlands-Shelstad transfer factor . Using maximal tori and identifications of Weyl groups one can define a canonical map from semisimple conjugacy classes in to semisimple conjugacy classes in . An element is called strongly regular if its centralizer is a torus. A conjugacy class in is called -strongly regular if its image under is strongly regular. There are a bunch of properties that satisfies, including -equivariance. Before we state the variance property that the transfer factor needs to satisfy, we need to understand better. By Tate-Nakayama isomorphism, this is isomorphic to via functorial in , and we can actually extend to where we replace the domain by for every connected reductive (a family of maps of pointed sets which is a bijection in the case of non-Archimedean ). See Theorem 6.7 of Taibi’s note. We can think of as a pairing between an element of and an element of . The variance property now becomes (6.3). For strongly regular element , the set of -conjugacy classes which are stably conjugate to is parametrized by via . It doesn’t characterize it because it does not compare the values at unrelated matching pairs.
Now we have all the ingredients to characterize . See conjecture 6.8. One of the most important condition is that the character should be invariant under stable conjugacy and independent of the Whittaker datum chosen.
Reference for representation of -adic groups: Cartier, Renard