?? is "whittaker model" idea somehow how something like "automorphic rep" and "automorphic form" get conflated ??? .... ??? .... ?? well, but if so then phenomenon mentioned of "degenerate" rep without whittaker model seems very annoying .... ??? ...

?? vaguely reminding me of how multihomogeneous coordinate algebra of flag variety can be thought of as direct sum of the irreps ..... ???? ....


"The motivation for this local conjecture comes largely from global considerations. We review the theory of Artin L-functions from a more sophisticated standpoint than in Section 1.8. In that section, we considered the theory of Artin L-functions attached to Galois representations. There is another theory of L-functions, namely, the Hecke-Tate theory, which we considered in Section 3.1, and there is some overlap between the theories of these two classes. Let F be a global field and let A be its adele ring, and let p : Gal(F-bar/F) -> GL(1, C) = C^* be a one-dimensional representation. Then p factors through Gal(F^ab/F), where F^ab is the maximal Abelian extension, and the composition of p with the reciprocity law homomorphism A^*/F^* -> Gal(F^ab/F) gives us a Hecke character whose L-function agrees with the Artin L-function of p.

Thus L-functions of some Hecke characters are also Artin L-functions. Not all are, however, because it follows from the "no small subgroups" argument (Exercise 3.1.1(a)) that the image of any continuous homomorphism p : Gal(F-bar/F) -> C^* is necessarily finite. Thus any Hecke character of infinite order, such as the ones we employed in Section 1.9, does not arise in this way. We see that although the two theories overlap, neither theory is contained in the other.

In order to obtain a theory that subsumes both these overlapping theories in a unified framework, Weil (1951) introduced a topological group W_F called the (absolute) Weil group of F, which is a substitute for the Galois group."

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"The result of Langlands is stated precisely in Tate (1979) Theorem 3.4.1, Deligne (1973b) Theorem 4.1, or Langlands (1970a) Theorem 1. If Fv is a local field, and if pv : WFv ->� GL(n, C) is a representation, then there are defined local L- and e-factors L(s, pv) and e(s, pv, fa), the latter depending on the choice of an additive character fa. These must satisfy certain axioms, the most important of which is a compatibility with induction. If pv is one dimensional, then in view of the isomorphism Wp? = F*, pv is essentially a quasicharacter of Fvx, and L(s, pv) and �(s, pV9fa) agree with the Tate factors that were defined in Section 3.1. The consistency of the conditions that �(s, pv,fa) must satisfy is by no means obvious because some representations might be expressible as linear combinations of representations induced from one-dimensional characters in more than one way, and when this occurs, an identity between Gauss sums must be satisfied. This consistency is the content of the result of Langlands."

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"The local Langlands conjecture asserts that if F is a local field and p : WF -> GL(n,C) is an irreducible representation, then there exists a supercuspidal representation pi of GL(n, F) whose L- and e-factors agree with those of p. (If n = 2, we defined these L- and e-factors in Section 3.5 and Section 4.7.) It is assumed that if x is a quasicharacter of F^* , then L(s, p) = L{s, x) and e(s, x#p, pitchfork) = e(s, pi, x, pitchfork). (We are suppressing the subscripts v from the notation, of course.) It is a consequence of Proposition 4.7.6 that at most one representation n can have this characterization. This representation is denoted pi(p).

The local Langlands conjecture thus asserts that the supercuspidal representations of GL(n, F), where F is anon-Archimedean local field, are in bijection with the irreducible n-dimensional representations of W_F- We can expand the scope of the conjecture to include all irreducible admissible representations of GL(n, F), which are in rough bijection with all semisimple representations of W_F, including the reducible ones - for example, if n = 2, the principal series representations are parametrized by the pairs of quasicharacters of F^* = C_F, which correspond to representations of W_F that are the direct sum of a pair of quasicharacters. One must be somewhat careful because this scheme does not account for the special representations. To obtain a precise bijection, one employs not the Weil group,but a slightly larger group, the Weil-Deligne group (Deligne (1973b), Tate (1979) and Borel (1979)). We avoid this issue by considering only supercuspidal representations."