By Armand Borel

ISBN-10: 0821814354

ISBN-13: 9780821814352

ISBN-10: 0821814370

ISBN-13: 9780821814376

ISBN-10: 0821814745

ISBN-13: 9780821814741

ISBN-10: 8319613213

ISBN-13: 9788319613210

ISBN-10: 8319772672

ISBN-13: 9788319772672

Includes sections on Reductive teams, representations, Automorphic types and representations

**Read or Download Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Volume XXXIII Part 1 PDF**

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**Additional resources for Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Volume XXXIII Part 1**

**Example text**

Hence V2 ⊗ Vp−1 λ, μ ∈ F× q , Rp−3 ⊕ (Vp−1 ⊗ det). 5. Assume p > 2 and let r and s be f -tuples such that 0 ≤ rj , sj ≤ p − 1 for all j. (i) For 0 ≤ i ≤ f − 1 and all integers a, b we have: i dimFp HomΓ (Vs ⊗ detb , V2Fr ⊗ Vr ⊗ deta ) ≤ 1. i (ii) We have HomΓ (Vs ⊗ detb , V2Fr ⊗ Vr ⊗ deta ) = 0 if and only if sj = rj for all j = i and one of the following holds: (a) si = ri + 2 and b ≡ a (q − 1) (b) si = ri and b ≡ a + pi (q − 1) (c) si = ri − 2 and b ≡ a + 2pi (q − 1) (d) f = 1, p = 3, s0 = r0 = p − 1 and b ≡ a (q − 1).

Let τ be a subrepresentation of V2p−2−r . Assume that there exists an integer k ∈ {0, · · · , f } such that, if Vr(ε) ⊗ dete(ε) occurs in cosocΓ τ , then |ε| = k. Then the graded pieces of the cosocle ﬁltration of τ are given by: τi ∼ = Vr(ε) ⊗ dete(ε) . ε∈Σr (τ ) |ε|=k−i Proof. 7 together with ε ≺ δ ⇒ |ε| = |δ| − 1. 10. Let λ, λ ∈ I(x0 , · · · , xf −1 ) (see Chapter 3). We say λ and λ are compatible if, whenever λi (xi ) ∈ {p − 2 − xi − ±1, xi ± 1} and λi (xi ) ∈ {p − 2 − xi − ±1, xi ± 1} for the same i, then the signs of the ±1 are the same in λi (xi ) and λi (xi ).

Let δ be the extension corresponding to δ via Ext1P (χ, χ) ∼ = Hom(P, Fp ). By inducing to G we obtain an exact sequence: × 0 / IndG P χ / IndG P δ / IndG P χ / 0. By evaluating functions at identity, we see that this sequence splits if and only if δ = 0. This proves (i) as π IndG P χ. Let v1 , v2 be the basis of the underlying vector 8. EXTENSIONS OF PRINCIPAL SERIES 47 space of δ such that for all g ∈ P , we have gv1 = χ(g)v1 + χ(g)δ(g)v2 and gv2 = χ(g)v2 . Denote by U (resp. U s ) in this proof (and only in this proof) the unipotent subgroup of P (resp.

### Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics. Volume XXXIII Part 1 by Armand Borel

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