Local L and epsilon factors in Hecke eigenvalues

Abstract

Abstract. Formulas (Theorems 4.2 and 5.1) which express the local L-factor and the local epsilon factor of an irreducible admissible representation of GL d over a non-archimedean local field in terms of the eigenvalues of some explicitly given Hecke operators are derived.

Whittaker functions associated to newforms for GL(n) over p-adic fields

“…, µ n−1 are complex numbers whose i-th elementary symmetric polynomial equals to q i(i−1)/2−i λ i , for 1 ≤ i ≤ n − 1, and µ n = 0. By [9] Theorem 4.2, we have…”

“…, λ n−1 (see section 3 for precise definition). Kondo and Yasuda [9] showed the relation between these Hecke eigenvalues and the L-factor of π. We therefore obtain an explicit formula for W on T 1 in terms of the L-factor of π (Theorem 4.1).…”

Non-abelian $p$-adic Rankin-Selberg $L$-functions and non-vanishing of central $L$-values

“…In particular, by (58) and t p • v 0 = v 0 the lattice L λ,O is stable under the renormalized action of t p . Recall the definition of d x in (6) and define the lattices…”

“…due to the arbitrary ramification we allow: The known explicit formulae for the valuation of Whittaker functions (cf. [58,61,62]) imply the vanishing of the ordinary projection of the essential vectors in these cases.…”

Local newforms for the general linear groups over a non-archimedean local field

“…In the appendix of the paper[17] by the second and third authors, they introduce a notion of mirahoric representations (see Section A.1.6 of[17]). Let us recall the definition.…”