Parahoric Restriction for GSp(4)

Abstract

Parahoric restriction is the parahoric analogue of Jacquet's functor. Fix an arbitrary parahoric subgroup of the group GSp(4, F ) of symplectic similitudes of genus two over a local number field F/Q p . We determine the parahoric restriction of the non-cuspidal irreducible smooth representations in terms of explicit character values. IntroductionFor a reductive connected group G over a local number field F/Q p , with group of Fvalued points G = G(F ), fix a compact parahoric subgroup P ⊆ G with pro-unipotent radical P + . The parahoric restriction functor between categories of admissible representationsis the parahoric analogue of Jacquet's functor of parabolic restriction. It assigns to an admissible representation (ρ, V ) of G the action of the Levi quotient P/P + on the space of invariants under P + . The functor is exact and factors over semisimplification, so it is sufficient to study irreducible admissible representations.For the group G = GSp(4) of symplectic similitudes of genus two, we determine the parahoric restriction of non-cuspidal irreducible admissible representations in terms of explicit character values for the finite Levi quotient P/P + . This has applications in the theory of Siegel modular forms of genus two, invariant under principal congruence subgroups of squarefree level. Main resultFor a proper parabolic subgroup of G = GSp(4, F ) fix a cuspidal irreducible admissible complex linear representation σ of its Levi quotient. Let ρ be a subquotient of the normalized parabolic induction of σ to G. Then ρ is non-cuspidal and every non-cuspidal irreducible admissible representation of G arises this way.

Serre weights for $GSp_4$ over totally real fields

“…We also define the subgroup K + P of K P which consists of all elements whose reduction modulo v belong to N i (F v ) (cf. p.151 of [75]). For any irreducible admissible representation Π = (Π, V ) of GSp 4 (F v ) as in Section 2.1 of [75] we define the parahoric restriction for K P by…”

“…p.151 of [75]). For any irreducible admissible representation Π = (Π, V ) of GSp 4 (F v ) as in Section 2.1 of [75] we define the parahoric restriction for K P by…”

“…For GL n we also define the parahoric restriction r n for K n = GL n (O Fv ) and K + n := I n + π v M n (O Fv ) in a similar manner (cf. Section 2 of [75]). By definition we also have a representation of…”

“…Proof. See the third column of Table 5 in p.158 of [75] for (XIa) and Table 6 in loc.cit. for (VII), (VIIIa) and (IXa).…”

Dimension formulas for Siegel modular forms of level $4$

“…The resulting representation of K/Γ(p) ∼ = Sp(4, F 2 ) (where F 2 is the field with two elements) is called the hyperspecial parahoric restriction of π and denoted by r K (π). It has been calculated for all π in [23,24].…”