After extending the theory of Rankin-Selberg local factors to pairs of -modular representations of Whittaker type, of general linear groups over a nonArchimedean local field, we study the reduction modulo of -adic local factors and their relation to these -modular local factors. While the -modular local γ -factor we associate with such a pair turns out to always coincide with the reduction modulo of the -adic γ -factor of any Whittaker lifts of this pair, the local L-factor exhibits a more interesting behaviour, always dividing the reduction modulo-of the -adic L-factor of any Whittaker lifts, but with the possibility of a strict division occurring. We completely describe -modular L-factors in the generic case and obtain two simple-to-state nice formulae: Let π, π be generic -modular representations; then, writing π b , π b for their banal parts, we haveUsing this formula, we obtain the inductivity relations for local factors of generic representations. Secondly, we show that L(X, π, π ) = GCD(r (L(X, τ, τ ))), B Robert Kurinczuk
Let F be a non-archimedean local field of residue characteristic p, let Ĝ be a split connected reductive group over Z[ 1 p ] with an action of W F , and let G L denote the semidirect product Ĝ ⋊ W F . We construct a moduli space of Langlands parameters W F → G L , and show that it is locally of finite type and flat over Z[ 1 p ], and that it is a reduced local complete intersection. We give parameterizations of the connected components of this space over algebraically closed fields of characteristic zero and characteristic ℓ = p, as well as of the components of the space over Z ℓ and (conjecturally) over Z[ 1 p ]. Finally, we study the functions on this space that are invariant under conjugation by Ĝ (or, equivalently, the GIT quotient of this space by Ĝ) and give a complete description of this ring of functions after inverting an explicit finite set of primes depending only on G L .
For a classical group over a non-archimedean local field of odd residual characteristic p, we prove that two cuspidal types, defined over an algebraically closed field $${\mathbf {C}}$$ C of characteristic different from p, intertwine if and only if they are conjugate. This completes work of the first and third authors who showed that every irreducible cuspidal $${\mathbf {C}}$$ C -representation of a classical group is compactly induced from a cuspidal type. We generalize Bushnell and Henniart’s notion of endo-equivalence to semisimple characters of general linear groups and to self-dual semisimple characters of classical groups, and introduce (self-dual) endo-parameters. We prove that these parametrize intertwining classes of (self-dual) semisimple characters and conjecture that they are in bijection with wild Langlands parameters, compatibly with the local Langlands correspondence.
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