 By Armand Borel

Comprises sections on Reductive teams, representations, Automorphic kinds and representations.

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Extra resources for Automorphic Forms, Representations, and L-functions

Example text

0 s−k−2 0 0 . . 0 −|x0 + 2m| In the cases where j = 0 it simpliﬁes to a diagonal matrix of the form −diag(|xj + 2m|−µ , . . , |xj + 2m|−µ , −(k + 1 − s)|xj + 2m|−µ , |xj + 2m|−µ , . . , |xj + 2m|−µ ) where the entry (k + 1 − s)(xj + 2m)−µ appears precisely at the j-th position. A summation of these matrices over the one-dimensional lattice 2Zej gives actually a well-deﬁned invertible matrix for all z ∈ IRej \2Zej . Hence, all a-points of cot1,s,0 (z, 2Zej ) that lie on the xj axis are isolated.

15) which is s-monogenic in the variable a in 0 ≤ a < γ < ω . In particular, it is real analytic in B(0, γ) and can be thus represented in B(0, γ) by its Taylor series. Using the same argument as in the previous part of the proof, we obtain h(a) ≤ (|m|+|l|+k−s+1)! 11). (k + 1)k+3+|m|+|l|−s a ω k+|m|+|l|+s−2 . 1. ···lk ! After having applied the multinomial formula and an index shift we thus obtain g(z) ≤ ∞ |m|+k+2−s r=0 γ=2 ≤ L (r + γ) a + b ω k+|m|+3−s (k+1)R ω r (k + 1)k+3+|m|−s a + b ω k+|m|+3−s R R.

Xj + 2m|−µ , −(k + 1 − s)|xj + 2m|−µ , |xj + 2m|−µ , . . , |xj + 2m|−µ ) where the entry (k + 1 − s)(xj + 2m)−µ appears precisely at the j-th position. A summation of these matrices over the one-dimensional lattice 2Zej gives actually a well-deﬁned invertible matrix for all z ∈ IRej \2Zej . Hence, all a-points of cot1,s,0 (z, 2Zej ) that lie on the xj axis are isolated. Since tan1,s,0 (z; 2Zej ) stems from cot1,s,0 (z, 2Zej ) by a applying a shift in the argument in the direction of the xj -axis, it follows directly that also the a-points of tan1,s,0 (z; 2Zej ) that lie on the xj axis are all isolated.