Let be a WD representation of over an algebraically closed field . We define
local -factor: .
conductor:
local -factor: depends on an additive character (Deligne, Tate’s ‘Local Constant’ in Fröhlich’s algebraic number fields: L-functions and Galois properties, Bushnell-Hennart)
Global -function: which converges to a holomorphic function on . Note that . and .
Example: Riemann zeta function, zeta function for elliptic curves
Remark: The global -function determines the local -factors for a.e. , so by Cebotarev it determines the semisimplification of .
Next we introduce the Gamma factor (local -factor at ). For reference see Deligne.
Introduce and . For each , define . Let be the multiplicity of in . If factors through and if , define by and where is the image of complex conjugation in . Note that these are integers rather than half integers if . This condition holds for etale cohomology of algebraic variety. Define . If , then define and proceed in the same way. Finally if doesn’t factor through , then define .
For the definition of -factor at infinity and the rest of the note, see here.
A Cartan subalgebra of a Lie algebra is a nilpotent subalgebra equal to its own normalizer.
Fact: If , reductive over , the Cartan subalgebras of are the Lie algebras of maximal tori.
Let (the component of at infinity) be the maximal compact subgroup (unique in the archimedean case, see here for a proof).
Example: Let . For , we have then . For , we have then .
For and . Then , and where the embedding is given by mapping to .
For and unitary group for a nondegenerate Hermitian form of signature , then and .
The upshot is the quotient are nice spaces.
Fix a decomposition , is the Lie algebra of a Borel. Roots are the weight (generalized eigenvalues) of on . Positive roots are the weight for the action on .
PBW: . Let by projection. Let be defined by for and thought of as polynomial in .
Harish-Chandra isomorphism: The composite induces an isomorphism onto .
Thus given any representation of with an induced homomorphism (infinitesimal weight) , we get a homomorphism , determined by a -orbit of a homomorphism or an element of , known as the Harish-Chandra parameter of .
We say a cuspidal automorphic representation is algebraic if the HC-parameter of lies in .
Serge Lang’s theorem: over extends to a smooth reductive group scheme over (smooth affine group scheme whose geometric fibers are connected reductive) is equivalent to quasi-split and split after an unramified extension.
If is a maximal torus of and is a maximal split subtorus of . In this case . Similarly we define where and similarly for .
Compute if and otherwise. Thus and similarly for . Now we just need to calculate the coset.
Note: The cocharacter group because and split over an unramified extension of .
Fact (reference: see Casselman’s note on spherical functions): and . Thus .
The Satake map by .
Consider .
Satake’s theorem: induces an isomorphism from the Hecke algebra of to the -invariant of that of , hence is commutative (not true if we choose other maximal compact).
For the rest of the note, see here.