Automorphic Forms, Representations, and L-Functions by W Casselman, Armand Borel, W. Casselman

By W Casselman, Armand Borel, W. Casselman

This was once the convention on $L$-functions and automorphic types. the 2 volumes at the moment are classics.

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The argument in Step 1 of the proof of the proposition amounts to the sort of argument behind [BS, Cor. 14] combined with Sylow’s Theorem. It is essentially identical to the proof of Lemma 2 in [Ch 95], it is just written in a different language. , at the primes 3 and 2 with H replaced by (E6 × SL3 )/µ3 and a half-spin group of type D8 respectively. This argument also appears for E6 relative to the prime 3 in [MPW, p. 153]. 9. The Rost invariant. 10) t rE 8 → H 3 (k, Q/Z(2)) H 1 (k, C × µ5 ) −−−∗−→ H 1 (k, E8 ) −−−− for every extension k/k0 .

Hence Oz has a k-point and z is in the image of the map 1 Hfppf (k, N ) → H 1 (k, G) by [DG 70, p. 373, Prop. 6b] (an fppf analogue of Prop. 37 in [Se 02]). 4. Example (char k = 0). Let G be a semisimple group. , G is of type A1 . Essentially, this is because the regular semisimple elements in V are an open subvariety. 4]. Conversely, if there is an open G-orbit in P(V ), it contains [v] for some regular semisimple element v. This v is contained in a maximal toral subalgebra of V , and by conjugacy of tori, we may assume that this subalgebra is the Lie algebra of a maximal torus T in G.

T∗ (A, α) is the neutral class in H 1 (k, F4 ), if and only if α is the reduced norm of an element of A× by [J 68, p. 416, Th. 20] or [McC, Th. 6]. [A] for uniquely determined λ1 , λ2 ∈ R3 (k0 ). But the algebra J(A, 1) is also split for every A. It follows that λ1 is zero. This proves that g3 spans Invnorm (F4 , Z/3Z). 5, we have found just three interesting invariants of F4 , namely g3 , f3 , and f5 . 7. Open problem. 4], [PeR 94, Q. 1, p. 205]) Is the map g3 × f3 × f5 : H 1 (∗, F4 ) → H 3 (∗, Z/3Z) × H 3 (∗, Z/2Z) × H 5 (∗, Z/2Z) injective?

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