By Jacquet H., Langlands R.P.

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The most goal of those lectures is first to in brief survey the basic con nection among the illustration thought of the symmetric workforce Sn and the idea of symmetric features and moment to teach how combinatorial equipment that come up evidently within the concept of symmetric features result in effective algorithms to precise quite a few prod ucts of representations of Sn when it comes to sums of irreducible representations.

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4) r These two expressions are equal for all choice of n, p, ρ, and ν . If ρ = ν and the conductor of νρ−1 is pm the gaussian sum η(ρν −1 , ̟ r ) is zero unless r = −m−ℓ. 4) reduces to η(ρν −1 , ̟ −m−ℓ )z0−p−m−ℓ Cn−m−ℓ (ν)Cp−m−ℓ (ρ−1 ν0−1 ). 3) is equal to η(σ −1 ν, ̟ n ) η(ρ−1 σ −1 ν0−1 ̟ p )z0−p Cp+n (σ). ρ−1 ν(−1) σ Replacing ρ by ρ−1 ν0−1 we obtain the first part of the proposition. If ρ = ν then δ(ρν −1 ) = 1. Moreover, as is well-known and easily verified, η(ρν −1 , ̟ r ) = 1 if r ≥ −ℓ, η(ρν −1 , ̟ −ℓ−1 ) = |̟|(|̟| − 1)−1 and η(ρν −1 , ̟ r ) = 0 if r ≤ −ℓ − 2.

22. Let π be an irreducible representation of GF . It is absolutely cuspidal if and only if for every vector v there is an ideal a in F such that π a 1 0 x 1 v dx = 0. It is clear that the condition cannot be satisfied by a finite dimensional representation. Suppose that π is infinite dimensional and in the Kirillov form. If ϕ is in V then π a 1 x 0 1 ϕ dx = 0 if and only if ψ(ax) dx = 0 ϕ(a) a for all a. If this is so the character x → ψ(ax) must be non-trivial on a for all a in the support of ϕ.

If π = σ(µ1 , µ2 ) and µ1 µ−1 2 = αF we can suppose that π is the restriction of ρ(µ1 , µ2 ) to BS (µ1 , µ2 ). The vectors in B(µ1 , µ2 ) invariant under GL(2, OF ) clearly do not lie in Bs (µ1 , µ2 ) so that the restriction of π to GL(2, OF ) does not contain the trivial representation. All that we have left to do is to show that the restiction of an absolutely cuspidal representation to GL(2, OF ) does not contain the trivial representation. Suppose the infinite-dimensional irreducible representation π is given in the Kirillov form with respect to an additive character ψ such that OF is the largest ideal on which ψ is trivial.