By H. Jacquet, R. P. Langlands

Show description

Read or Download Automorphic Forms on GL(2): Part 1 PDF

Similar science & mathematics books

Aspects of Constructibility

Keith Devlin - ordinary nationwide Public Radio commentator and member of the Stanford college employees - writes in regards to the genetic development of mathematical pondering and the main head-scratching math difficulties of the day. And he by some means manages to make it enjoyable for the lay reader.

Consumer Driven Cereal Innovation: Where Science Meets Industry

Customer pushed Cereal Innovation: the place technological know-how Meets incorporates a choice of papers from oral and poster shows, in addition to the entire abstracts from the 1st Spring assembly geared up through Cereals&Europe, the ecu element of AACC foreign. those complaints speak about the key innovation demanding situations the cereal is dealing with to satisfy shoppers calls for and expectancies.

Additional info for Automorphic Forms on GL(2): Part 1

Example text

If µ1 = χαF and µ2 = χαF the representations on B(µ1 , µ2 )/Bs(µ1 , µ2 ) and Bf (µ2 , µ1 ) are both equivalent to the representations g → χ(detg). The space W (µ1 , µ2 ; ψ) has been defined for all pairs µ1 , µ2 . 4 (i) For all pairs µ1 , µ2 W (µ1 , µ2 ; ψ) = W (µ2 , µ1 ; ψ) (ii) In particular if µ1 µ−1 = α−1 2 F the representation of GF on W (µ1 , µ2 ; ψ) is equivalent to ρ(µ1 , µ2 ). If Φ is a function on F 2 define Φι by Φι (x, y) = Φ(y, x). To prove the proposition we show that, if Φ is in S(F 2 ), µ1 (detg) |detg|1/2 θ µ1 , µ2 ; r(g)Φι = µ2 (detg) |detg|1/2 θ µ2 , µ1 ; r(g)Φ .

Since A = λA0 the hermitian form (ϕ1 , ϕ2 ) is equal to λ F× ϕ1 (a)ϕ2 (a) d× a. 2. Let π be an absolutely cuspidal representation of G F for which the quasi-character ω defined by π a 0 0 a = ω(a)I is a character. (i) If π is in the Kirillov form the hermitian form F× ϕ1 (a)ϕ2 (a) d× a is invariant. (ii) If |z| = 1 then |C(ν, z)| = 1 and if Res = 1/2 | (s, π, ψ)| = 1. Since |z0 | = 1 the second relation of part (ii) follows from the first and the relation (s, π, ψ) = C(ν0−1 , q s−1/2 z0−1 ). If n is in Z and ν is a character of UF let m ϕ( ) = δn,m ν( )ν0 ( ) for m in Z and in UF .

It reduces to µ1 (a) × Φι (at, t−1 )µ1 (t)µ−1 2 (t) d t = µ2 (a) × Φ(at, t−1 )µ2 (t)µ−1 2 (t) d t. The left side equals µ1 (a) × Φ(t−1 , at)µ1 (t)µ−1 2 (t) d t which, after changing the variable of integration, one sees is equal to the right side. Chapter 1 52 −1 If µ1 µ−1 2 is not αF or αF so that ρ(µ1 , µ2 ) is irreducible we let π(µ1 , µ2 ) be any representation in the class of ρ(µ1 , µ2 ). If ρ(µ1 , µ2 ) is reducible it has two constituents one finite dimensional and one infinite dimensional.

Download PDF sample

Rated 4.05 of 5 – based on 27 votes