Eisenstein Series Arising from Jordan Algebras

Abstract

We describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure. * MSC:11F70,22E55,22E50

An exceptional Siegel–Weil formula and poles of the Spin L-function of

“…Now, by an easy check left to the reader, non-isomorphic O give non-isomorphic Θ(1). Moreover, using our description of Θ(1) in Proposition 8.1 and [HS20,Theorem 6.4], one sees that all those possible Θ(1) sum to E. This proves the theorem. 2…”

“…Proof. Comparing Proposition 8.1 with [HS20,Theorem 6.4], one sees that Θ(1) is isomorphic, as an abstract representation, to a summand of E. In view of the multiplicity-1 result in Proposition 8.1(i), it follows that Θ(1) is equal to that irreducible summand, as a subspace of the space of automorphic forms. Now recall that the dual pair D 1 × G D arises from an embedding of D into an octonion algebra O.…”

“…The inclusion Π → I(−5) composed with the restriction of functions from G to G D gives a non-zero D 1 -invariant map Π → I D (−1), which clearly factors through Θ(1). By [Wei03] and [HS20], I D (−1) has a composition series of length 2. The unique irreducible submodule Σ has N D -rank 2.…”

“…Let E D (s, f ) be the degenerate Eisenstein series attached to this maximal parabolic subgroup, where s ∈ R and f varies over all standard sections of the corresponding degenerate principal series representation I D (s). In [HS20], it was proved that E D (s, f ) has at most a simple pole at s = 1, and the residual representation…”

An exceptional Siegel-Weil formula and poles of the Spin L-function of $PGSp_6$

“…For a flat section Φ ∈ I D (s), let E D (s, Φ) be the associated Eisenstein series. Then E D (s, Φ) has at most simple poles at s = 1, 3 or 5 and the corresponding residual representations are completely described in [HS,Theorem 6.4]. Set…”